1 00:00:02,290 --> 00:00:04,150 NICOLA MARZARI: OK, good morning. 2 00:00:04,150 --> 00:00:07,075 And welcome to lecture 6. 3 00:00:07,075 --> 00:00:10,600 We're still working on electronic structure methods. 4 00:00:10,600 --> 00:00:14,560 And today in particular, we'll finish up our introduction 5 00:00:14,560 --> 00:00:18,040 of the Hartree-Fock methods, if you want the cornerstone 6 00:00:18,040 --> 00:00:20,110 of quantum chemistry. 7 00:00:20,110 --> 00:00:23,650 Really developed in the late '20s, 8 00:00:23,650 --> 00:00:26,290 we have Douglas Hartree here from the University 9 00:00:26,290 --> 00:00:28,780 of Cambridge and Fock, I think, from the University 10 00:00:28,780 --> 00:00:30,670 of St. Petersburg. 11 00:00:30,670 --> 00:00:34,300 And then we'll also go into Density-Functional theory that, 12 00:00:34,300 --> 00:00:37,450 in many ways, is a much more recent approach. 13 00:00:37,450 --> 00:00:41,470 The theory itself was developed in the '60s 14 00:00:41,470 --> 00:00:45,310 by Walter Kohn in collaboration with Pierre Hohenberg 15 00:00:45,310 --> 00:00:47,560 and in collaboration with Lu Sham. 16 00:00:47,560 --> 00:00:51,160 But really, I would say it's only in the mid '70s 17 00:00:51,160 --> 00:00:53,500 and then really in the mid '80s that it 18 00:00:53,500 --> 00:00:57,370 started to take off as actually a practical approach 19 00:00:57,370 --> 00:00:59,920 to study the electronic structure, especially 20 00:00:59,920 --> 00:01:03,520 of solids, and has become very popular, popular to the point 21 00:01:03,520 --> 00:01:07,180 that the Nobel Prize for chemistry in '98 22 00:01:07,180 --> 00:01:13,270 was given to Walter Kohn for DFT and John Pople, a quantum 23 00:01:13,270 --> 00:01:17,370 chemist, for the development of quantum chemistry methods. 24 00:01:21,510 --> 00:01:26,760 Let me remind you before we go on into the lecture of the two 25 00:01:26,760 --> 00:01:30,810 main conclusions of last class. 26 00:01:30,810 --> 00:01:34,830 The first one was the appropriate matrix formulation 27 00:01:34,830 --> 00:01:36,900 of the Schrodinger equation that is something 28 00:01:36,900 --> 00:01:39,060 that is very powerful and very useful 29 00:01:39,060 --> 00:01:41,820 when we actually need to solve a differential 30 00:01:41,820 --> 00:01:43,920 equation on a computer. 31 00:01:43,920 --> 00:01:48,780 And the hypothesis there was that we had chosen a basis set 32 00:01:48,780 --> 00:01:50,730 that I indicate here with phi. 33 00:01:50,730 --> 00:01:54,960 That would be the basis set in which I would expand my ground 34 00:01:54,960 --> 00:01:56,070 stateway function. 35 00:01:56,070 --> 00:01:59,070 And so once the basis set is chosen-- 36 00:01:59,070 --> 00:02:02,940 it could be sines and cosines with different wavelengths. 37 00:02:02,940 --> 00:02:06,300 It could be localized wave functions like Gaussians. 38 00:02:06,300 --> 00:02:10,110 It could be just points on a discrete grid. 39 00:02:10,110 --> 00:02:14,780 But once my basis set is chosen, and once I know what is 40 00:02:14,780 --> 00:02:16,530 my electronic structure problem-- that is, 41 00:02:16,530 --> 00:02:20,040 once I know what is the potential in my Hamiltonian 42 00:02:20,040 --> 00:02:21,090 operator-- 43 00:02:21,090 --> 00:02:25,420 this here, these integrals, are just numbers. 44 00:02:25,420 --> 00:02:27,840 So solving the Schrodinger equation-- that is, 45 00:02:27,840 --> 00:02:32,670 finding the key E eigenvalues and the corresponding 46 00:02:32,670 --> 00:02:33,870 eigenvectors-- 47 00:02:33,870 --> 00:02:38,130 is nothing else than solving this linear algebra problem 48 00:02:38,130 --> 00:02:42,690 where, again, these coefficients here form a vector that really 49 00:02:42,690 --> 00:02:46,560 gives me what are the terms in the expansion of my wave 50 00:02:46,560 --> 00:02:49,410 function, in terms of orthonormal basis. 51 00:02:49,410 --> 00:02:53,070 And so linear algebra problems, we have a matrix here. 52 00:02:53,070 --> 00:02:57,700 And it will have, if it's an order n matrix, n values 53 00:02:57,700 --> 00:02:59,580 for which the determinant is 0. 54 00:02:59,580 --> 00:03:01,290 And those are the eigenvalues for which 55 00:03:01,290 --> 00:03:02,730 we can find the solution. 56 00:03:02,730 --> 00:03:04,980 So this is important not only in electronic structure, 57 00:03:04,980 --> 00:03:08,250 but in a lot of applied computational approaches. 58 00:03:08,250 --> 00:03:12,000 And the other very useful principle here 59 00:03:12,000 --> 00:03:14,610 that I have listed is the variational principle. 60 00:03:14,610 --> 00:03:18,360 That is, again, it is possible to define a functional-- 61 00:03:18,360 --> 00:03:20,370 that is, if you want an algorithm-- 62 00:03:20,370 --> 00:03:23,640 that takes as input a generic function 63 00:03:23,640 --> 00:03:26,190 and gives us output, a number. 64 00:03:26,190 --> 00:03:28,110 And the functional that we use is 65 00:03:28,110 --> 00:03:29,850 the functional that I've written here 66 00:03:29,850 --> 00:03:33,570 with the integral expectation value of the Hamiltonian 67 00:03:33,570 --> 00:03:36,360 on an arbitrary wave function divided by basically 68 00:03:36,360 --> 00:03:38,010 a normalization term. 69 00:03:38,010 --> 00:03:39,300 And it can be proven-- 70 00:03:39,300 --> 00:03:41,490 and it was given to us an exercise-- 71 00:03:41,490 --> 00:03:44,550 that this quantity is always greater 72 00:03:44,550 --> 00:03:46,770 or equal than the ground state. 73 00:03:46,770 --> 00:03:48,570 And this is very powerful because, if we 74 00:03:48,570 --> 00:03:52,230 have no idea on what the solution for our ground state 75 00:03:52,230 --> 00:03:55,740 problem is, well, we can just try out a few. 76 00:03:55,740 --> 00:03:58,890 And our best solution-- although, 77 00:03:58,890 --> 00:04:01,560 we will never know if it's the exact one or not-- 78 00:04:01,560 --> 00:04:05,370 will be the one that gives the lowest expectation value 79 00:04:05,370 --> 00:04:07,010 for this term. 80 00:04:07,010 --> 00:04:11,280 And here is an example on how we could actually 81 00:04:11,280 --> 00:04:16,860 go on and try to solve a very simple problem that is finding 82 00:04:16,860 --> 00:04:22,230 the eigenstates and eigenvalues for a hydrogen atom using 83 00:04:22,230 --> 00:04:24,150 the variational principle. 84 00:04:24,150 --> 00:04:28,920 And instead of trying actually different wave functions 85 00:04:28,920 --> 00:04:32,250 at a time, what we can do that is very meaningful 86 00:04:32,250 --> 00:04:36,400 is choose a parametric format for the wave function. 87 00:04:36,400 --> 00:04:39,810 So you see, I'm writing here a generic wave function 88 00:04:39,810 --> 00:04:45,090 that decays as an exponential as a function of the r distance 89 00:04:45,090 --> 00:04:47,640 from the nucleus from the center of the electron. 90 00:04:47,640 --> 00:04:50,470 So if you wanted this parameter here, 91 00:04:50,470 --> 00:04:54,150 alpha determines how steep the decay of this wave 92 00:04:54,150 --> 00:04:55,180 function here. 93 00:04:55,180 --> 00:04:57,480 So by writing it in this way, I'm 94 00:04:57,480 --> 00:05:00,330 not considering any more just one way function. 95 00:05:00,330 --> 00:05:04,620 But I am considering an entire family of functions 96 00:05:04,620 --> 00:05:06,300 with different decays. 97 00:05:06,300 --> 00:05:10,990 And so what I can do is try this family of wave functions. 98 00:05:10,990 --> 00:05:14,460 And so what I would do is stick this expression 99 00:05:14,460 --> 00:05:16,710 in this expectation value. 100 00:05:16,710 --> 00:05:19,200 And then actually, the constant, C, 101 00:05:19,200 --> 00:05:21,450 is what we call a normalization constant. 102 00:05:21,450 --> 00:05:24,960 It always cancels out, you see, from the integral below 103 00:05:24,960 --> 00:05:26,130 and the integral above. 104 00:05:26,130 --> 00:05:28,720 This C squared would just go away. 105 00:05:28,720 --> 00:05:32,610 And so overall, the variational principle and expectation value 106 00:05:32,610 --> 00:05:35,670 is a parametric function of alpha. 107 00:05:35,670 --> 00:05:39,390 And the optimal alpha will be the alpha 108 00:05:39,390 --> 00:05:43,050 that gives me the minimum value for this number. 109 00:05:43,050 --> 00:05:47,040 Again, we can't go below the ground state 110 00:05:47,040 --> 00:05:49,120 for this expectation value. 111 00:05:49,120 --> 00:05:52,390 So the lower that we can go, the better. 112 00:05:52,390 --> 00:05:56,430 And if we use this particular choice of wave function-- 113 00:05:56,430 --> 00:05:59,880 and this is, again, a very simple analysis problem 114 00:05:59,880 --> 00:06:03,090 that you could actually work out by yourself. 115 00:06:03,090 --> 00:06:06,510 I've actually written here all the terms of the normalization 116 00:06:06,510 --> 00:06:07,800 integral. 117 00:06:07,800 --> 00:06:11,190 What is its value as a function of this parameter once you 118 00:06:11,190 --> 00:06:13,810 do it properly in spherical coordinates, 119 00:06:13,810 --> 00:06:16,290 considering that this is really the radial distance, 120 00:06:16,290 --> 00:06:19,080 and this is the expectation value of the kinetic energy, 121 00:06:19,080 --> 00:06:22,440 and this is the expectation value of-- you see minus 1 122 00:06:22,440 --> 00:06:23,430 over r. 123 00:06:23,430 --> 00:06:25,620 That is the potential for the electron 124 00:06:25,620 --> 00:06:27,750 in the field of a proton. 125 00:06:27,750 --> 00:06:30,660 And if you minimize this expression-- very easy-- 126 00:06:30,660 --> 00:06:33,630 what you obtain is alpha equal to 1. 127 00:06:33,630 --> 00:06:37,980 So the optimal solution is then alpha equal to 1. 128 00:06:37,980 --> 00:06:41,580 It turns out that, by solving directly the differential 129 00:06:41,580 --> 00:06:43,390 equation, the Schrodinger equation, 130 00:06:43,390 --> 00:06:47,820 we actually know that this exponential of minus r 131 00:06:47,820 --> 00:06:50,410 is actually the exact solution. 132 00:06:50,410 --> 00:06:52,290 And so what happens, in this case, 133 00:06:52,290 --> 00:06:55,560 is that this parametric family of wave functions 134 00:06:55,560 --> 00:06:57,300 is the right form. 135 00:06:57,300 --> 00:07:01,020 That is, it contains actually the exact the ground state 136 00:07:01,020 --> 00:07:03,300 among all its possible forms. 137 00:07:03,300 --> 00:07:05,190 And that exact ground state is reached 138 00:07:05,190 --> 00:07:06,750 when alpha is equal to 1. 139 00:07:06,750 --> 00:07:11,310 And the variational principle will give us alpha equal to 1 140 00:07:11,310 --> 00:07:14,850 without having any need of solving 141 00:07:14,850 --> 00:07:18,660 the very complex radial Schrodinger equation that I've 142 00:07:18,660 --> 00:07:23,250 shown at a certain point in the class on Tuesday 143 00:07:23,250 --> 00:07:26,280 where we really need to solve for the spherical harmonics 144 00:07:26,280 --> 00:07:29,880 and find the Legendre polynomial, Laguerre terms 145 00:07:29,880 --> 00:07:32,200 in the radial part, and so on and so forth. 146 00:07:32,200 --> 00:07:33,870 So this becomes very simple. 147 00:07:33,870 --> 00:07:36,330 And for this reason, the variational principle 148 00:07:36,330 --> 00:07:37,830 is very powerful. 149 00:07:37,830 --> 00:07:41,370 And we'll actually see how Hartree and Fock use 150 00:07:41,370 --> 00:07:46,710 that to find out a way to tackle the problem of many electrons 151 00:07:46,710 --> 00:07:50,550 interacting because, up to now, we have really only thought 152 00:07:50,550 --> 00:07:54,240 of the problem of one electron in a potential. 153 00:07:54,240 --> 00:07:57,990 But the problem of many interacting electrons 154 00:07:57,990 --> 00:08:01,330 increases its complexity very quickly. 155 00:08:01,330 --> 00:08:05,940 So I'm just showing here what would be, say, the Schrodinger 156 00:08:05,940 --> 00:08:09,690 equation for a two-electron atom, say something typically 157 00:08:09,690 --> 00:08:13,460 like the atom of helium. 158 00:08:13,460 --> 00:08:18,950 You have a nucleus that has really two protons. 159 00:08:18,950 --> 00:08:25,270 And then you have two electrons around the nucleus, 160 00:08:25,270 --> 00:08:26,540 going around. 161 00:08:26,540 --> 00:08:29,470 And so what is the Hamiltonian, the quantum mechanical 162 00:08:29,470 --> 00:08:31,340 Hamiltonian for this problem? 163 00:08:31,340 --> 00:08:35,840 Well, what we have is the kinetic energy terms here. 164 00:08:35,840 --> 00:08:40,600 So we have a Laplacian that is a second derivative in space 165 00:08:40,600 --> 00:08:43,150 for each one of the coordinates. 166 00:08:43,150 --> 00:08:45,310 So you see this is the wave function. 167 00:08:45,310 --> 00:08:48,460 The wave function is an amplitude 168 00:08:48,460 --> 00:08:52,660 that is a function of the combined position of r1 169 00:08:52,660 --> 00:08:55,780 describing the first electron and r2 describing 170 00:08:55,780 --> 00:08:57,010 the second electron. 171 00:08:57,010 --> 00:08:59,950 So you have these two second derivatives. 172 00:08:59,950 --> 00:09:04,750 And then there is the term that deals 173 00:09:04,750 --> 00:09:09,310 with the attraction between electron 1, 174 00:09:09,310 --> 00:09:12,310 say this electron here, and the nucleus. 175 00:09:12,310 --> 00:09:15,940 We have an attraction term here, Coulombic attraction, 176 00:09:15,940 --> 00:09:21,430 that goes basically as 2/r or as Z/r if we think of the nucleus 177 00:09:21,430 --> 00:09:24,610 as having charge Z. And so there is attraction here 178 00:09:24,610 --> 00:09:26,770 of the first electron to the nucleus 179 00:09:26,770 --> 00:09:30,610 and then attraction of the second electron to the nucleus. 180 00:09:30,610 --> 00:09:35,010 And then the last term is actually a repulsive term-- 181 00:09:35,010 --> 00:09:37,720 so there should be actually a plus sign here-- 182 00:09:37,720 --> 00:09:40,060 between the two electrons. 183 00:09:40,060 --> 00:09:42,220 And so you see it depends-- 184 00:09:42,220 --> 00:09:45,130 again, Coulombic comparison between the electrons-- 185 00:09:45,130 --> 00:09:50,800 on the distance between r1 and r2, the instantaneous position. 186 00:09:50,800 --> 00:09:53,830 So you see, this is how the Hamiltonian becomes, 187 00:09:53,830 --> 00:09:58,120 in the case of an atom that has two electrons and everything 188 00:09:58,120 --> 00:10:03,340 here, the final generalization still for one atom 189 00:10:03,340 --> 00:10:04,810 but with many electrons. 