1 00:00:00,090 --> 00:00:02,490 The following content is provided under a Creative 2 00:00:02,490 --> 00:00:04,030 Commons license. 3 00:00:04,030 --> 00:00:06,330 Your support will help MIT OpenCourseware 4 00:00:06,330 --> 00:00:10,690 continue to offer high-quality, educational resources for free. 5 00:00:10,690 --> 00:00:13,320 To make a donation or view additional materials 6 00:00:13,320 --> 00:00:17,250 from hundreds of MIT courses, visit MIT OpenCourseware 7 00:00:17,250 --> 00:00:18,220 at ocw.mit.edu. 8 00:00:22,659 --> 00:00:23,825 TROY VAN VOORHIS: All right. 9 00:00:23,825 --> 00:00:25,100 Well, good morning, everyone. 10 00:00:25,100 --> 00:00:28,990 Hope you had some at least rest over the long weekend. 11 00:00:32,049 --> 00:00:36,290 Looking forward to getting back to talking about quantum. 12 00:00:36,290 --> 00:00:39,070 So last time, which was a whole week ago-- 13 00:00:39,070 --> 00:00:41,680 long time ago-- we started talking 14 00:00:41,680 --> 00:00:43,660 about electronic structure. 15 00:00:43,660 --> 00:00:47,012 We talked about the Born Oppenheimer approximation 16 00:00:47,012 --> 00:00:48,970 and how that gives us an electronic Schrodinger 17 00:00:48,970 --> 00:00:51,282 equation, how we look for-- 18 00:00:51,282 --> 00:00:53,740 what we're looking for are the eigenvalues and eigenvectors 19 00:00:53,740 --> 00:00:55,240 of that Schrodinger equation, how 20 00:00:55,240 --> 00:00:59,590 we can get lots of interesting information out of that. 21 00:00:59,590 --> 00:01:01,270 And then we broke it down so that we 22 00:01:01,270 --> 00:01:03,784 saw that there were going to be two main knobs that we were 23 00:01:03,784 --> 00:01:05,200 going to have to turn when we were 24 00:01:05,200 --> 00:01:06,520 making our approximations here. 25 00:01:06,520 --> 00:01:11,170 We were going out to look at our atomic orbital basis sets 26 00:01:11,170 --> 00:01:12,760 and how we compute the energy. 27 00:01:12,760 --> 00:01:15,070 And we spent the last half of class 28 00:01:15,070 --> 00:01:17,485 last time talking about choosing basis sets. 29 00:01:17,485 --> 00:01:20,080 We came up with some understanding 30 00:01:20,080 --> 00:01:23,500 of these strange names that are written at the bottom here. 31 00:01:23,500 --> 00:01:25,900 And then today, we're going to talk about some 32 00:01:25,900 --> 00:01:28,090 of these ways of computing the energy. 33 00:01:28,090 --> 00:01:30,160 And then I'm going to flip over, and we're 34 00:01:30,160 --> 00:01:33,160 going to talk about computational chemistry tools 35 00:01:33,160 --> 00:01:37,510 and how we sort of assemble that into doing a calculation. 36 00:01:37,510 --> 00:01:39,130 But again, the idea with basis sets 37 00:01:39,130 --> 00:01:43,180 was that we were sort of questing 38 00:01:43,180 --> 00:01:46,660 after this mythical complete basis set limit where the basis 39 00:01:46,660 --> 00:01:49,060 set was so big that it was infinitely 40 00:01:49,060 --> 00:01:51,280 big, that making it bigger didn't 41 00:01:51,280 --> 00:01:52,807 change your answer at all. 42 00:01:52,807 --> 00:01:54,140 And we never actually get there. 43 00:01:54,140 --> 00:01:58,091 But if we get close enough, we decide that's good. 44 00:01:58,091 --> 00:01:58,590 All right. 45 00:01:58,590 --> 00:02:01,980 So now we're going to talk today about computing the energy. 46 00:02:01,980 --> 00:02:04,080 And thankfully, most of the methods 47 00:02:04,080 --> 00:02:06,390 for computing the energy start from something 48 00:02:06,390 --> 00:02:09,870 that I believe you are familiar with already from 5.61. 49 00:02:09,870 --> 00:02:12,710 You guys talked about the Hartree-Fock approximation? 50 00:02:12,710 --> 00:02:13,220 Yeah? 51 00:02:13,220 --> 00:02:13,780 Yeah. 52 00:02:13,780 --> 00:02:14,430 OK. 53 00:02:14,430 --> 00:02:16,860 So-- or maybe we talked about-- 54 00:02:16,860 --> 00:02:21,280 did we talk about singled did we talk about determinants? 