190 00:10:04,810 --> 00:10:06,520 So in the case where, again, you have 191 00:10:06,520 --> 00:10:10,750 a nucleus with many electrons orbiting around, 192 00:10:10,750 --> 00:10:13,540 what you will have is a wave function 193 00:10:13,540 --> 00:10:21,530 that is a combined amplitude of n different spatial variables. 194 00:10:21,530 --> 00:10:27,340 So if we have, say, an atom like iron that has 26 electrons, 195 00:10:27,340 --> 00:10:31,630 well, that wave function is a combined amplitude 196 00:10:31,630 --> 00:10:34,360 of 78 coordinates. 197 00:10:34,360 --> 00:10:36,460 And the Hamiltonian operator, again, 198 00:10:36,460 --> 00:10:40,180 is our second derivative for each electron in there 199 00:10:40,180 --> 00:10:43,960 as an attractive term for each electron that 200 00:10:43,960 --> 00:10:49,390 is attracted to the nucleus that now has a charge Z equal to 26. 201 00:10:49,390 --> 00:10:54,490 And then each electron-- so this sum i goes over each electron. 202 00:10:54,490 --> 00:10:59,140 Each electron will be repelling each other electron. 203 00:10:59,140 --> 00:11:03,120 So there is an overall repulsive term. 204 00:11:03,120 --> 00:11:05,310 And we actually call this term here-- 205 00:11:05,310 --> 00:11:06,900 this is the difficult term. 206 00:11:06,900 --> 00:11:10,470 We call this term here a two-body term 207 00:11:10,470 --> 00:11:14,310 because it depends on the simultaneous possession of two 208 00:11:14,310 --> 00:11:15,300 electrons. 209 00:11:15,300 --> 00:11:17,530 Well, if you want, the first the terms 210 00:11:17,530 --> 00:11:21,450 here are called one-body terms because they act only 211 00:11:21,450 --> 00:11:23,850 on one electron at a time, quantum 212 00:11:23,850 --> 00:11:27,000 kinetic energy and attraction. 213 00:11:27,000 --> 00:11:31,830 And truly, this Schrodinger equation 214 00:11:31,830 --> 00:11:35,790 becomes overly complex already by the time 215 00:11:35,790 --> 00:11:37,330 when we have two electrons. 216 00:11:37,330 --> 00:11:41,760 So we can't solve analytically even the helium atom, OK? 217 00:11:41,760 --> 00:11:44,342 We can solve everything about the hydrogen atom. 218 00:11:44,342 --> 00:11:46,050 And we can solve the Schrodinger equation 219 00:11:46,050 --> 00:11:47,970 for a number of very simple problems. 220 00:11:47,970 --> 00:11:52,600 But already, for the helium atom, we can't do that. 221 00:11:52,600 --> 00:11:55,890 And so in the '40s and '50s, people 222 00:11:55,890 --> 00:11:58,230 developed variational approaches. 223 00:11:58,230 --> 00:12:01,330 They were making an uncertain hypothesis on this wave 224 00:12:01,330 --> 00:12:04,410 function, expanding this wave function 225 00:12:04,410 --> 00:12:08,790 in a series of other functions, depending parametrically 226 00:12:08,790 --> 00:12:10,110 on a number of coefficients-- 227 00:12:10,110 --> 00:12:13,740 2, 3, 5, 8, 200 coefficients. 228 00:12:13,740 --> 00:12:17,670 And the more coefficients there were, the more flexible 229 00:12:17,670 --> 00:12:19,350 that wave function was. 230 00:12:19,350 --> 00:12:23,910 And so systematically, by adding more and more flexibility, 231 00:12:23,910 --> 00:12:26,370 you could go lower and lower. 232 00:12:26,370 --> 00:12:28,860 And hopefully, you were converging. 233 00:12:28,860 --> 00:12:31,380 And say for something like the helium atom, 234 00:12:31,380 --> 00:12:37,350 well, the total energy of two electrons in the helium atom 235 00:12:37,350 --> 00:12:43,950 is something like 5.8, 5.8 Rydbergs. 236 00:12:43,950 --> 00:12:47,190 And you could get very, very close, 237 00:12:47,190 --> 00:12:51,300 down to a fraction of hundreds of electron volts 238 00:12:51,300 --> 00:12:54,000 just by 3 or 5 parameters. 239 00:12:54,000 --> 00:12:56,430 But then as you go to more and more electrons, 240 00:12:56,430 --> 00:13:00,720 the problem really has a complexity that explodes. 241 00:13:00,720 --> 00:13:03,970 And in all of this, I want to remind you, 242 00:13:03,970 --> 00:13:07,710 I'm not using any more the international system of units. 243 00:13:07,710 --> 00:13:10,650 So I'm not using meters and seconds. 244 00:13:10,650 --> 00:13:13,290 But I'm using what are called atomic units. 245 00:13:13,290 --> 00:13:17,160 And there is a handout on your Stellar web page reminding 246 00:13:17,160 --> 00:13:21,930 that the unit of energy is what we call the Hartree that 247 00:13:21,930 --> 00:13:26,880 corresponds to another set of unit that are often used. 248 00:13:26,880 --> 00:13:30,510 Say a Hartree corresponds to Rydberg. 249 00:13:30,510 --> 00:13:33,750 And the Rydberg is 13.6 electron volts. 250 00:13:33,750 --> 00:13:37,020 And electron volt is the energy of an electron that 251 00:13:37,020 --> 00:13:39,570 feels a potential difference of 1 volt. 252 00:13:39,570 --> 00:13:42,090 And 1 electron volt is often considered 253 00:13:42,090 --> 00:13:45,210 23.5 kilocalories per mole. 254 00:13:45,210 --> 00:13:47,230 In the electronic structure literature, 255 00:13:47,230 --> 00:13:49,020 you'll see all these numbers. 256 00:13:49,020 --> 00:13:53,790 Often, chemists use the kilocalorie per mole unit. 257 00:13:53,790 --> 00:13:56,910 Physicists tend to use the electron volts. 258 00:13:56,910 --> 00:14:00,990 And again, I remind you that the average kinetic energy 259 00:14:00,990 --> 00:14:05,850 of an atom at room temperature is 0.04 electron volts. 260 00:14:05,850 --> 00:14:10,920 And the binding of water dimer is 0.29 electron volts. 261 00:14:10,920 --> 00:14:13,380 So the electron volts is really the order 262 00:14:13,380 --> 00:14:20,350 of magnitude of the binding of weakly attached molecules. 263 00:14:23,600 --> 00:14:24,290 OK. 264 00:14:24,290 --> 00:14:26,340 So now, the problem-- 265 00:14:26,340 --> 00:14:29,120 and that's where the Hartree and Hartree-Fock solution become 266 00:14:29,120 --> 00:14:30,560 important-- 267 00:14:30,560 --> 00:14:33,680 is to deal with a realistic system. 268 00:14:33,680 --> 00:14:36,380 That is, we need to deal with molecules 269 00:14:36,380 --> 00:14:38,390 or we need to deal with solids. 270 00:14:38,390 --> 00:14:40,130 So we need to deal, in principle, 271 00:14:40,130 --> 00:14:45,180 with a Schrodinger equation in which we have a lot of nuclei. 272 00:14:45,180 --> 00:14:50,210 I'm denoting here with capital R the position of the nuclei. 273 00:14:50,210 --> 00:14:51,920 Again, if you want to find out what 274 00:14:51,920 --> 00:14:54,680 is the structure of benzene, if you want to find out 275 00:14:54,680 --> 00:14:56,660 what is the electron structure of silicon, 276 00:14:56,660 --> 00:14:59,510 you need to keep in mind that, say for benzene, you 277 00:14:59,510 --> 00:15:02,990 will have six Coulombic attractive centers that 278 00:15:02,990 --> 00:15:06,320 are the six carbon nuclei in a ring 279 00:15:06,320 --> 00:15:10,320 and also the six 1/r protons around. 280 00:15:10,320 --> 00:15:14,570 So the position of those nuclei determines 281 00:15:14,570 --> 00:15:18,870 what is the potential acting on your electrons. 282 00:15:18,870 --> 00:15:21,500 So in principle, the Schrodinger equation 283 00:15:21,500 --> 00:15:24,590 will be something like I've written here. 284 00:15:24,590 --> 00:15:27,680 But we can make this fundamental simplification 285 00:15:27,680 --> 00:15:31,580 that will always be the case for the classes that follow. 286 00:15:31,580 --> 00:15:35,540 That is, we never treat the nuclei as quantum particles, 287 00:15:35,540 --> 00:15:36,140 OK? 288 00:15:36,140 --> 00:15:40,300 In principle, also the nuclei are quantum particles. 289 00:15:40,300 --> 00:15:42,290 So they would have their own wavelength. 290 00:15:42,290 --> 00:15:46,230 They come in as variables into the wave function. 291 00:15:46,230 --> 00:15:48,890 And you would need to calculate, say, things 292 00:15:48,890 --> 00:15:51,500 like the quantum kinetic energy of the nucleus. 293 00:15:51,500 --> 00:15:54,230 That is the Laplacian, the second derivative 294 00:15:54,230 --> 00:15:56,000 of the overall wave function with respect 295 00:15:56,000 --> 00:15:57,620 to the nuclear coordinates. 296 00:15:57,620 --> 00:16:00,470 In practice, remember the Bernoulli relation. 297 00:16:00,470 --> 00:16:05,590 The nuclei are so heavy that their wavelength is very, very 298 00:16:05,590 --> 00:16:06,090 small. 299 00:16:06,090 --> 00:16:09,650 So we can truly treat them as classical particles. 300 00:16:09,650 --> 00:16:13,790 And the electrons are so much faster than the nuclei 301 00:16:13,790 --> 00:16:18,350 that even if the nuclei move, the electrons can always, 302 00:16:18,350 --> 00:16:21,800 much faster than the nuclei, follow this movement 303 00:16:21,800 --> 00:16:25,820 and reorganize themselves as to be getting the lowest energy 304 00:16:25,820 --> 00:16:26,990 state possible. 305 00:16:26,990 --> 00:16:30,230 Basically, you have to imagine this picture of the molecule 306 00:16:30,230 --> 00:16:32,420 vibrating at room temperature. 307 00:16:32,420 --> 00:16:35,690 And the nucleus then will have a kinetic energy. 308 00:16:35,690 --> 00:16:38,100 All the mass of the molecule is in the nuclei. 309 00:16:38,100 --> 00:16:40,730 So all the effects of temperature, if you want, 310 00:16:40,730 --> 00:16:42,680 are in the vibration of the nuclei. 311 00:16:42,680 --> 00:16:46,520 But these vibrations are very slow from the point 312 00:16:46,520 --> 00:16:48,150 of view of the electrons. 313 00:16:48,150 --> 00:16:50,840 So the electrons see that the nucleus is slowly moving 314 00:16:50,840 --> 00:16:55,310 and rearrange themselves as to being in the ground state 315 00:16:55,310 --> 00:16:58,640 for that instantaneous configuration of the nuclei. 316 00:16:58,640 --> 00:17:01,610 Obviously, if the nucleus starts moving very, very 317 00:17:01,610 --> 00:17:06,329 fast by any chance, well, then the electrons can't do this. 318 00:17:06,329 --> 00:17:08,130 They can't follow any more. 319 00:17:08,130 --> 00:17:10,099 And so if you want, you can start 320 00:17:10,099 --> 00:17:12,140 having electronic excitation. 321 00:17:12,140 --> 00:17:14,780 That is the electrons are not anymore 322 00:17:14,780 --> 00:17:17,960 on the lowest energy state possible for that given 323 00:17:17,960 --> 00:17:19,640 configuration. 324 00:17:19,640 --> 00:17:22,339 That doesn't really happen, especially 325 00:17:22,339 --> 00:17:25,220 if we are just considering a molecule a solid at room 326 00:17:25,220 --> 00:17:26,210 temperature. 327 00:17:26,210 --> 00:17:30,650 But it could happen, say, if we have an external potential that 328 00:17:30,650 --> 00:17:32,240 changes very fast. 329 00:17:32,240 --> 00:17:35,780 If we shine a laser light on a molecule, 330 00:17:35,780 --> 00:17:39,590 then light is nothing less than a electromagnetic field. 331 00:17:39,590 --> 00:17:42,830 And laser light will typically have a frequency 332 00:17:42,830 --> 00:17:46,190 that is fast and comparable to the frequency of the electrons. 333 00:17:46,190 --> 00:17:49,205 And so all these adiabatic approximations break down. 334 00:17:49,205 --> 00:17:52,340 And lo and behold, we can actually excite with the laser 335 00:17:52,340 --> 00:17:54,950 the electron in a higher energy state. 336 00:17:54,950 --> 00:17:56,960 But for what you are seeing in this class, 337 00:17:56,960 --> 00:18:01,670 we'll always think of the electrons in a ground state. 338 00:18:01,670 --> 00:18:05,060 That this is what people call often the adiabatic 339 00:18:05,060 --> 00:18:08,480 or the Born-Oppenheimer approximation. 340 00:18:08,480 --> 00:18:14,330 These two terms, in most cases, are used in the same way. 341 00:18:14,330 --> 00:18:20,480 Although, chemists tend to make a subtle distinction about what 342 00:18:20,480 --> 00:18:23,940 adiabatic means and Born-Oppenheimer means. 343 00:18:23,940 --> 00:18:26,810 And adiabatic really refers to the coupling 344 00:18:26,810 --> 00:18:30,440 between the different potential surfaces for the electrons, 345 00:18:30,440 --> 00:18:32,630 depending on the velocity of the nuclei. 346 00:18:32,630 --> 00:18:35,270 And Born-Oppenheimer implies that there 347 00:18:35,270 --> 00:18:37,580 is no influence of the ionic motion of one 348 00:18:37,580 --> 00:18:39,050 single electronic surface. 349 00:18:39,050 --> 00:18:40,940 But I mean, this is fairly technical. 350 00:18:40,940 --> 00:18:45,020 Just remember that sometimes these two terms actually mean 351 00:18:45,020 --> 00:18:47,930 something very different and very specific. 352 00:18:47,930 --> 00:18:48,650 OK. 353 00:18:48,650 --> 00:18:53,960 So this has now become our most general expression 354 00:18:53,960 --> 00:19:00,920 for the Hamiltonian and also for the energy of a set of nuclei 355 00:19:00,920 --> 00:19:02,150 and a set of electrons. 356 00:19:02,150 --> 00:19:06,050 That is really our picture of a molecule or a solid. 357 00:19:06,050 --> 00:19:12,130 And again, we will have the nuclei generating 358 00:19:12,130 --> 00:19:16,120 Coulombic attractive potentials in every position where 359 00:19:16,120 --> 00:19:16,970 they are. 360 00:19:16,970 --> 00:19:19,720 So there is what we call an electron nucleus term. 361 00:19:19,720 --> 00:19:22,300 That is an attractive term in which 362 00:19:22,300 --> 00:19:26,020 we have a sum over each and every electron 363 00:19:26,020 --> 00:19:30,070 because each and every electron feels the potential of all 364 00:19:30,070 --> 00:19:31,180 the nuclei. 365 00:19:31,180 --> 00:19:38,060 And the sum of all the nuclei refers to a sum of Z/r terms. 366 00:19:38,060 --> 00:19:41,050 So in whole space, wherever there is a nucleus, 367 00:19:41,050 --> 00:19:43,420 there is a 1/r term. 368 00:19:43,420 --> 00:19:50,020 And this sum over nuclei is the overall potential 369 00:19:50,020 --> 00:19:51,550 for the electronic system. 370 00:19:51,550 --> 00:19:55,000 And each electron feels this potential. 371 00:19:55,000 --> 00:19:57,760 And this is the fundamental attractive term. 372 00:19:57,760 --> 00:20:00,460 So electrons are attracted to the nuclei. 373 00:20:00,460 --> 00:20:03,830 But also, electrons repel each other. 374 00:20:03,830 --> 00:20:05,560 And this is the other term. 375 00:20:05,560 --> 00:20:08,620 So you see each and every electron 376 00:20:08,620 --> 00:20:11,920 has a charge 1 in atomic units and has 377 00:20:11,920 --> 00:20:15,700 a Coulombic repulsion with each and every other electron. 