55 00:02:21,280 --> 00:02:21,780 Yes. 56 00:02:21,780 --> 00:02:22,279 All right. 57 00:02:22,279 --> 00:02:23,190 So there we go. 58 00:02:23,190 --> 00:02:25,920 So then we talked about a determinant. 59 00:02:25,920 --> 00:02:29,160 The idea of a determinant is that you 60 00:02:29,160 --> 00:02:31,274 have a set of orbitals, they're independent, 61 00:02:31,274 --> 00:02:32,940 and then you compute the average energy. 62 00:02:32,940 --> 00:02:35,429 And so, in one form or another, I 63 00:02:35,429 --> 00:02:37,470 would write the energy of a determinant something 64 00:02:37,470 --> 00:02:37,969 like this. 65 00:02:37,969 --> 00:02:41,520 I would say that there's a piece here 66 00:02:41,520 --> 00:02:45,567 which is the energy of each electron by itself. 67 00:02:45,567 --> 00:02:47,025 So these are one-electron energies. 68 00:02:50,100 --> 00:02:52,870 So for every electron in my system, if I have n electrons, 69 00:02:52,870 --> 00:02:56,020 I add up the energies of each of those electrons. 70 00:02:56,020 --> 00:02:59,590 But then, because my Hamiltonian has electron-electron repulsion 71 00:02:59,590 --> 00:03:03,220 terms, I end up with some additional contributions 72 00:03:03,220 --> 00:03:04,960 that can't be assigned to just one 73 00:03:04,960 --> 00:03:08,170 electron or another electron. 74 00:03:08,170 --> 00:03:10,540 And those terms are here in this sum. 75 00:03:10,540 --> 00:03:16,330 They're pairwise so for every mu and nu summing 76 00:03:16,330 --> 00:03:20,470 over independent pairs, I have a coulomb, term j, 77 00:03:20,470 --> 00:03:22,700 and an exchange, term k. 78 00:03:22,700 --> 00:03:25,990 And what these things show me is that there 79 00:03:25,990 --> 00:03:33,560 is an average repulsion here between the electrons. 80 00:03:33,560 --> 00:03:34,880 The electrons are independent. 81 00:03:34,880 --> 00:03:36,710 They can't avoid each other. 82 00:03:36,710 --> 00:03:37,930 But they still interact. 83 00:03:37,930 --> 00:03:39,880 There's still the electron-electron repulsion 84 00:03:39,880 --> 00:03:41,650 between electrons here. 85 00:03:41,650 --> 00:03:44,150 So that's an energy expression that, in one form or another, 86 00:03:44,150 --> 00:03:47,300 should be somewhat familiar, I think, from things 87 00:03:47,300 --> 00:03:49,340 you guys have done before. 88 00:03:49,340 --> 00:03:52,730 The idea of Hartree-Fock is relatively simple, 89 00:03:52,730 --> 00:03:54,170 which is we have an energy, then. 90 00:03:54,170 --> 00:03:56,150 This is true for any Slater determinant, 91 00:03:56,150 --> 00:03:57,222 any set of orbitals. 92 00:03:57,222 --> 00:03:58,430 So I have some orbitals here. 93 00:04:01,570 --> 00:04:10,100 And the idea of Hartree-Fock is to choose the orbitals 94 00:04:10,100 --> 00:04:11,090 to minimize the energy. 95 00:04:19,000 --> 00:04:21,370 This independent particle model energy, 96 00:04:21,370 --> 00:04:22,370 I want to minimize that. 97 00:04:22,370 --> 00:04:23,770 I'm going to choose the orbitals that minimize that. 98 00:04:23,770 --> 00:04:24,850 And the way I'm going to do that is I'm 99 00:04:24,850 --> 00:04:26,270 going to write my orbitals-- 100 00:04:26,270 --> 00:04:31,630 psi mu-- as a sum over some index, i, 101 00:04:31,630 --> 00:04:38,870 some coefficients c i mu, times my basis functions, phi i. 102 00:04:38,870 --> 00:04:47,160 And I'll just emphasize that what I have here are 103 00:04:47,160 --> 00:04:48,960 functions from my AO basis, the things 104 00:04:48,960 --> 00:04:50,430 that I talked about last time, like 105 00:04:50,430 --> 00:04:57,417 the actual 1s-like, 2s-like, 2p-like, 3p-like orbitals. 106 00:04:57,417 --> 00:04:59,250 And then on the other side, what I have here 107 00:04:59,250 --> 00:05:02,050 are my molecular-like orbitals. 108 00:05:02,050 --> 00:05:05,040 The molecular orbitals are linear combinations 109 00:05:05,040 --> 00:05:09,780 of my atomic orbital basis functions. 