378 00:20:15,700 --> 00:20:17,800 And then, of course, for each electron, 379 00:20:17,800 --> 00:20:21,820 we have the quantum kinetic energy here. 380 00:20:21,820 --> 00:20:25,840 And so I listed here in the Hamiltonian all these terms. 381 00:20:25,840 --> 00:20:29,950 We have the quantum kinetic energy here. 382 00:20:29,950 --> 00:20:33,260 We have the electron-electron repulsion. 383 00:20:33,260 --> 00:20:37,000 And we have the electron-nucleus attraction. 384 00:20:37,000 --> 00:20:39,160 And then there is a last term that 385 00:20:39,160 --> 00:20:41,380 is truly a classical term because it 386 00:20:41,380 --> 00:20:43,460 involves only the nuclei. 387 00:20:43,460 --> 00:20:45,940 And so it's a repulsive term that is 388 00:20:45,940 --> 00:20:48,100 the nucleus-nucleus repulsion. 389 00:20:48,100 --> 00:20:52,150 So if you want to think for a moment of a hydrogen molecule, 390 00:20:52,150 --> 00:20:57,520 say, what you would there is a nucleus, another nucleus, 391 00:20:57,520 --> 00:21:02,140 and then a wave function of all the electrons around. 392 00:21:02,140 --> 00:21:06,910 And there is a classic term that is the electrostatic repulsion 393 00:21:06,910 --> 00:21:08,630 between the two nuclei. 394 00:21:08,630 --> 00:21:14,260 And there is a quantum term of repulsion between the electrons 395 00:21:14,260 --> 00:21:16,720 and an attraction between the electron and the nuclei. 396 00:21:16,720 --> 00:21:20,950 And basically, all these electrostatic terms-- 397 00:21:20,950 --> 00:21:22,570 nucleus-nucleus, electron-nucleus, 398 00:21:22,570 --> 00:21:26,770 and electron-electron-- almost balance themselves. 399 00:21:26,770 --> 00:21:29,620 Each and every one is very large. 400 00:21:29,620 --> 00:21:32,620 But then the combination of these three almost 401 00:21:32,620 --> 00:21:34,720 cancels itself out. 402 00:21:34,720 --> 00:21:39,340 And that's why actually the binding energy of a molecule 403 00:21:39,340 --> 00:21:42,830 is much, much smaller than any of the energy, say, 404 00:21:42,830 --> 00:21:46,150 of two charges at that distance repelling each other 405 00:21:46,150 --> 00:21:50,800 or even just the energy of a core electron 406 00:21:50,800 --> 00:21:52,210 very close to its nucleus. 407 00:21:52,210 --> 00:21:54,670 And that's why also electronic structure calculations are 408 00:21:54,670 --> 00:21:58,240 very delicate because what you need to find out 409 00:21:58,240 --> 00:22:04,390 is a total energy that is the combination of terms that 410 00:22:04,390 --> 00:22:08,960 largely cancel each other out. 411 00:22:08,960 --> 00:22:11,500 And so you need to be very accurate in order 412 00:22:11,500 --> 00:22:15,340 to actually decide if something like a hydrogen molecule 413 00:22:15,340 --> 00:22:20,880 binds together or breaks apart. 414 00:22:20,880 --> 00:22:21,510 OK. 415 00:22:21,510 --> 00:22:25,740 Now, this is truly a problem of greater 416 00:22:25,740 --> 00:22:27,570 computational complexity. 417 00:22:27,570 --> 00:22:30,220 And as I said over and over again, 418 00:22:30,220 --> 00:22:34,140 we can't really deal, even computationally, 419 00:22:34,140 --> 00:22:38,820 with an object that has all the information content of a wave 420 00:22:38,820 --> 00:22:39,730 function. 421 00:22:39,730 --> 00:22:43,200 So let me actually go through this 422 00:22:43,200 --> 00:22:47,160 into the next slide, in which I've written out explicitly 423 00:22:47,160 --> 00:22:49,680 the example of the iron atom. 424 00:22:49,680 --> 00:22:51,930 So that has, again, 26 electrons. 425 00:22:51,930 --> 00:22:53,700 So the electromagnetic wave function 426 00:22:53,700 --> 00:22:57,240 will in itself have 78 variables. 427 00:22:57,240 --> 00:22:59,250 And if you think about how many numbers 428 00:22:59,250 --> 00:23:03,600 do we need to store this object with any kind of precision, 429 00:23:03,600 --> 00:23:09,600 well, suppose that we even limit ourselves to a very, very core 430 00:23:09,600 --> 00:23:15,510 sampling, only say 10 values around either nucleus. 431 00:23:15,510 --> 00:23:18,060 Well, even to give this wave function, 432 00:23:18,060 --> 00:23:21,580 to give this amplitude with this very core sampling, 433 00:23:21,580 --> 00:23:24,090 we would need 10 to the 78 numbers. 434 00:23:24,090 --> 00:23:27,690 So basically, there is no way we can numerically deal 435 00:23:27,690 --> 00:23:30,280 with the complexity of the wave function. 436 00:23:30,280 --> 00:23:34,590 And this is where the power of the variational principle 437 00:23:34,590 --> 00:23:38,640 and the ideas of Hartree-Fock came together. 438 00:23:38,640 --> 00:23:44,250 And we'll first discuss the first idea of Hartree 439 00:23:44,250 --> 00:23:46,200 in dealing with this problem. 440 00:23:46,200 --> 00:23:49,770 Remember, what we have is a set of interacting electrons. 441 00:23:49,770 --> 00:23:54,612 And I like to compare each one of you to an electron. 442 00:23:54,612 --> 00:23:56,070 So what you need to think, and what 443 00:23:56,070 --> 00:23:58,005 the complexity of this problem is, 444 00:23:58,005 --> 00:24:02,430 is that at every instant in time each one of you 445 00:24:02,430 --> 00:24:07,810 is interacting or actually repelling with each one else. 446 00:24:07,810 --> 00:24:10,650 So this is the complexity of a many-body problem. 447 00:24:10,650 --> 00:24:13,740 In order to understand what's happening, 448 00:24:13,740 --> 00:24:18,390 each thing needs to know what everyone else is doing. 449 00:24:18,390 --> 00:24:21,930 And Hartree introduced the concept 450 00:24:21,930 --> 00:24:25,110 of independent particles and effective potential. 451 00:24:25,110 --> 00:24:27,480 This is something that comes over not 452 00:24:27,480 --> 00:24:29,940 only in electronic structure, but it comes over 453 00:24:29,940 --> 00:24:32,310 in a lot of problems where we actually 454 00:24:32,310 --> 00:24:37,980 need to deal with a very large number of interacting elements 455 00:24:37,980 --> 00:24:40,290 and interacting particles. 456 00:24:40,290 --> 00:24:45,900 And the general idea is that we can approximate and try 457 00:24:45,900 --> 00:24:50,130 to solve this problem by not considering what 458 00:24:50,130 --> 00:24:53,130 each electron instantly does. 459 00:24:53,130 --> 00:24:56,550 But we can solve the problem by treating 460 00:24:56,550 --> 00:25:01,620 what one electron would do in a field that, 461 00:25:01,620 --> 00:25:06,420 on average, represents what all the other electrons would 462 00:25:06,420 --> 00:25:07,470 be doing. 463 00:25:07,470 --> 00:25:10,990 So if we want to think, say, what I would be doing-- 464 00:25:10,990 --> 00:25:14,430 I shouldn't try to find the solution that instantaneously 465 00:25:14,430 --> 00:25:17,550 knows about what each and everyone else is doing. 466 00:25:17,550 --> 00:25:22,080 But I could try to find a solution for myself interacting 467 00:25:22,080 --> 00:25:26,220 electrostatically with the average charge distribution 468 00:25:26,220 --> 00:25:28,090 that everyone else does. 469 00:25:28,090 --> 00:25:30,330 So instead of having to know what 470 00:25:30,330 --> 00:25:32,670 is the instantaneous position of all 471 00:25:32,670 --> 00:25:34,710 the other interacting electrons, I 472 00:25:34,710 --> 00:25:38,130 could make an approximation that I could actually really 473 00:25:38,130 --> 00:25:41,490 just try to deal with the way I interact 474 00:25:41,490 --> 00:25:44,980 with an average distribution of everyone else. 475 00:25:44,980 --> 00:25:48,600 So this is, if you want, the concept of mean field 476 00:25:48,600 --> 00:25:50,640 or effective potential. 477 00:25:50,640 --> 00:25:53,850 We are averaging over all the variables. 478 00:25:53,850 --> 00:25:56,700 And there is actually a mathematical way 479 00:25:56,700 --> 00:26:00,390 to do this that I'll introduce in this moment. 480 00:26:00,390 --> 00:26:04,230 But if you want, the Hartree solution really leads to this. 481 00:26:04,230 --> 00:26:06,330 It leads to a Schrodinger equation 482 00:26:06,330 --> 00:26:09,120 in which we are actually trying to solve the problem 483 00:26:09,120 --> 00:26:12,030 of a single electron at a time. 484 00:26:12,030 --> 00:26:16,410 But that electron feels the average electrostatic charge 485 00:26:16,410 --> 00:26:20,220 distribution of all the other electrons. 486 00:26:20,220 --> 00:26:24,360 And one can actually obtain this directly 487 00:26:24,360 --> 00:26:26,520 from the variational principle. 488 00:26:26,520 --> 00:26:31,770 That is, one can make an answer for the wave function. 489 00:26:31,770 --> 00:26:36,630 What is written here is this most generic wave function. 490 00:26:36,630 --> 00:26:39,420 And one can make what turns out to be actually 491 00:26:39,420 --> 00:26:42,120 a fairly severe approximation. 492 00:26:42,120 --> 00:26:46,980 That is, we can say that this many-body wave function can 493 00:26:46,980 --> 00:26:53,027 actually be written as a product of single particle orbitals. 494 00:26:53,027 --> 00:26:55,110 So you see, what's happening, when we are actually 495 00:26:55,110 --> 00:27:01,380 making this hypothesis, is that varying r1 will change 496 00:27:01,380 --> 00:27:03,420 the amplitude of the wave function 497 00:27:03,420 --> 00:27:06,930 independently from what happens to r2 and rn. 498 00:27:06,930 --> 00:27:10,180 These have all become independent variable. 499 00:27:10,180 --> 00:27:14,970 There is no combined effect what r2 and r1 are 500 00:27:14,970 --> 00:27:17,430 doing-- so your any couple, any triplet, 501 00:27:17,430 --> 00:27:19,270 and so on and so forth. 502 00:27:19,270 --> 00:27:22,530 So if you want, if we fix all the other variables, 503 00:27:22,530 --> 00:27:26,790 we can independently look at what each one of these orbitals 504 00:27:26,790 --> 00:27:27,730 is doing. 505 00:27:27,730 --> 00:27:29,670 And you can think just what would 506 00:27:29,670 --> 00:27:31,330 come from a Taylor expansion. 507 00:27:31,330 --> 00:27:32,490 This is an approximation. 508 00:27:32,490 --> 00:27:35,610 It's not a true solution, OK? 509 00:27:35,610 --> 00:27:39,300 Suppose that we were dealing with two variables. 510 00:27:39,300 --> 00:27:42,820 Well, I mean, something like-- 511 00:27:42,820 --> 00:27:45,790 think of a generic wave function that could be written 512 00:27:45,790 --> 00:27:53,320 as, say, the exponential of the square root of r1 plus r2 that 513 00:27:53,320 --> 00:27:58,310 can't be decoupled in the product of two wave functions. 514 00:27:58,310 --> 00:28:01,660 So the product of two single-particle wave functions 515 00:28:01,660 --> 00:28:04,240 is something simple that doesn't capture 516 00:28:04,240 --> 00:28:07,600 the complexity of all the possible two-body wave 517 00:28:07,600 --> 00:28:08,360 functions. 518 00:28:08,360 --> 00:28:11,300 And so in this, it's an approximation. 519 00:28:11,300 --> 00:28:14,000 And so Hartree made this approximation 520 00:28:14,000 --> 00:28:17,260 and then asked himself, what happens 521 00:28:17,260 --> 00:28:21,400 if I actually throw this approximation 522 00:28:21,400 --> 00:28:23,980 in the variational principle? 523 00:28:23,980 --> 00:28:26,560 Now, the difference is that, instead 524 00:28:26,560 --> 00:28:31,360 of having just a function that varies parametrically, now what 525 00:28:31,360 --> 00:28:33,550 we can really vary in our valuational 526 00:28:33,550 --> 00:28:40,460 principle are the shapes of all these single-particle orbitals. 527 00:28:40,460 --> 00:28:43,660 So what we are asking ourselves is 528 00:28:43,660 --> 00:28:45,970 a function of this former throwing 529 00:28:45,970 --> 00:28:48,760 into the variational principle will give me 530 00:28:48,760 --> 00:28:53,840 a condition that each of these orbitals need to satisfy. 531 00:28:53,840 --> 00:28:56,890 So if you want, you really need to now calculate 532 00:28:56,890 --> 00:29:01,060 with functional analysis what are the differential equations 533 00:29:01,060 --> 00:29:04,960 that each of these orbitals, phi 1 to phi n, 534 00:29:04,960 --> 00:29:08,770 needs to satisfy so that the overall expectation 535 00:29:08,770 --> 00:29:13,240 value of the energy is minimum for a wave function written 536 00:29:13,240 --> 00:29:17,050 in this restricted class of product 537 00:29:17,050 --> 00:29:20,080 of single-particle orbitals. 538 00:29:20,080 --> 00:29:22,730 And so when you actually work out 539 00:29:22,730 --> 00:29:27,520 the fairly complex functional analysis of this problem-- 540 00:29:27,520 --> 00:29:30,130 that's actually described, if you're interested, 541 00:29:30,130 --> 00:29:31,750 in one of the references that we have 542 00:29:31,750 --> 00:29:34,390 given you, the Bransden and Joachain book 543 00:29:34,390 --> 00:29:37,570 on the physics of atoms and molecules-- 544 00:29:37,570 --> 00:29:40,900 what you obtain for the specific case, 545 00:29:40,900 --> 00:29:46,180 again of a Hamiltonian operator in which the potential is given 546 00:29:46,180 --> 00:29:50,470 by a linear combination of attractive Coulombic potential, 547 00:29:50,470 --> 00:29:52,660 is a set of equations. 548 00:29:52,660 --> 00:29:55,630 That is, what you obtain is a new set 549 00:29:55,630 --> 00:29:57,820 of differential equations. 550 00:29:57,820 --> 00:30:01,990 Instead of having, if you want, one single Schrodinger equation 551 00:30:01,990 --> 00:30:04,510 for a many-body wave function-- 552 00:30:04,510 --> 00:30:07,420 what you obtain if you are dealing with n particles 553 00:30:07,420 --> 00:30:13,090 is n different differential equations, each one 554 00:30:13,090 --> 00:30:17,710 being a differential equation for only a single particle wave 555 00:30:17,710 --> 00:30:19,060 function. 556 00:30:19,060 --> 00:30:21,190 This is still fairly complex. 557 00:30:21,190 --> 00:30:23,890 But the complexity of this has gone 558 00:30:23,890 --> 00:30:28,300 from being a complexity of a wave function of 78 variables 559 00:30:28,300 --> 00:30:31,990 to, say, the complexity of 26 equations 560 00:30:31,990 --> 00:30:34,360 in three variables each, if we are 561 00:30:34,360 --> 00:30:36,340 working in three dimensions. 562 00:30:36,340 --> 00:30:38,680 And actualyl, the form of this equation 563 00:30:38,680 --> 00:30:40,900 is very intriguing because this really 564 00:30:40,900 --> 00:30:45,400 looks like a Schrodinger equation for one electron. 565 00:30:45,400 --> 00:30:49,120 You see, here is the quantum kinetic energy 566 00:30:49,120 --> 00:30:51,190 term for this electron. 567 00:30:51,190 --> 00:30:53,950 And then there is the interaction 568 00:30:53,950 --> 00:30:57,820 between this single electron with the Coulombic distribution 569 00:30:57,820 --> 00:30:59,140 of nuclei. 570 00:30:59,140 --> 00:31:00,640 And then there is a term-- 571 00:31:00,640 --> 00:31:05,650 you see, this appropriately is called the Hartree term-- 572 00:31:05,650 --> 00:31:13,070 in which this electron, i, is actually feeling a Coulombic 573 00:31:13,070 --> 00:31:15,890 repulsion-- this is the Coulombic term-- 574 00:31:15,890 --> 00:31:19,160 of a charge density distribution. 575 00:31:19,160 --> 00:31:22,640 Remember that if we take this square model of a wave 576 00:31:22,640 --> 00:31:26,090 function, we obtain the probability of finding 577 00:31:26,090 --> 00:31:27,470 an electron somewhere. 578 00:31:27,470 --> 00:31:29,660 And the charge density is nothing less than that. 579 00:31:29,660 --> 00:31:32,720 It's the probability of finding an electron somewhere. 580 00:31:32,720 --> 00:31:35,600 And so you see, what we have is, for the electron 581 00:31:35,600 --> 00:31:41,240 that we have denoted as i, the interaction between the wave 582 00:31:41,240 --> 00:31:46,850 function and a potential, as usual a potential, in which we 583 00:31:46,850 --> 00:31:49,580 have the Coulombic term. 584 00:31:49,580 --> 00:31:52,040 And the repulsion, the Coulombic repulsion 585 00:31:52,040 --> 00:31:55,580 is between the electron, i, and the charged density 586 00:31:55,580 --> 00:32:00,170 distribution of each and every other electron, j. 587 00:32:00,170 --> 00:32:04,070 So you see, this sum goes over all the other electrons. 588 00:32:04,070 --> 00:32:06,350 So my many-body Schrodinger equation 589 00:32:06,350 --> 00:32:10,670 has become a series of equations that we 590 00:32:10,670 --> 00:32:13,700 call single-particle equations, a differential equation 591 00:32:13,700 --> 00:32:15,680 for each and every electron. 592 00:32:15,680 --> 00:32:18,020 And those differential equations, 593 00:32:18,020 --> 00:32:20,360 as I said, have been obtained formally 594 00:32:20,360 --> 00:32:25,130 just by applying this answer to the variational principle, 595 00:32:25,130 --> 00:32:28,400 are in the form of a Schrodinger-like equation 596 00:32:28,400 --> 00:32:31,670 with a kinetic energy, an attractive potential. 597 00:32:31,670 --> 00:32:35,720 And now, we have only a mean field interaction 598 00:32:35,720 --> 00:32:38,870 between the electrons because electron i 599 00:32:38,870 --> 00:32:42,410 doesn't instantaneously need to know what each 600 00:32:42,410 --> 00:32:44,180 and every other electron does. 601 00:32:44,180 --> 00:32:49,250 But it actually only interacts with the average charge density 602 00:32:49,250 --> 00:32:53,690 distribution that is given by the square model of j. 603 00:32:53,690 --> 00:32:54,560 OK. 604 00:32:54,560 --> 00:32:56,510 So this is a great simplification. 605 00:32:56,510 --> 00:33:00,950 And it actually allowed some of the first calculations, 606 00:33:00,950 --> 00:33:02,390 say, on atoms. 607 00:33:02,390 --> 00:33:05,180 Actually, this was developed in the late '20s. 608 00:33:05,180 --> 00:33:08,450 And very quickly, it was realized what was wrong. 609 00:33:08,450 --> 00:33:09,800 And we'll see that in a moment. 610 00:33:09,800 --> 00:33:15,080 That is the lack of correlation because there 611 00:33:15,080 --> 00:33:18,380 is a specific role that electrons are fermions. 612 00:33:18,380 --> 00:33:20,270 But really, this was the first time 613 00:33:20,270 --> 00:33:24,710 in which we had a workable differential equation 614 00:33:24,710 --> 00:33:26,750 for our many-body system. 615 00:33:26,750 --> 00:33:31,790 And the most important conclusion 616 00:33:31,790 --> 00:33:34,790 of this, and where the complexity 617 00:33:34,790 --> 00:33:37,520 of the many-body problem comes back in, 618 00:33:37,520 --> 00:33:41,840 is that this new operator, this new overall Hartree 619 00:33:41,840 --> 00:33:44,510 operator acting on the wave function, 620 00:33:44,510 --> 00:33:48,440 has become, as we say, self-consistent. 621 00:33:48,440 --> 00:33:52,610 That is, the operator itself depends 622 00:33:52,610 --> 00:33:54,940 on what the other electrons do. 623 00:33:54,940 --> 00:33:58,610 So it depends on what the solution 624 00:33:58,610 --> 00:34:02,390 to all the other differential equations are. 625 00:34:02,390 --> 00:34:06,440 So if you want to know what electron i does 626 00:34:06,440 --> 00:34:08,900 in the mean field of all the other electrons, 627 00:34:08,900 --> 00:34:12,350 you need to know what is the wave function of each 628 00:34:12,350 --> 00:34:14,510 and every other electron. 629 00:34:14,510 --> 00:34:17,030 But in order to solve the wave function of each 630 00:34:17,030 --> 00:34:18,590 and every other electron, you will 631 00:34:18,590 --> 00:34:22,620 need to know also what electron i is doing. 632 00:34:22,620 --> 00:34:25,070 And so really, this is very different. 633 00:34:25,070 --> 00:34:29,239 The Hamiltonian operator here is not anymore given 634 00:34:29,239 --> 00:34:31,820 at the beginning of the problem, but actually 635 00:34:31,820 --> 00:34:35,719 needs to be found because the operator-- 636 00:34:35,719 --> 00:34:38,239 the action on one electron depends 637 00:34:38,239 --> 00:34:40,022 on what the other electrons do. 638 00:34:40,022 --> 00:34:42,230 And in order to find out what the other electrons do, 639 00:34:42,230 --> 00:34:43,790 we need to solve the set. 640 00:34:43,790 --> 00:34:45,170 So this set of-- 641 00:34:45,170 --> 00:34:46,730 say, the case of iron again-- 642 00:34:46,730 --> 00:34:53,159 26 differential equations needs to be solved simultaneously, 643 00:34:53,159 --> 00:34:53,659 OK? 644 00:34:53,659 --> 00:34:57,440 So we actually solve it iteratively. 645 00:34:57,440 --> 00:35:02,510 We start with a guess for what the wave function would be. 646 00:35:02,510 --> 00:35:04,970 And we try to find a combined solution. 647 00:35:04,970 --> 00:35:08,330 I'll describe in a moment what the algorithm is. 648 00:35:08,330 --> 00:35:11,480 So this is how the many-body complexity comes in. 649 00:35:11,480 --> 00:35:15,170 That is, we need to solve a differential equation for which 650 00:35:15,170 --> 00:35:17,960 we don't even know, at the beginning, what 651 00:35:17,960 --> 00:35:24,150 is the operator acting on our single-particle wave function. 652 00:35:24,150 --> 00:35:27,410 And so the concept of self-consistency 653 00:35:27,410 --> 00:35:30,440 and of iterative solution will basically 654 00:35:30,440 --> 00:35:33,950 be always present in all the electronic structure 655 00:35:33,950 --> 00:35:37,520 approaches that we are going to see in this class. 656 00:35:37,520 --> 00:35:41,120 And it's actually very simple to figure out 657 00:35:41,120 --> 00:35:43,880 what could be an actual algorithm 658 00:35:43,880 --> 00:35:46,160 to try to get to the solution. 659 00:35:46,160 --> 00:35:50,300 That is, we need to start with an arbitrary guess for all 660 00:35:50,300 --> 00:35:53,360 those 26 orbitals. 661 00:35:53,360 --> 00:35:56,810 And once we have that arbitrary guess, 662 00:35:56,810 --> 00:36:00,260 we can construct the charge density of every electron. 663 00:36:00,260 --> 00:36:04,700 And so we can construct what would be the operator acting 664 00:36:04,700 --> 00:36:07,610 on each one of the phi i. 665 00:36:07,610 --> 00:36:14,000 That is, we can construct this term here. 666 00:36:14,000 --> 00:36:18,050 Once now we know what actually our differential equation is, 667 00:36:18,050 --> 00:36:19,890 we can solve it. 668 00:36:19,890 --> 00:36:24,140 So we find what are the ground states of each of those 26 669 00:36:24,140 --> 00:36:25,800 differential equations. 670 00:36:25,800 --> 00:36:28,280 And now, with those ground states, 671 00:36:28,280 --> 00:36:30,890 we can construct again a charge density of each 672 00:36:30,890 --> 00:36:32,420 and every single electron. 673 00:36:32,420 --> 00:36:33,710 We put it together. 674 00:36:33,710 --> 00:36:36,110 We have a new Hartree operator. 675 00:36:36,110 --> 00:36:40,130 And we can solve these differential equations again. 676 00:36:40,130 --> 00:36:44,690 And we keep iterating this until what 677 00:36:44,690 --> 00:36:48,680 we obtain is a Hartree operator in each differential equation 678 00:36:48,680 --> 00:36:50,550 that doesn't change anymore. 679 00:36:50,550 --> 00:36:55,190 And so we have now a set of self-consistent orbitals. 680 00:36:55,190 --> 00:36:58,940 In most cases, you actually don't get to convergence. 681 00:36:58,940 --> 00:37:02,510 So a lot of the algorithmic advances 682 00:37:02,510 --> 00:37:04,610 that has been done in the 20th century 683 00:37:04,610 --> 00:37:06,800 is actually to do with this problem. 684 00:37:06,800 --> 00:37:10,310 That is, we need to find ways to get this procedure 685 00:37:10,310 --> 00:37:13,850 to converge to an actually self-consistent point. 686 00:37:13,850 --> 00:37:16,920 But the concept is all here. 687 00:37:16,920 --> 00:37:19,400 And if you want, the simplest thing 688 00:37:19,400 --> 00:37:22,940 that you can do to make sure that you're 689 00:37:22,940 --> 00:37:25,220 going to iterate to self-consistence 690 00:37:25,220 --> 00:37:28,200 is to move very slowly. 691 00:37:28,200 --> 00:37:31,490 That is, whenever you have a set of solutions, 692 00:37:31,490 --> 00:37:34,910 you don't want to construct your Hartree operator, 693 00:37:34,910 --> 00:37:37,670 this new charge density, with the solution. 694 00:37:37,670 --> 00:37:42,620 But you want just to modify a little bit your previous charge 695 00:37:42,620 --> 00:37:46,550 densities to go in the direction of the new charge density 696 00:37:46,550 --> 00:37:48,050 that you were calculating. 697 00:37:48,050 --> 00:37:53,600 So you somehow try to minimize the change in the operators 698 00:37:53,600 --> 00:37:55,650 from one iteration to the other. 699 00:37:55,650 --> 00:38:00,800 And that tends to be actually fairly functional in a lot 700 00:38:00,800 --> 00:38:02,840 of problems. 701 00:38:02,840 --> 00:38:06,080 Now, Hartree was a very interesting character, 702 00:38:06,080 --> 00:38:10,340 again growing up in Cambridge at the turn of the century. 703 00:38:10,340 --> 00:38:13,460 He became an expert in differential equations 704 00:38:13,460 --> 00:38:15,920 during the First World War because basically people 705 00:38:15,920 --> 00:38:20,480 had the problem of sending cannonballs across the lines. 706 00:38:20,480 --> 00:38:23,530 So there was a lot of differential equations. 707 00:38:23,530 --> 00:38:29,840 And so mid '20s, he developed this general idea 708 00:38:29,840 --> 00:38:32,420 of the Hartree equation that really what we 709 00:38:32,420 --> 00:38:35,210 call coupled integral differential 710 00:38:35,210 --> 00:38:37,220 equation because there are derivatives 711 00:38:37,220 --> 00:38:38,750 and there are integrals. 712 00:38:38,750 --> 00:38:41,390 And so what you need to do is now solve 713 00:38:41,390 --> 00:38:44,840 this equation that can't really be solved analytically. 714 00:38:44,840 --> 00:38:47,180 And luckily, that was the time in which 715 00:38:47,180 --> 00:38:50,930 people were developing, if you want, the first computers. 716 00:38:50,930 --> 00:38:53,750 No electronic in there-- so computing 717 00:38:53,750 --> 00:38:55,580 machine, but mechanical. 718 00:38:55,580 --> 00:38:58,520 And one of the first computers was at MIT. 719 00:38:58,520 --> 00:39:01,970 This is actually a picture from the MIT archives. 720 00:39:01,970 --> 00:39:05,420 There was an electrical engineer named Vannevar Bush, 721 00:39:05,420 --> 00:39:08,780 to which Building 13 the Bush Building, is named 722 00:39:08,780 --> 00:39:14,780 that developed one of the first mechanical differential 723 00:39:14,780 --> 00:39:16,400 equation solvers. 724 00:39:16,400 --> 00:39:19,580 And so Hartree came at the end of the '20s to MIT 725 00:39:19,580 --> 00:39:21,680 to actually solve the Hartree equation, 726 00:39:21,680 --> 00:39:25,760 I guess, in some building here exactly on that machine, 727 00:39:25,760 --> 00:39:29,360 and then went back to Cambridge. 728 00:39:29,360 --> 00:39:33,850 And so this, I thought, was an interesting note. 729 00:39:33,850 --> 00:39:36,010 So what do we obtain when we actually 730 00:39:36,010 --> 00:39:38,570 solve the Hartree equation? 731 00:39:38,570 --> 00:39:41,950 Well, the most fundamental concept 732 00:39:41,950 --> 00:39:45,700 is that we are losing some information 733 00:39:45,700 --> 00:39:48,670 on what's happening instantaneously 734 00:39:48,670 --> 00:39:50,770 to all the electrons. 735 00:39:50,770 --> 00:39:53,170 This is what we call, generically speaking, 736 00:39:53,170 --> 00:39:54,310 a correlation. 737 00:39:54,310 --> 00:39:57,863 And I'll give you the technical definition in a few slides. 738 00:39:57,863 --> 00:39:59,530 But basically, this is what's happening. 739 00:39:59,530 --> 00:40:03,070 Let's consider the case of the helium atom. 740 00:40:03,070 --> 00:40:06,010 That is the simplest case in which you 741 00:40:06,010 --> 00:40:08,000 have more than one electron. 742 00:40:08,000 --> 00:40:10,150 So what you have is two electrons. 743 00:40:10,150 --> 00:40:13,930 What does the Hartree equation tell us about-- 744 00:40:13,930 --> 00:40:16,180 or what do the Hartree equations tell us 745 00:40:16,180 --> 00:40:18,020 about these two electrons? 746 00:40:18,020 --> 00:40:20,560 Well, we have two equations. 747 00:40:20,560 --> 00:40:23,950 And in each of them, one electron 748 00:40:23,950 --> 00:40:28,900 is going to feel an average electrostatic repulsion 749 00:40:28,900 --> 00:40:30,910 from the other electron. 750 00:40:30,910 --> 00:40:34,210 So what we have is that this right electron here 751 00:40:34,210 --> 00:40:37,120 is attracted to the nucleus and is 752 00:40:37,120 --> 00:40:45,430 repelled by a spherically symmetrical average charge 753 00:40:45,430 --> 00:40:46,720 distribution. 754 00:40:46,720 --> 00:40:48,140 So this is what's happening. 755 00:40:48,140 --> 00:40:50,950 This is what Hartree tells us. 756 00:40:50,950 --> 00:40:57,510 But in reality, electrons instantaneously 757 00:40:57,510 --> 00:41:02,610 try to keep themselves as far apart as possible 758 00:41:02,610 --> 00:41:04,840 because electrons repel each other. 759 00:41:04,840 --> 00:41:10,260 So in a simplified way, you could think of electron 1 760 00:41:10,260 --> 00:41:17,340 and electron 2 trying to orbit the nucleus as much as possible 761 00:41:17,340 --> 00:41:20,100 in a position of phases. 762 00:41:20,100 --> 00:41:23,730 So the true two interacting electrons 763 00:41:23,730 --> 00:41:28,650 try to be as far as possible at every moment in time 764 00:41:28,650 --> 00:41:34,860 during their revolution around the helium nucleus. 765 00:41:34,860 --> 00:41:37,620 But this instantaneous correlation-- 766 00:41:37,620 --> 00:41:41,610 that is, the fact that the wave functions tries 767 00:41:41,610 --> 00:41:46,530 to keep the electron as far away as possible-- is lost 768 00:41:46,530 --> 00:41:50,760 in the Hartree equations because what we do is we 769 00:41:50,760 --> 00:41:54,600 are really having one electron interacting 770 00:41:54,600 --> 00:41:58,800 with the average position of the other electron. 771 00:41:58,800 --> 00:42:02,070 And so in the Hartree equation, there 772 00:42:02,070 --> 00:42:07,140 are a lot of terms that have to do with our initial electron, 773 00:42:07,140 --> 00:42:12,030 red, being too close to the green charge density 774 00:42:12,030 --> 00:42:13,600 distribution. 775 00:42:13,600 --> 00:42:16,710 So if you want, the wave function-- that is, 776 00:42:16,710 --> 00:42:19,560 the overall solution of the Hartree equation-- 777 00:42:19,560 --> 00:42:24,180 tends to have too much electrostatic repulsion 778 00:42:24,180 --> 00:42:26,490 between electron 1 and electron 2. 779 00:42:26,490 --> 00:42:31,890 And that's why ultimately the energy of this Hartree function 780 00:42:31,890 --> 00:42:34,380 is higher than the true solution. 781 00:42:34,380 --> 00:42:37,890 This is what is missing in the Hartree equation, the fact 782 00:42:37,890 --> 00:42:40,800 that what you want is a lot of correlation. 783 00:42:40,800 --> 00:42:44,610 That is, electrons want to keep each other apart as 784 00:42:44,610 --> 00:42:45,840 much as possible. 785 00:42:45,840 --> 00:42:50,520 But that, if you want, is really an instantaneous solution. 786 00:42:50,520 --> 00:42:53,280 It's what people call dynamical correlation. 787 00:42:53,280 --> 00:42:56,170 Electrons want to keep apart from each other. 788 00:42:56,170 --> 00:42:57,900 But if you start looking at a mean field 789 00:42:57,900 --> 00:43:00,390 solution in which only one interacts 790 00:43:00,390 --> 00:43:02,640 with the average charge density, you 791 00:43:02,640 --> 00:43:05,610 have lost the possibility of having 792 00:43:05,610 --> 00:43:10,350 this instantaneous non-symmetric distribution. 793 00:43:10,350 --> 00:43:14,050 And so in general, this is what we call correlation. 794 00:43:14,050 --> 00:43:19,330 And this is what is missing in the Hartree picture. 795 00:43:19,330 --> 00:43:22,780 There is another set of very fundamental concepts that 796 00:43:22,780 --> 00:43:26,350 I'll describe in a moment that the wave function-- 797 00:43:26,350 --> 00:43:29,620 the answer for the Hartree wave function-- 798 00:43:29,620 --> 00:43:34,360 doesn't satisfy a fundamental rule for wave functions. 799 00:43:34,360 --> 00:43:36,280 We say it's not anti-symmetric. 800 00:43:36,280 --> 00:43:38,845 And I'll show you in a moment what it is. 801 00:43:38,845 --> 00:43:41,590 That is, if you want, it doesn't respect 802 00:43:41,590 --> 00:43:46,750 a fundamental constraint on the shape of functions. 803 00:43:46,750 --> 00:43:48,610 And so that's, if you want, an error. 804 00:43:48,610 --> 00:43:52,570 And that's another source of error in our final estimate 805 00:43:52,570 --> 00:43:54,100 of the energy. 806 00:43:54,100 --> 00:43:57,850 And in particular, what it doesn't do-- 807 00:43:57,850 --> 00:44:02,170 it does not remove what is the accidental degeneracy, the fact 808 00:44:02,170 --> 00:44:06,100 that there is the same energy for electrons that 809 00:44:06,100 --> 00:44:10,510 have the same principle quantum number n and the same angular 810 00:44:10,510 --> 00:44:13,180 momentum number l. 811 00:44:13,180 --> 00:44:15,940 But really, what is most important 812 00:44:15,940 --> 00:44:19,930 is this lack of a physical constraint 813 00:44:19,930 --> 00:44:22,880 and this lack of correlation. 814 00:44:22,880 --> 00:44:27,020 Very soon, I mean probably the same year 815 00:44:27,020 --> 00:44:31,250 or a year later, Hartree, and independently Fock, 816 00:44:31,250 --> 00:44:35,210 realized that one could actually find a better solution 817 00:44:35,210 --> 00:44:38,960 to the problem satisfying one, again, 818 00:44:38,960 --> 00:44:42,410 of the fundamental rules of nature that 819 00:44:42,410 --> 00:44:46,550 had been discovered in the '20s during the development 820 00:44:46,550 --> 00:44:48,740 of quantum mechanics. 821 00:44:48,740 --> 00:44:52,040 And so one of these rules was what 822 00:44:52,040 --> 00:44:56,700 is called the spin-statistic correlation. 823 00:44:56,700 --> 00:44:59,510 And this is really very general. 824 00:44:59,510 --> 00:45:03,260 First of all, there is a division 825 00:45:03,260 --> 00:45:06,020 in elementary particles that says 826 00:45:06,020 --> 00:45:09,350 that all elementary particles can be called 827 00:45:09,350 --> 00:45:12,320 either fermions or bosons. 828 00:45:12,320 --> 00:45:15,290 And so things like electrons are actually fermions. 829 00:45:15,290 --> 00:45:17,780 They have a half-integer spin. 830 00:45:17,780 --> 00:45:20,930 But there is a fundamental difference 831 00:45:20,930 --> 00:45:23,660 between fermions and bosons. 832 00:45:23,660 --> 00:45:31,070 And in particular, they satisfy different statistical rules 833 00:45:31,070 --> 00:45:34,670 for an ensemble of many interacting electrons 834 00:45:34,670 --> 00:45:36,440 or for many bosons. 835 00:45:36,440 --> 00:45:38,360 And this rule-- so this is nothing else 836 00:45:38,360 --> 00:45:41,990 that-- again, another rule like in classical mechanics-- 837 00:45:41,990 --> 00:45:43,790 you have Newton's equation of motion. 838 00:45:43,790 --> 00:45:46,550 In quantum mechanics, you have a rule 839 00:45:46,550 --> 00:45:51,800 that wave functions that described fermions-- that 840 00:45:51,800 --> 00:45:56,030 is, a wave function that describes electrons 841 00:45:56,030 --> 00:45:58,490 needs to have this overall shape. 842 00:45:58,490 --> 00:46:05,720 That is, it needs to change sign when we invert two variables. 843 00:46:05,720 --> 00:46:09,980 So we have this general form for the wave function. 844 00:46:09,980 --> 00:46:12,770 We exchange two variables. 845 00:46:12,770 --> 00:46:18,080 And what needs to happen is that the wave function 846 00:46:18,080 --> 00:46:19,910 needs to change sign. 847 00:46:19,910 --> 00:46:21,620 And this has something to do-- 848 00:46:21,620 --> 00:46:23,480 this can actually be demonstrated. 849 00:46:23,480 --> 00:46:25,930 But it's what people call quantum field theory. 850 00:46:25,930 --> 00:46:27,680 So it's an advanced concept. 851 00:46:27,680 --> 00:46:29,480 But it's a very simple rule. 852 00:46:29,480 --> 00:46:32,315 And it's a very simple symmetry of the wave functioning. 853 00:46:32,315 --> 00:46:36,170 In the same way, if you want, you have in a crystal, 854 00:46:36,170 --> 00:46:39,500 you have physical symmetries for what 855 00:46:39,500 --> 00:46:41,210 could be some of your properties, 856 00:46:41,210 --> 00:46:44,540 like an elastic tensor or a piezoelectric coefficient. 857 00:46:44,540 --> 00:46:47,960 Well, what you have is a fundamental symmetry for a wave 858 00:46:47,960 --> 00:46:49,730 function describing electrons. 859 00:46:49,730 --> 00:46:53,540 It needs to change sign when you invert two coordinates. 860 00:46:53,540 --> 00:47:05,150 And so what the Hartree solution didn't have 861 00:47:05,150 --> 00:47:08,510 was exactly this anti-symmetry requirement. 862 00:47:08,510 --> 00:47:12,500 Remember, the Hartree solution was just the product 863 00:47:12,500 --> 00:47:15,560 of single-particle orbitals. 864 00:47:15,560 --> 00:47:18,710 But Hartree very quickly realized that you can actually 865 00:47:18,710 --> 00:47:22,430 satisfy this symmetry requirement 866 00:47:22,430 --> 00:47:27,350 if instead of taking just the product of n orbitals, 867 00:47:27,350 --> 00:47:33,170 you take the sum of the product of n orbitals 868 00:47:33,170 --> 00:47:37,470 where you interchange the variables in all 869 00:47:37,470 --> 00:47:38,940 the possible ways-- 870 00:47:38,940 --> 00:47:42,330 putting a plus or minus sign in front, 871 00:47:42,330 --> 00:47:45,300 depending on how much interchanges you had. 872 00:47:45,300 --> 00:47:48,420 And I think it's very simple to think of this problem if you 873 00:47:48,420 --> 00:47:50,910 have, say, only two electrons. 874 00:47:50,910 --> 00:47:53,020 So you have only two orbitals. 875 00:47:53,020 --> 00:47:59,190 And so we could call these two orbital, say, alpha and beta. 876 00:47:59,190 --> 00:48:04,170 And so what we would have is the Hartree solution 877 00:48:04,170 --> 00:48:09,270 that is the product of the alpha orbital function of the first r 878 00:48:09,270 --> 00:48:11,700 variable and the beta orbital function 879 00:48:11,700 --> 00:48:13,420 of the second variable. 880 00:48:13,420 --> 00:48:17,760 And this doesn't change sign if we exchange 1 with 2. 881 00:48:17,760 --> 00:48:19,260 It becomes a different function. 882 00:48:19,260 --> 00:48:23,430 It doesn't become the same function with the sign changed. 883 00:48:23,430 --> 00:48:26,970 But you can see that what we can do, without increasing really 884 00:48:26,970 --> 00:48:29,670 the complexity of the problem-- that is, still dealing 885 00:48:29,670 --> 00:48:33,120 with just the need of describing two orbitals, what we could do 886 00:48:33,120 --> 00:48:36,840 is take as an ansatz for the wave 887 00:48:36,840 --> 00:48:40,560 function describing two electrons something that 888 00:48:40,560 --> 00:48:51,090 is actually alpha 1 B2 minus B1 alpha 2, OK? 889 00:48:51,090 --> 00:48:55,740 So we have still two orbitals that we need to figure out. 890 00:48:55,740 --> 00:48:58,110 We need to figure out what is the shape of alpha 891 00:48:58,110 --> 00:49:00,090 and what is the shape of beta. 892 00:49:00,090 --> 00:49:02,820 But now, we are using as an ansatz 893 00:49:02,820 --> 00:49:05,220 for this two-electron wave function 894 00:49:05,220 --> 00:49:08,790 something that actually changes sign 895 00:49:08,790 --> 00:49:12,120 when you exchange 1 with 2. 896 00:49:12,120 --> 00:49:16,410 So this is the trial wave function of Hartree. 897 00:49:16,410 --> 00:49:19,740 And this is the trial wave function 898 00:49:19,740 --> 00:49:22,920 of what we call the Hartree-Fock method that is basically 899 00:49:22,920 --> 00:49:27,630 a trial wave function that has built-in anti-symmetry 900 00:49:27,630 --> 00:49:31,410 constraint for exchange of particles. 901 00:49:31,410 --> 00:49:36,090 And this can be generalized to the case of n particles. 902 00:49:36,090 --> 00:49:41,220 And really, what we call a sum of n terms 903 00:49:41,220 --> 00:49:43,230 with all the possible permutations, 904 00:49:43,230 --> 00:49:45,810 with all the possible signs, is nothing less 905 00:49:45,810 --> 00:49:47,160 than a determinant. 906 00:49:47,160 --> 00:49:49,620 If you think at what a determinant is, 907 00:49:49,620 --> 00:49:54,880 well, this determinant is going to have one element. 908 00:49:54,880 --> 00:49:57,930 It's going to have a sum of terms that each of them 909 00:49:57,930 --> 00:50:01,770 are rated as one term from each column. 910 00:50:01,770 --> 00:50:04,650 And we are taking all the possible permutations. 911 00:50:04,650 --> 00:50:08,640 And we are taking a plus sign or a minus sign 912 00:50:08,640 --> 00:50:11,280 in the sum of all these terms, depending 913 00:50:11,280 --> 00:50:13,080 on how many permutations they are. 914 00:50:13,080 --> 00:50:16,800 This is just the linear algebra definition of a determinant. 915 00:50:16,800 --> 00:50:21,930 And you see, again, a determinant of two functions 916 00:50:21,930 --> 00:50:24,570 is-- that's what I've written here in green, 917 00:50:24,570 --> 00:50:26,010 right here in the corner. 918 00:50:26,010 --> 00:50:34,890 If we are asking what alpha 1 determinant beta 919 00:50:34,890 --> 00:50:45,360 1 beta 2 and alpha and alpha-- 920 00:50:45,360 --> 00:50:49,600 so beta 1 and alpha 2. 921 00:50:54,380 --> 00:51:00,030 So this would be the specific expression of this determinant 922 00:51:00,030 --> 00:51:01,960 for the case of two particles. 923 00:51:01,960 --> 00:51:04,500 And if you just solved that determinant, 924 00:51:04,500 --> 00:51:06,610 you have only these two terms. 925 00:51:06,610 --> 00:51:10,110 So we haven't increased, in going into the Hartree-Fock 926 00:51:10,110 --> 00:51:12,360 method, the complexity of the problem 927 00:51:12,360 --> 00:51:17,190 that we need to solve because we still need to find n functions, 928 00:51:17,190 --> 00:51:19,200 where n is the number of electrons. 929 00:51:19,200 --> 00:51:21,810 And we need to find out the appropriate differential 930 00:51:21,810 --> 00:51:25,380 equation that descend from the variational principle 931 00:51:25,380 --> 00:51:29,940 once we stick this determinant into the variational principle. 932 00:51:29,940 --> 00:51:33,780 And again, it's not very complex functional analysis. 933 00:51:33,780 --> 00:51:36,990 And the Bransden Joachain describes that in detail. 934 00:51:36,990 --> 00:51:41,370 But with this new solution, with this ansatz, what 935 00:51:41,370 --> 00:51:44,400 we find is a new set of differential equations 936 00:51:44,400 --> 00:51:48,120 that look a lot like the Hartree equation 937 00:51:48,120 --> 00:51:49,380 that we had written before-- 938 00:51:49,380 --> 00:51:51,790 I'll go back to this in a moment-- 939 00:51:51,790 --> 00:51:54,250 but have an additional term. 940 00:51:54,250 --> 00:51:59,620 So what we find are, again, single-particle equations. 941 00:51:59,620 --> 00:52:02,730 So we have an integral differential equation 942 00:52:02,730 --> 00:52:08,880 for each single-particle orbital lambda that is written in red. 943 00:52:08,880 --> 00:52:13,440 And now, what we have is a set of additional terms. 944 00:52:13,440 --> 00:52:17,100 So we still have the quantum kinetic energy 945 00:52:17,100 --> 00:52:18,450 for that electron. 946 00:52:18,450 --> 00:52:20,220 And we still have the interaction 947 00:52:20,220 --> 00:52:22,740 between that electron and the collection 948 00:52:22,740 --> 00:52:26,140 of attractive Coulombic potentials. 949 00:52:26,140 --> 00:52:30,130 We still have the Hartree electrostatic term 950 00:52:30,130 --> 00:52:34,600 in which the electron lambda interacts 951 00:52:34,600 --> 00:52:38,230 with the charge density phi star mu times phi 952 00:52:38,230 --> 00:52:42,670 mu of every other electron mu. 953 00:52:42,670 --> 00:52:45,530 And I'll come to what goes into this sum in a moment. 954 00:52:45,530 --> 00:52:47,500 But basically, we have electron lambda 955 00:52:47,500 --> 00:52:50,560 interacting with the charge density of electron mu. 956 00:52:50,560 --> 00:52:53,770 And obviously, it's a Coulombic electrostatic repulsion. 957 00:52:53,770 --> 00:52:55,960 So there is a 1/r term. 958 00:52:55,960 --> 00:53:01,510 But now, there is a new term with a minus sign 959 00:53:01,510 --> 00:53:06,430 that comes out only from the anti-symmetry requirement. 960 00:53:06,430 --> 00:53:09,430 And that is what is called the exchange term 961 00:53:09,430 --> 00:53:13,880 and is the new player in the Hartree-Fock equation. 962 00:53:13,880 --> 00:53:16,690 And it's a little bit exotic because, if you want, 963 00:53:16,690 --> 00:53:19,960 now we don't have any more an operator that 964 00:53:19,960 --> 00:53:24,100 could be, say, a local operator, a charge density distribution 965 00:53:24,100 --> 00:53:25,720 acting on an orbital. 966 00:53:25,720 --> 00:53:28,880 But it's really become a known local operator 967 00:53:28,880 --> 00:53:33,060 because the orbital on which I'm acting 968 00:53:33,060 --> 00:53:37,150 has gone inside the orbital sign. 969 00:53:37,150 --> 00:53:42,220 So this new term here that we call the exchange term is 970 00:53:42,220 --> 00:53:47,290 a purely quantum mechanical term that comes out exclusively 971 00:53:47,290 --> 00:53:50,980 by the anti symmetry requirements on what 972 00:53:50,980 --> 00:53:53,860 happens to a fermionic wave function 973 00:53:53,860 --> 00:53:57,410 when we invert coefficients. 974 00:53:57,410 --> 00:54:01,340 And there is another fundamental distinction between the Hartree 975 00:54:01,340 --> 00:54:05,120 and the Hartree-Fock equations. 976 00:54:05,120 --> 00:54:09,590 This sum here-- you see this time over mu-- 977 00:54:09,590 --> 00:54:13,220 in the Hartree equation was running 978 00:54:13,220 --> 00:54:16,850 over all the electrons but the one 979 00:54:16,850 --> 00:54:19,280 that we were considering, OK? 980 00:54:19,280 --> 00:54:23,390 And now instead, this sum mu is running 981 00:54:23,390 --> 00:54:26,880 on all the possible electrons. 982 00:54:26,880 --> 00:54:29,660 So what you have in this term here 983 00:54:29,660 --> 00:54:33,830 is what is technically called self-interaction. 984 00:54:33,830 --> 00:54:35,870 Suppose that for a moment you were 985 00:54:35,870 --> 00:54:42,050 trying to solve the hydrogen atom with just one electron. 986 00:54:42,050 --> 00:54:45,230 Well, what happens in the Hartree equation 987 00:54:45,230 --> 00:54:47,000 for the hydrogen atom-- 988 00:54:47,000 --> 00:54:49,700 there is obviously no exchange term. 989 00:54:49,700 --> 00:54:52,970 And there would be also no Hartree term 990 00:54:52,970 --> 00:54:55,790 because, in the Hartree equation that you've seen before, 991 00:54:55,790 --> 00:54:58,580 what you have is a sum over all the other electrons. 992 00:54:58,580 --> 00:55:00,300 But there are no other electrons. 993 00:55:00,300 --> 00:55:01,670 So this term is not there. 994 00:55:01,670 --> 00:55:04,160 And what you trivially recover is 995 00:55:04,160 --> 00:55:08,450 the single-particle Schrodinger equation for the hydrogen atom. 996 00:55:08,450 --> 00:55:11,390 In the Hartree-Fock equations, now even 997 00:55:11,390 --> 00:55:14,480 for the hydrogen atom, what you have 998 00:55:14,480 --> 00:55:16,880 is that now you have this term. 999 00:55:16,880 --> 00:55:21,650 You have actually an unphysical self-interaction. 1000 00:55:21,650 --> 00:55:27,335 That is, you have electron 1 interacting with itself. 1001 00:55:27,335 --> 00:55:29,210 And that's really not interacting with itself 1002 00:55:29,210 --> 00:55:32,030 in a local way. 1003 00:55:32,030 --> 00:55:36,290 But it's interacting with its own charge distribution. 1004 00:55:36,290 --> 00:55:39,140 This is actually a sort of unphysical thing. 1005 00:55:39,140 --> 00:55:45,320 But what happens exclusively in the Hartree-Fock formulation 1006 00:55:45,320 --> 00:55:47,480 is that there is a second term. 1007 00:55:47,480 --> 00:55:49,700 There is the exchange term. 1008 00:55:49,700 --> 00:55:53,270 And the exchange term-- you see, there is a minus sign-- 1009 00:55:53,270 --> 00:55:57,950 cancels for the hydrogen atom the self-interaction term 1010 00:55:57,950 --> 00:55:59,240 exactly. 1011 00:55:59,240 --> 00:56:03,170 If you just think-- if you have only one electron, 1012 00:56:03,170 --> 00:56:05,780 you have that there is only one phi. 1013 00:56:05,780 --> 00:56:06,980 So the lambda goes away. 1014 00:56:06,980 --> 00:56:08,130 The mu goes away. 1015 00:56:08,130 --> 00:56:10,670 So here, the electron is interacting 1016 00:56:10,670 --> 00:56:12,780 with its charge distribution. 1017 00:56:12,780 --> 00:56:14,360 But then this term here-- 1018 00:56:14,360 --> 00:56:17,600 just remove all the sum and remove mu and lambda-- 1019 00:56:17,600 --> 00:56:20,480 is canceled out thanks to the minus sign 1020 00:56:20,480 --> 00:56:22,400 by the exchange term. 1021 00:56:22,400 --> 00:56:26,510 So Hartree-Fock formalism is actually 1022 00:56:26,510 --> 00:56:30,110 what we call self-interaction corrected, OK? 1023 00:56:30,110 --> 00:56:33,230 An electron, even if it's a mean field picture, 1024 00:56:33,230 --> 00:56:35,330 doesn't interact with itself. 1025 00:56:35,330 --> 00:56:39,170 And that's actually a very beautiful symmetry property 1026 00:56:39,170 --> 00:56:42,170 that other approaches like density functional theory 1027 00:56:42,170 --> 00:56:43,790 do not satisfy. 1028 00:56:43,790 --> 00:56:48,120 And truly, a lot of the problems that come out 1029 00:56:48,120 --> 00:56:51,560 in density functional theory that otherwise perform really 1030 00:56:51,560 --> 00:56:55,590 well have to do with this self-interaction problem. 1031 00:56:55,590 --> 00:56:58,970 And so those problems are very significant, 1032 00:56:58,970 --> 00:57:02,480 if you think for a moment, of problems like dissociation 1033 00:57:02,480 --> 00:57:05,090 of a molecule in atoms. 1034 00:57:05,090 --> 00:57:07,610 So when you really consider how-- 1035 00:57:07,610 --> 00:57:09,290 and we'll discuss this in detail-- 1036 00:57:09,290 --> 00:57:14,660 how the energy changes along a process in which the electron 1037 00:57:14,660 --> 00:57:17,870 needs to localize itself from a shared bonds 1038 00:57:17,870 --> 00:57:21,380 to a localized bond and that self interaction problem 1039 00:57:21,380 --> 00:57:24,980 kills density function theory and would actually not 1040 00:57:24,980 --> 00:57:28,130 be present to begin with in Hartree-Fock. 1041 00:57:28,130 --> 00:57:31,730 So this is actually very important. 1042 00:57:31,730 --> 00:57:32,900 There is another thing. 1043 00:57:32,900 --> 00:57:35,750 Actually, the Hartree-Fock equation 1044 00:57:35,750 --> 00:57:41,090 has a beauty in the fact that the operator acting 1045 00:57:41,090 --> 00:57:44,120 on the single-particle orbitals does not 1046 00:57:44,120 --> 00:57:50,160 depend on which orbital you are looking at because here we 1047 00:57:50,160 --> 00:57:52,660 have a sum over all the states. 1048 00:57:52,660 --> 00:57:56,790 So this does not depend on which electron you are looking at. 1049 00:57:56,790 --> 00:57:58,920 That's different from the Hartree equation. 1050 00:57:58,920 --> 00:58:00,330 In the Hartree equation, there is 1051 00:58:00,330 --> 00:58:03,150 a Hartree term where this sum would 1052 00:58:03,150 --> 00:58:05,880 exclude the electron itself. 1053 00:58:05,880 --> 00:58:07,980 So the Hartree equations are actually 1054 00:58:07,980 --> 00:58:13,740 more complex to solve because the operator changes depending 1055 00:58:13,740 --> 00:58:16,350 on which electron it's acting on, 1056 00:58:16,350 --> 00:58:21,420 while the Hartree-Fock equation has the same operator. 1057 00:58:21,420 --> 00:58:23,490 And so what we really need to find out, 1058 00:58:23,490 --> 00:58:26,550 if we are solving the iron atom, what 1059 00:58:26,550 --> 00:58:31,560 are the 26 lowest energy solutions 1060 00:58:31,560 --> 00:58:34,320 for all the electrons in here. 1061 00:58:34,320 --> 00:58:37,300 So that's a very good thing. 1062 00:58:37,300 --> 00:58:39,480 But there are terms-- 1063 00:58:39,480 --> 00:58:42,120 there are integrals that are actually very 1064 00:58:42,120 --> 00:58:44,820 expensive still to calculate. 1065 00:58:44,820 --> 00:58:50,220 And ultimately, it's terms like this that give us the scaling 1066 00:58:50,220 --> 00:58:51,660 cost of the equation. 1067 00:58:51,660 --> 00:58:55,110 Often, what we saw in such a problem, we want to know 1068 00:58:55,110 --> 00:58:56,535 what is it scaling cost. 1069 00:58:56,535 --> 00:58:59,070 That is, how much our calculation 1070 00:58:59,070 --> 00:59:01,740 becomes more expensive, say, if we double 1071 00:59:01,740 --> 00:59:04,050 the size of the system because that basically 1072 00:59:04,050 --> 00:59:06,660 tells us how large we can go. 1073 00:59:06,660 --> 00:59:10,020 And because of these integrals that 1074 00:59:10,020 --> 00:59:13,320 involve three orbitals and the fact that then to calculate 1075 00:59:13,320 --> 00:59:16,620 an energy of one more sum, really the scaling 1076 00:59:16,620 --> 00:59:20,550 cost of the Hartree-Fock equations, in principle, 1077 00:59:20,550 --> 00:59:22,690 goes as the fourth power. 1078 00:59:22,690 --> 00:59:25,470 So if I'm starting a system with two electrons, 1079 00:59:25,470 --> 00:59:28,530 and then I want to study a system with four electrons, 1080 00:59:28,530 --> 00:59:31,590 well, the cost has gone up 16 times. 1081 00:59:31,590 --> 00:59:37,630 And a fourth power, like any power, kills you very rapidly. 1082 00:59:37,630 --> 00:59:41,280 So that's why we can't study a molecule like benzene 1083 00:59:41,280 --> 00:59:42,390 with Hartree-Fock. 1084 00:59:42,390 --> 00:59:45,000 But we can't study really something 1085 00:59:45,000 --> 00:59:48,300 like DNA with Hartree-Fock or really none 1086 00:59:48,300 --> 00:59:50,700 of the standard electronic structure methods. 1087 00:59:50,700 --> 00:59:53,970 And a lot of effort that goes into developing 1088 00:59:53,970 --> 00:59:57,600 a linear scaling methods, that is methods 1089 00:59:57,600 --> 01:00:00,510 in which the computational cost of your calculation 1090 01:00:00,510 --> 01:00:03,600 doubles if you double the number of electrons. 1091 01:00:03,600 --> 01:00:07,620 And at the end, nature is linear scaling 1092 01:00:07,620 --> 01:00:09,900 because really you can imagine that the wave 1093 01:00:09,900 --> 01:00:12,750 function of the electrons here doesn't have anything 1094 01:00:12,750 --> 01:00:15,360 to do with the wave function of the electrons there. 1095 01:00:15,360 --> 01:00:19,290 So there is really, when you go far away, no 1096 01:00:19,290 --> 01:00:22,590 exchange of information between wave functions 1097 01:00:22,590 --> 01:00:24,490 in different parts of space. 1098 01:00:24,490 --> 01:00:28,200 And so there is ultimately a linear scaling nature. 1099 01:00:28,200 --> 01:00:32,920 But our algorithms, in general, are not yet there. 1100 01:00:32,920 --> 01:00:38,460 And we'll discuss a little bit in some of the later classes 1101 01:00:38,460 --> 01:00:39,660 how to solve this problem. 1102 01:00:42,470 --> 01:00:49,710 Before going on, I wanted to show one set of very simple 1103 01:00:49,710 --> 01:00:54,840 conclusions actually of having a wave function with the proper 1104 01:00:54,840 --> 01:00:57,330 symmetry-- that is, having a wave function written 1105 01:00:57,330 --> 01:00:59,490 as a Slater determinant-- 1106 01:00:59,490 --> 01:01:04,350 because that form gives us automatically 1107 01:01:04,350 --> 01:01:06,120 what is called the Pauli principle. 1108 01:01:06,120 --> 01:01:08,340 If you remember what the Pauli principle is, 1109 01:01:08,340 --> 01:01:12,510 it's that you can't have two fermions-- two electrons, 1110 01:01:12,510 --> 01:01:16,060 in particular-- in the same quantum state. 1111 01:01:16,060 --> 01:01:22,200 So you can't have two electrons having, say, the same orbital. 1112 01:01:22,200 --> 01:01:25,590 And that's obvious because, in a determinant, 1113 01:01:25,590 --> 01:01:28,590 two electrons having the same orbital 1114 01:01:28,590 --> 01:01:33,640 would mean that two columns in the determinant are identical. 1115 01:01:33,640 --> 01:01:36,930 And when two columns in a determinant are identical, 1116 01:01:36,930 --> 01:01:40,170 the determinant linearly dependent. 1117 01:01:40,170 --> 01:01:43,290 And so the solution is 0. 1118 01:01:43,290 --> 01:01:46,350 So a lot of good things came out Hartree-Fock. 1119 01:01:46,350 --> 01:01:49,680 In particular, one could start solving atoms. 1120 01:01:49,680 --> 01:01:53,140 And one would recover, say, the shell structure of atoms. 1121 01:01:53,140 --> 01:01:56,280 So if you would obtain the Hartree-Fock solution 1122 01:01:56,280 --> 01:01:58,710 for something like an argon atom, 1123 01:01:58,710 --> 01:02:02,650 and then say plot the overall charge density of the system, 1124 01:02:02,650 --> 01:02:08,710 well, it would start to look like this as we 1125 01:02:08,710 --> 01:02:11,860 move from the center outwards. 1126 01:02:11,860 --> 01:02:14,760 So it would clearly show the fundamentals 1127 01:02:14,760 --> 01:02:16,630 of the periodic table nature of things. 1128 01:02:16,630 --> 01:02:20,170 That is, it would show a 1s shell. 1129 01:02:20,170 --> 01:02:23,920 And then it would show a 2s and a 2p shell. 1130 01:02:23,920 --> 01:02:26,772 And this is something that some of the other approaches, 1131 01:02:26,772 --> 01:02:28,480 like the Thomas Fermi approach that we'll 1132 01:02:28,480 --> 01:02:31,210 see in a moment that were being developed at the time, 1133 01:02:31,210 --> 01:02:32,620 didn't have. 1134 01:02:32,620 --> 01:02:35,170 In general, Hartree-Fock is very good 1135 01:02:35,170 --> 01:02:39,130 to describe atomic properties. 1136 01:02:39,130 --> 01:02:45,940 And what is most important is a well-defined approximation 1137 01:02:45,940 --> 01:02:48,070 in the variational principle. 1138 01:02:48,070 --> 01:02:50,740 Remember, one of the fundamental powers 1139 01:02:50,740 --> 01:02:53,050 of the variational principle is that if we 1140 01:02:53,050 --> 01:02:57,280 make our wave function ansatz, our [INAUDIBLE] wave 1141 01:02:57,280 --> 01:03:01,580 function more and more flexible, we become better and better. 1142 01:03:01,580 --> 01:03:05,800 So Hartree-Fock is a certain hypothesis 1143 01:03:05,800 --> 01:03:07,780 and gives us certain energies. 1144 01:03:07,780 --> 01:03:11,560 But what we can do is make our wave functions 1145 01:03:11,560 --> 01:03:13,510 more and more flexible-- 1146 01:03:13,510 --> 01:03:17,740 write them not only just as a single determinant, 1147 01:03:17,740 --> 01:03:21,530 but a single determinant plus something else. 1148 01:03:21,530 --> 01:03:25,150 And the solution that we'll find will be computationally more 1149 01:03:25,150 --> 01:03:28,430 expensive to find, but is going to be better. 1150 01:03:28,430 --> 01:03:34,150 So Hartree-Fock, in principle, can be improved indefinitely. 1151 01:03:34,150 --> 01:03:37,420 That is something very powerful conceptually. 1152 01:03:37,420 --> 01:03:40,000 It's practically very complex because those 1153 01:03:40,000 --> 01:03:42,110 costs that we are scaling already 1154 01:03:42,110 --> 01:03:44,570 in a simple Hartree-Fock, like the fourth power, 1155 01:03:44,570 --> 01:03:46,390 keep going up. 1156 01:03:46,390 --> 01:03:49,210 What you will see in this differential theory 1157 01:03:49,210 --> 01:03:51,820 is that besides being a theory that 1158 01:03:51,820 --> 01:03:54,460 tends to give more accurate results 1159 01:03:54,460 --> 01:03:56,440 for a lot of physical properties, 1160 01:03:56,440 --> 01:03:58,750 it's something that also scales a bit better. 1161 01:03:58,750 --> 01:04:02,380 It scales as the third power of the size. 1162 01:04:02,380 --> 01:04:04,570 But it's a theory that can't really be 1163 01:04:04,570 --> 01:04:07,390 improved in any systematic way. 1164 01:04:07,390 --> 01:04:10,840 One can find ingenious ways to make it better. 1165 01:04:10,840 --> 01:04:15,160 But there isn't a brute force improvement strategy 1166 01:04:15,160 --> 01:04:19,310 like there is in Hartree-Fock. 1167 01:04:19,310 --> 01:04:23,060 The Hartree-Fock operator included the last term 1168 01:04:23,060 --> 01:04:25,760 that we have called the exchange term. 1169 01:04:25,760 --> 01:04:29,450 And so for every possible atom, for every possible molecule, 1170 01:04:29,450 --> 01:04:33,800 for every system, there is a well-defined Hartree-Fock 1171 01:04:33,800 --> 01:04:34,610 energy. 1172 01:04:34,610 --> 01:04:38,600 And this Hartree-Fock energy is going to be good, or very good, 1173 01:04:38,600 --> 01:04:39,830 or sort of so-so. 1174 01:04:39,830 --> 01:04:44,360 But it's always going to be higher than the true ground 1175 01:04:44,360 --> 01:04:46,490 state energy of our system. 1176 01:04:46,490 --> 01:04:48,830 And actually, what is technically 1177 01:04:48,830 --> 01:04:52,760 called the correlation energy is the difference 1178 01:04:52,760 --> 01:04:55,790 between the true energy of your system and the Hartree-Fock 1179 01:04:55,790 --> 01:04:56,880 energy. 1180 01:04:56,880 --> 01:04:59,210 So when people talk about correlation energy, 1181 01:04:59,210 --> 01:05:03,830 they refer to all the energy that is not captured 1182 01:05:03,830 --> 01:05:06,260 by a Hartree-Fock approach. 1183 01:05:06,260 --> 01:05:09,500 And in that sense, it's a well-defined quantity. 1184 01:05:09,500 --> 01:05:12,860 Although, it involves a generic term correlation 1185 01:05:12,860 --> 01:05:14,240 that can mean a lot of things. 1186 01:05:14,240 --> 01:05:16,175 And I'll show you a few examples. 1187 01:05:20,700 --> 01:05:23,880 There is one more thing that we can do. 1188 01:05:23,880 --> 01:05:27,600 We have never discussed up to now spin. 1189 01:05:27,600 --> 01:05:31,170 But in reality, an electron is described 1190 01:05:31,170 --> 01:05:35,080 by a wave function that doesn't have only space parts. 1191 01:05:35,080 --> 01:05:37,200 So in order to describe an electron, 1192 01:05:37,200 --> 01:05:40,815 we don't only describe what the distribution of its wave 1193 01:05:40,815 --> 01:05:43,530 function is in space, but we also 1194 01:05:43,530 --> 01:05:47,700 specify what is the spin of the electron. 1195 01:05:47,700 --> 01:05:51,030 And that has to do basically with spin 1196 01:05:51,030 --> 01:05:55,470 being an operator that can be simultaneously diagonalized 1197 01:05:55,470 --> 01:05:57,180 with a set of-- 1198 01:05:57,180 --> 01:05:58,830 well, it becomes complex. 1199 01:05:58,830 --> 01:06:00,570 But it's another quantity. 1200 01:06:00,570 --> 01:06:02,280 You can think of it as a color. 1201 01:06:02,280 --> 01:06:04,770 We need to specify if our electron is red or blue. 1202 01:06:04,770 --> 01:06:06,990 Or in particular, we need to specify 1203 01:06:06,990 --> 01:06:10,830 what is its spin, what is its projection with respect 1204 01:06:10,830 --> 01:06:12,310 to an axis. 1205 01:06:12,310 --> 01:06:14,635 And so in this, you need to think 1206 01:06:14,635 --> 01:06:17,040 of a wave function of an electron 1207 01:06:17,040 --> 01:06:20,580 not having only a spatial distribution. 1208 01:06:20,580 --> 01:06:25,000 But it has another property besides the spatial variables 1209 01:06:25,000 --> 01:06:26,700 that is called the spin. 1210 01:06:26,700 --> 01:06:30,930 And you can make a sort of approximation-- 1211 01:06:30,930 --> 01:06:35,650 that is what is called a restricted Hartree-Fock 1212 01:06:35,650 --> 01:06:36,510 scheme-- 1213 01:06:36,510 --> 01:06:41,550 in which you decide that an electron of spin 1214 01:06:41,550 --> 01:06:44,730 up and an electron of spin down will 1215 01:06:44,730 --> 01:06:48,360 have the same spatial part, so the same wave 1216 01:06:48,360 --> 01:06:49,980 function in space. 1217 01:06:49,980 --> 01:06:52,920 And their wave function differs only 1218 01:06:52,920 --> 01:06:55,620 because you describe an electron with spin up 1219 01:06:55,620 --> 01:06:57,480 and an electron with spin down. 1220 01:06:57,480 --> 01:07:00,630 Again, this corresponds to the classical periodic table 1221 01:07:00,630 --> 01:07:01,300 picture. 1222 01:07:01,300 --> 01:07:03,420 You are constructing the periodic table. 1223 01:07:03,420 --> 01:07:07,650 You go, say, from hydrogen-- one electron in the 1s level-- 1224 01:07:07,650 --> 01:07:12,210 to helium-- one electron in the 1s level with spin up 1225 01:07:12,210 --> 01:07:16,110 and another electron in the same level with the spin down. 1226 01:07:16,110 --> 01:07:19,440 Actually, if you think, the periodic table 1227 01:07:19,440 --> 01:07:22,320 itself is not a truth. 1228 01:07:22,320 --> 01:07:27,390 It's just a Hartree-Fock picture of electrons, OK? 1229 01:07:27,390 --> 01:07:31,590 In principle, you shouldn't be able to talk about 1230 01:07:31,590 --> 01:07:33,750 single-particle quantities-- 1231 01:07:33,750 --> 01:07:37,950 1s, 2s quantities-- because, in reality, if you have iron, 1232 01:07:37,950 --> 01:07:40,290 you have a many-body wave function 1233 01:07:40,290 --> 01:07:45,220 that is an overall function of all the electrons. 1234 01:07:45,220 --> 01:07:49,200 It's only when you enter into a Hartree-Fock picture 1235 01:07:49,200 --> 01:07:53,730 that you can have a well-defined concept as a single orbital 1236 01:07:53,730 --> 01:07:57,120 for an electron and what's the energy for that electron. 1237 01:07:57,120 --> 01:07:59,700 So if you want, what we think of this beautiful thing 1238 01:07:59,700 --> 01:08:03,370 as the periodic table is nothing else than the Hartree-Fock 1239 01:08:03,370 --> 01:08:06,000 solution of the atoms. 1240 01:08:06,000 --> 01:08:08,490 And again, we can make the approximation 1241 01:08:08,490 --> 01:08:13,740 in which we fill up every orbital, every spatial part 1242 01:08:13,740 --> 01:08:16,170 with two electrons with the same spin. 1243 01:08:16,170 --> 01:08:18,899 That tends to be a very good approximation 1244 01:08:18,899 --> 01:08:23,220 for a lot of problems, say a lot of bound systems. 1245 01:08:23,220 --> 01:08:27,630 And we'll see the case of the hydrogen molecule in a moment. 1246 01:08:27,630 --> 01:08:32,399 But you could actually make your wave function more flexible, 1247 01:08:32,399 --> 01:08:36,630 saying that say orbitals don't need to be paired. 1248 01:08:36,630 --> 01:08:39,270 That is, an electron with a spin up and an electron with a spin 1249 01:08:39,270 --> 01:08:41,500 down, even if they are very close in energy, 1250 01:08:41,500 --> 01:08:46,380 can have two wave functions in which the space part differs. 1251 01:08:46,380 --> 01:08:55,830 And this is really an ansatz that contains this in itself. 1252 01:08:55,830 --> 01:08:59,300 So an unrestricted Hartree-Fock solution 1253 01:08:59,300 --> 01:09:01,880 will always give you a lower energy 1254 01:09:01,880 --> 01:09:04,100 than a restricted solution. 1255 01:09:04,100 --> 01:09:06,740 And we'll see in a moment an example. 1256 01:09:06,740 --> 01:09:10,340 And this is the case of the dissociation of a hydrogen 1257 01:09:10,340 --> 01:09:11,609 molecule. 1258 01:09:11,609 --> 01:09:17,899 So when we go back and try to understand what is the bonding 1259 01:09:17,899 --> 01:09:20,029 between, say, two hydrogen atoms-- 1260 01:09:20,029 --> 01:09:22,250 and we had seen in one of the first lectures, 1261 01:09:22,250 --> 01:09:25,160 we discussed about potentials-- 1262 01:09:25,160 --> 01:09:29,600 that is, what is the energy of a system as a function 1263 01:09:29,600 --> 01:09:31,479 of the nuclear distances. 1264 01:09:31,479 --> 01:09:32,960 And this is what we are doing here. 1265 01:09:32,960 --> 01:09:35,569 We are trying to look at the energy of the system 1266 01:09:35,569 --> 01:09:38,720 as a function of the hydrogen-hydrogen distance. 1267 01:09:38,720 --> 01:09:41,240 And there will be an equilibrium distance that corresponds 1268 01:09:41,240 --> 01:09:42,979 to the minimum of the energy. 1269 01:09:42,979 --> 01:09:45,500 And this is what classical potential tried to do. 1270 01:09:45,500 --> 01:09:47,960 They tried to replicate what is the energy 1271 01:09:47,960 --> 01:09:50,420 of a system as a function of the nuclear coordinates. 1272 01:09:50,420 --> 01:09:52,850 And they tend to do very well, as Professor Ceder has 1273 01:09:52,850 --> 01:09:58,020 told you, closer to the region where they have been fitted. 1274 01:09:58,020 --> 01:10:00,650 If we have created a potential around here, 1275 01:10:00,650 --> 01:10:03,140 we tend to be able to reproduce things very well. 1276 01:10:03,140 --> 01:10:05,730 Obviously, it's very easy to even find out 1277 01:10:05,730 --> 01:10:08,420 the potential that reproduced all these curves. 1278 01:10:08,420 --> 01:10:11,750 But when you start to have more than two atoms interacting, 1279 01:10:11,750 --> 01:10:14,960 there are all these problems of bond-breaking and bond-forming 1280 01:10:14,960 --> 01:10:18,500 that can't really be given by classical potential 1281 01:10:18,500 --> 01:10:22,670 and can be given by quantum mechanical calculations. 1282 01:10:22,670 --> 01:10:25,730 And so in principle, we have an ideal solution, 1283 01:10:25,730 --> 01:10:27,620 that is if we were able. 1284 01:10:27,620 --> 01:10:30,110 And nowadays, with numerical accuracy, 1285 01:10:30,110 --> 01:10:33,290 we have basically been able to solve almost perfectly 1286 01:10:33,290 --> 01:10:33,930 this problem. 1287 01:10:33,930 --> 01:10:36,860 We are able to find out what is the total energy 1288 01:10:36,860 --> 01:10:41,060 of the system as a function of the nuclear coordinates. 1289 01:10:41,060 --> 01:10:43,280 And then for this specific problem, 1290 01:10:43,280 --> 01:10:46,880 the Coulombic potentials at different distances, 1291 01:10:46,880 --> 01:10:50,210 we can find a Hartree-Fock solution. 1292 01:10:50,210 --> 01:10:53,450 And with only two electrons, we can 1293 01:10:53,450 --> 01:10:55,730 find a restricted Hartree-Fock solution 1294 01:10:55,730 --> 01:10:58,820 in which we say, well, these two electrons are 1295 01:10:58,820 --> 01:11:03,530 going to have the same orbital part in the wave function. 1296 01:11:03,530 --> 01:11:08,000 They just differ in having a spin up and spin down term. 1297 01:11:08,000 --> 01:11:10,130 And that makes them orthogonal. 1298 01:11:10,130 --> 01:11:13,970 So certain things are going to happen in the exchange term. 1299 01:11:13,970 --> 01:11:19,010 But then we plotted this energy as a function of the distance. 1300 01:11:19,010 --> 01:11:21,590 And this is what we have. 1301 01:11:21,590 --> 01:11:24,850 And then we can release this condition. 1302 01:11:24,850 --> 01:11:27,940 We can say these two electrons don't 1303 01:11:27,940 --> 01:11:32,000 need to have the same orbital part for the wave functions. 1304 01:11:32,000 --> 01:11:33,970 They can have different parts. 1305 01:11:33,970 --> 01:11:36,310 And we can do that calculation. 1306 01:11:36,310 --> 01:11:39,730 And what we obtain is the unrestricted Hartree-Fock 1307 01:11:39,730 --> 01:11:41,620 solution. 1308 01:11:41,620 --> 01:11:45,820 And you see two fundamental things coming out from here. 1309 01:11:45,820 --> 01:11:50,250 First of all is that all these Hartree-Fock approaches 1310 01:11:50,250 --> 01:11:52,080 give you an energy that is obviously 1311 01:11:52,080 --> 01:11:54,540 larger than the exact energy. 1312 01:11:54,540 --> 01:11:56,280 It's the variational principle. 1313 01:11:56,280 --> 01:11:58,560 We can get lower and lower. 1314 01:11:58,560 --> 01:12:01,170 But we will never be able-- and that's very good-- 1315 01:12:01,170 --> 01:12:04,350 to go below the true energy. 1316 01:12:04,350 --> 01:12:07,380 So the more flexible we make our wave function 1317 01:12:07,380 --> 01:12:10,570 with both Hartree-Fock methods, the more 1318 01:12:10,570 --> 01:12:14,010 we'll be able to recover this last electron 1319 01:12:14,010 --> 01:12:16,350 volt of correlation energy. 1320 01:12:16,350 --> 01:12:19,530 So this is all where our effort is going. 1321 01:12:19,530 --> 01:12:21,780 But you see, Hartree-Fock is already 1322 01:12:21,780 --> 01:12:26,520 doing extremely well in giving us the equilibrium distance. 1323 01:12:26,520 --> 01:12:28,770 I mean, this is the Hartree-Fock equilibrium distance. 1324 01:12:28,770 --> 01:12:30,920 And this is the exact equilibrium distance. 1325 01:12:30,920 --> 01:12:33,550 So it's doing a good job. 1326 01:12:33,550 --> 01:12:37,290 What restricted Hartree-Fock is not doing well 1327 01:12:37,290 --> 01:12:40,770 is giving us the dissociation energy. 1328 01:12:40,770 --> 01:12:46,470 So restricted Hartree-Fock works very well around the minimum. 1329 01:12:46,470 --> 01:12:48,330 Well, you really should have this sort 1330 01:12:48,330 --> 01:12:50,400 of physical picture of your ground 1331 01:12:50,400 --> 01:12:54,960 state being given by a bonding combination of 1s orbitals. 1332 01:12:54,960 --> 01:12:57,870 Really, this is what the covalent bond for a hydrogen 1333 01:12:57,870 --> 01:13:02,820 molecule is, is the two 1s orbitals covalently 1334 01:13:02,820 --> 01:13:04,330 overlapping. 1335 01:13:04,330 --> 01:13:07,170 And so restricted Hartree-Fock does very well here. 1336 01:13:07,170 --> 01:13:11,430 And it's basically identical to unrestricted Hartree-Fock. 1337 01:13:11,430 --> 01:13:16,290 But formally, unrestricted will always be lower than restricted 1338 01:13:16,290 --> 01:13:20,910 because it contains the restricted solution because, 1339 01:13:20,910 --> 01:13:24,690 in order to have a restricted solution, 1340 01:13:24,690 --> 01:13:27,780 you just need to have the orbital part for the two 1341 01:13:27,780 --> 01:13:29,190 electrons to be identical. 1342 01:13:29,190 --> 01:13:31,570 But because it can also be different, 1343 01:13:31,570 --> 01:13:35,430 it will always be lower than the restricted Hartree-Fock. 1344 01:13:35,430 --> 01:13:38,850 And you see, when we break this system apart-- 1345 01:13:38,850 --> 01:13:45,720 when we want to go from a bound hydrogen molecule to two atoms, 1346 01:13:45,720 --> 01:13:49,380 the restricted Hartree-Fock is doing very poorly. 1347 01:13:49,380 --> 01:13:52,590 You'll see, it'll give us an energy 1348 01:13:52,590 --> 01:13:56,800 that is a very poor predictor of the dissociation energy. 1349 01:13:56,800 --> 01:13:59,880 The dissociation energy-- the true disassociation energy 1350 01:13:59,880 --> 01:14:02,610 of the system is the distance between the minimum 1351 01:14:02,610 --> 01:14:04,428 here and the 0 value. 1352 01:14:04,428 --> 01:14:05,970 That this is, it's the energy that we 1353 01:14:05,970 --> 01:14:08,640 need to spend to break apart the molecule. 1354 01:14:08,640 --> 01:14:12,520 Unrestricted Hartree-Fock will do very well. 1355 01:14:12,520 --> 01:14:15,570 I mean, obviously, it goes to 0 in this scale when 1356 01:14:15,570 --> 01:14:16,560 we are far apart. 1357 01:14:16,560 --> 01:14:19,110 And so we have this 1 electron volt error. 1358 01:14:19,110 --> 01:14:22,110 But the restricted Hartree-Fock is doing very poorly. 1359 01:14:22,110 --> 01:14:23,010 And why is that? 1360 01:14:23,010 --> 01:14:25,680 Well, basically because the restricted Hartree-Fock 1361 01:14:25,680 --> 01:14:30,600 is really doubly occupying the same spatial part 1362 01:14:30,600 --> 01:14:33,460 of the same bonding combination. 1363 01:14:33,460 --> 01:14:35,670 And that's good when the system is bound. 1364 01:14:35,670 --> 01:14:37,650 But when you break it apart, it's 1365 01:14:37,650 --> 01:14:41,670 very poor because what you really want is-- 1366 01:14:41,670 --> 01:14:47,190 in your solution, you want to mix in another determinant that 1367 01:14:47,190 --> 01:14:50,640 is given by the anti-bonding combination 1368 01:14:50,640 --> 01:14:54,340 because if you think of the bonding state 1369 01:14:54,340 --> 01:14:58,950 as always a pile-up of charge in between the atoms. 1370 01:14:58,950 --> 01:15:01,110 But when the molecule disassociates, 1371 01:15:01,110 --> 01:15:05,070 you really want to have a solution that has zero charge 1372 01:15:05,070 --> 01:15:06,840 density between the atoms. 1373 01:15:06,840 --> 01:15:10,740 And that looks much more like the anti-bonding combination 1374 01:15:10,740 --> 01:15:12,880 of 1s orbitals. 1375 01:15:12,880 --> 01:15:14,310 And so the restricted Hartree-Fock 1376 01:15:14,310 --> 01:15:15,880 doesn't have this freedom. 1377 01:15:15,880 --> 01:15:17,790 So it does it very poorly. 1378 01:15:17,790 --> 01:15:21,240 And unrestricted has the freedom of having 1379 01:15:21,240 --> 01:15:24,750 two orbitals that are different for the two electrons. 1380 01:15:24,750 --> 01:15:28,950 And so it just puts one orbital on one atom and another orbital 1381 01:15:28,950 --> 01:15:32,970 in another atom instead of having a single orbital doubly 1382 01:15:32,970 --> 01:15:34,440 occupied. 1383 01:15:34,440 --> 01:15:38,710 And so unrestricted Hartree-Fock is going to be much better, 1384 01:15:38,710 --> 01:15:44,100 especially for problems like the bond-breaking reactions 1385 01:15:44,100 --> 01:15:51,530 or for problems in which you have isolated spins, 1386 01:15:51,530 --> 01:15:53,450 so you have atoms-- 1387 01:15:53,450 --> 01:15:58,735 you have single electrons that are not paired. 1388 01:15:58,735 --> 01:16:00,360 There are actually a number of theorems 1389 01:16:00,360 --> 01:16:03,000 that can be derived from the Hartree-Fock equation. 1390 01:16:03,000 --> 01:16:04,410 I won't dwell on them. 1391 01:16:04,410 --> 01:16:08,220 They are generically called Koopmans' theorems. 1392 01:16:08,220 --> 01:16:11,100 And they have to do with calculations, say, 1393 01:16:11,100 --> 01:16:14,010 of quantities like the ionization energy 1394 01:16:14,010 --> 01:16:15,510 or the electron affinity. 1395 01:16:15,510 --> 01:16:16,890 What is the initiation energy? 1396 01:16:16,890 --> 01:16:20,250 It's the energy that you need to spend to remove 1397 01:16:20,250 --> 01:16:22,140 an electron from an atom. 1398 01:16:22,140 --> 01:16:24,960 Or the electron affinity is the energy 1399 01:16:24,960 --> 01:16:28,620 that you gain when an electron captures-- 1400 01:16:28,620 --> 01:16:31,930 sorry, when an atom captures an extra electron. 1401 01:16:31,930 --> 01:16:34,740 So how do you calculate them, say, in an electronic structure 1402 01:16:34,740 --> 01:16:35,580 calculation? 1403 01:16:35,580 --> 01:16:37,150 Say the ionization energy? 1404 01:16:37,150 --> 01:16:40,020 Well, it'll just be given by the difference, say 1405 01:16:40,020 --> 01:16:42,900 for the case of iron atoms, of the Hartree-Fock 1406 01:16:42,900 --> 01:16:45,960 solution with 26 electrons and the Hartree-Fock 1407 01:16:45,960 --> 01:16:48,210 solution with 25 electrons. 1408 01:16:48,210 --> 01:16:49,770 So you do these two calculations. 1409 01:16:49,770 --> 01:16:51,090 You take the difference. 1410 01:16:51,090 --> 01:16:53,340 And that will be our ionization energy. 1411 01:16:53,340 --> 01:16:54,990 And the affinity will be the difference 1412 01:16:54,990 --> 01:16:58,920 between the calculation with 27 or 26 electrons. 1413 01:16:58,920 --> 01:17:02,130 But you can actually do very well 1414 01:17:02,130 --> 01:17:05,670 without having to do two calculations, 1415 01:17:05,670 --> 01:17:11,220 but having just one calculation, if you make the hypothesis 1416 01:17:11,220 --> 01:17:14,670 that really your single-particle electrons do not 1417 01:17:14,670 --> 01:17:16,000 change in the process. 1418 01:17:16,000 --> 01:17:17,580 So if you make the hypothesis that 1419 01:17:17,580 --> 01:17:22,050 in going from 26 to 25 the shape of electron 1, electron 2, 1420 01:17:22,050 --> 01:17:24,090 electron 3 do not change-- 1421 01:17:24,090 --> 01:17:25,290 and that's an approximation. 1422 01:17:25,290 --> 01:17:27,070 They will change a little bit. 1423 01:17:27,070 --> 01:17:29,100 But if you make this approximation, 1424 01:17:29,100 --> 01:17:32,700 you can actually prove that the difference 1425 01:17:32,700 --> 01:17:36,030 between the system with 26 electrons 1426 01:17:36,030 --> 01:17:38,160 and the system with 25 electrons-- 1427 01:17:38,160 --> 01:17:40,890 the difference in energy is just given 1428 01:17:40,890 --> 01:17:44,460 by the eigenvalue of the 26th electron. 1429 01:17:44,460 --> 01:17:46,950 So basically, a single calculation 1430 01:17:46,950 --> 01:17:50,670 gives you already an estimate of ionization energies 1431 01:17:50,670 --> 01:17:52,063 and electron affinities. 1432 01:17:52,063 --> 01:17:53,730 Although, in principle, you could always 1433 01:17:53,730 --> 01:17:55,200 do two calculations. 1434 01:17:55,200 --> 01:17:58,830 But these, if you find them, are called the Koopmans' theorems. 1435 01:18:01,360 --> 01:18:03,520 What is missing in Hartree-Fock? 1436 01:18:03,520 --> 01:18:06,430 What is this correlation that we are trying to recover? 1437 01:18:06,430 --> 01:18:10,030 Well, often we think at it in two ways. 1438 01:18:10,030 --> 01:18:14,710 We can think that part of it is dynamical correlation. 1439 01:18:14,710 --> 01:18:18,040 It's what I described to you in the case of the helium atom. 1440 01:18:18,040 --> 01:18:21,370 That is, when we have two electrons interacting, 1441 01:18:21,370 --> 01:18:25,420 they like to keep each other as far away as possible 1442 01:18:25,420 --> 01:18:27,670 from each other instantaneously. 1443 01:18:27,670 --> 01:18:30,130 And because we have a mean field solution, 1444 01:18:30,130 --> 01:18:33,550 we are actually overestimating the Hartree energy. 1445 01:18:33,550 --> 01:18:36,670 We tend to put electrons too close to each other 1446 01:18:36,670 --> 01:18:38,830 because we have one electron interacting 1447 01:18:38,830 --> 01:18:41,210 with the average field of the other. 1448 01:18:41,210 --> 01:18:43,797 And so in that energy term, you have 1449 01:18:43,797 --> 01:18:45,380 that there are a lot of configurations 1450 01:18:45,380 --> 01:18:48,040 in which the electrons are too close to each other. 1451 01:18:48,040 --> 01:18:50,360 That raises the energy of your system. 1452 01:18:50,360 --> 01:18:52,690 So we call that dynamical correlation. 1453 01:18:52,690 --> 01:18:55,390 And these are heuristic terms. 1454 01:18:55,390 --> 01:18:58,840 And then there is another class of errors 1455 01:18:58,840 --> 01:19:02,980 that we are making that often are called static correlations. 1456 01:19:02,980 --> 01:19:05,560 And those have to do more with the fact 1457 01:19:05,560 --> 01:19:08,620 that a single determinant solution doesn't have 1458 01:19:08,620 --> 01:19:10,220 the flexibility that you need. 1459 01:19:10,220 --> 01:19:13,630 And this was the case of the breaking apart of the hydrogen 1460 01:19:13,630 --> 01:19:14,560 molecule. 1461 01:19:14,560 --> 01:19:19,660 You really want, when you you're breaking apart a molecule, 1462 01:19:19,660 --> 01:19:22,690 to have a two determinant kind of flexibility 1463 01:19:22,690 --> 01:19:26,650 with both bonding and anti-bonding combinations. 1464 01:19:26,650 --> 01:19:28,730 And all of this is missing. 1465 01:19:28,730 --> 01:19:33,310 And we can systematically build it up, 1466 01:19:33,310 --> 01:19:37,690 improving, say, the flexibility of the wave function. 1467 01:19:37,690 --> 01:19:42,520 And one of the conceptually simplest way, 1468 01:19:42,520 --> 01:19:45,340 but computationally more expensive ways, 1469 01:19:45,340 --> 01:19:48,430 is actually to look at the wave function 1470 01:19:48,430 --> 01:19:52,210 that now, instead of being given by a single determinant, 1471 01:19:52,210 --> 01:19:55,780 is given by a combination of determinants 1472 01:19:55,780 --> 01:19:59,830 with different coefficients in which, say, the determinants 1473 01:19:59,830 --> 01:20:05,230 have been constructed with a number of orbitals that 1474 01:20:05,230 --> 01:20:07,890 include also excited orbitals. 1475 01:20:07,890 --> 01:20:11,050 Our original Hartree-Fock determinant, say, 1476 01:20:11,050 --> 01:20:15,010 for iron was given by the 26th lowest solution. 1477 01:20:15,010 --> 01:20:16,060 And this is it. 1478 01:20:16,060 --> 01:20:17,920 But then you could add a second term 1479 01:20:17,920 --> 01:20:22,450 that contains 25 of those 26 lower solutions and then one 1480 01:20:22,450 --> 01:20:23,380 excited state. 1481 01:20:23,380 --> 01:20:25,160 Or you could do a number of things. 1482 01:20:25,160 --> 01:20:27,790 But basically, you could increase the variation 1483 01:20:27,790 --> 01:20:29,710 of flexibility of your problem. 1484 01:20:29,710 --> 01:20:32,020 And the more flexible you become, 1485 01:20:32,020 --> 01:20:34,370 the closer you get to the right solution. 1486 01:20:34,370 --> 01:20:36,850 But you pay an enormous price for this. 1487 01:20:36,850 --> 01:20:39,460 And this general approach would be called 1488 01:20:39,460 --> 01:20:41,110 configuration interaction. 1489 01:20:41,110 --> 01:20:45,040 That has actually this horrific scaling of n to the 7th. 1490 01:20:45,040 --> 01:20:47,260 And so you can really do it for 10 electrons. 1491 01:20:47,260 --> 01:20:50,410 But you can't do it for 11 electrons on your best 1492 01:20:50,410 --> 01:20:51,000 computer. 1493 01:20:51,000 --> 01:20:55,120 Probably now, we can get to 15 electrons or so. 1494 01:20:55,120 --> 01:20:57,460 And I think with this, I'll conclude. 1495 01:20:57,460 --> 01:21:00,980 So this was a panorama on Hartree-Fock methods. 1496 01:21:00,980 --> 01:21:02,860 One of the best books is the Jensen book 1497 01:21:02,860 --> 01:21:05,380 of computational chemistry in the literature. 1498 01:21:05,380 --> 01:21:07,120 And then with the next class, we'll 1499 01:21:07,120 --> 01:21:09,880 start looking at density functional theory and again 1500 01:21:09,880 --> 01:21:14,560 the Nobel Prize for chemistry in 1998. 1501 01:21:14,560 --> 01:21:19,720 Next Tuesday is-- being Monday Presidents' Day, 1502 01:21:19,720 --> 01:21:23,060 we'll have on Tuesday Monday's schedule of classes. 1503 01:21:23,060 --> 01:21:25,720 That means, in practice, that we have no class. 1504 01:21:25,720 --> 01:21:28,720 So you have time until Thursday of next week 1505 01:21:28,720 --> 01:21:29,950 to brood over this. 1506 01:21:29,950 --> 01:21:34,120 I've posted a few readings on the website. 1507 01:21:34,120 --> 01:21:37,300 If you are really wanting to more 1508 01:21:37,300 --> 01:21:40,030 about the functional theory, one of those readings 1509 01:21:40,030 --> 01:21:42,700 is the Kohanoff paper on fundamentals 1510 01:21:42,700 --> 01:21:44,230 of density financial theory. 1511 01:21:44,230 --> 01:21:46,690 And if you want to know more of Hartree-Fock, 1512 01:21:46,690 --> 01:21:50,090 we won't really see any more in the rest of the class. 1513 01:21:50,090 --> 01:21:52,360 You should go to one of the references. 1514 01:21:52,360 --> 01:21:55,380 And otherwise, see you next Thursday.