110 00:05:09,780 --> 00:05:12,949 So that's the general idea of Hartree-Fock there, 111 00:05:12,949 --> 00:05:14,990 is that we're just going to choose these orbitals 112 00:05:14,990 --> 00:05:18,230 to minimize the energy. 113 00:05:18,230 --> 00:05:20,810 And because the orbitals are defined by these coefficients, 114 00:05:20,810 --> 00:05:23,360 I can think of this energy as a function 115 00:05:23,360 --> 00:05:25,430 of the coefficients, c. 116 00:05:25,430 --> 00:05:27,290 So I can say, all right, that means 117 00:05:27,290 --> 00:05:29,600 that I have an independent particle model 118 00:05:29,600 --> 00:05:36,297 energy which depends on these coefficients, c. 119 00:05:36,297 --> 00:05:37,880 And if I want to minimize that energy, 120 00:05:37,880 --> 00:05:39,800 I then need to look for the places 121 00:05:39,800 --> 00:05:44,310 where the derivatives are equal to zero. 122 00:05:44,310 --> 00:05:47,880 So I take the derivative of that energy, set it equal to 0. 123 00:05:47,880 --> 00:05:52,920 Now as you might suspect, if I look at this expression, 124 00:05:52,920 --> 00:05:57,510 I have an orbital here, here, here, here, here, here, here, 125 00:05:57,510 --> 00:05:59,095 here, here, here. 126 00:05:59,095 --> 00:06:01,460 I have a lot of orbitals appearing in there. 127 00:06:01,460 --> 00:06:02,760 And then I'm going to have to use the chain rule, 128 00:06:02,760 --> 00:06:04,218 because every one of those orbitals 129 00:06:04,218 --> 00:06:06,010 depends on every one of the coefficients. 130 00:06:06,010 --> 00:06:07,290 So I'm going to have a chain rule expression. 131 00:06:07,290 --> 00:06:08,440 You might expect that there's going 132 00:06:08,440 --> 00:06:10,980 to be a lot of algebra involved in taking this derivative, 133 00:06:10,980 --> 00:06:12,150 and there is. 134 00:06:12,150 --> 00:06:15,540 This is where I get to use my favorite abbreviation, which 135 00:06:15,540 --> 00:06:17,320 is ASA. 136 00:06:17,320 --> 00:06:20,055 ASA stands for After Some Algebra. 137 00:06:20,055 --> 00:06:22,270 It means I'm not going to work through the algebra. 138 00:06:22,270 --> 00:06:23,145 There's some algebra. 139 00:06:23,145 --> 00:06:24,810 There's several pages of lecture notes 140 00:06:24,810 --> 00:06:26,490 where algebra is gone through. 141 00:06:26,490 --> 00:06:29,377 After you do the algebra, you end up 142 00:06:29,377 --> 00:06:31,710 with the left-hand equation reducing to an equation that 143 00:06:31,710 --> 00:06:32,770 looks like this. 144 00:06:32,770 --> 00:06:38,980 So there is a matrix, which I will call h, 145 00:06:38,980 --> 00:06:42,440 and then when I dot that matrix into a vector, c 146 00:06:42,440 --> 00:06:47,440 mu, the coefficient vector, I get some energy, e, 147 00:06:47,440 --> 00:06:53,600 mu, times the coefficient vector again. 148 00:06:53,600 --> 00:06:56,180 So that takes a lot of algebra to get to that, 149 00:06:56,180 --> 00:06:58,222 but you can get an equation that looks like this. 150 00:06:58,222 --> 00:07:00,513 And that's encouraging, because this is familiar to me. 151 00:07:00,513 --> 00:07:01,820 This is an eigenvalue equation. 152 00:07:09,140 --> 00:07:12,450 And that's nice because I remember eigenvalue equations 153 00:07:12,450 --> 00:07:12,950 from-- 154 00:07:12,950 --> 00:07:15,270 I mean, the Schrodinger equation is an eigenvalue equation. 155 00:07:15,270 --> 00:07:17,186 So it's nice that when I do this minimization, 156 00:07:17,186 --> 00:07:18,770 I get back an eigenvalue equation. 157 00:07:18,770 --> 00:07:20,180 That's nice. 158 00:07:20,180 --> 00:07:25,100 The one complication is that the Hamiltonian here 159 00:07:25,100 --> 00:07:27,570 depends on the coefficient, c. 160 00:07:27,570 --> 00:07:29,540 So it's some effective Hamiltonian. 161 00:07:29,540 --> 00:07:31,770 I'm intentionally collecting terms 162 00:07:31,770 --> 00:07:33,747 so that it looks like a Schrodinger equation, 163 00:07:33,747 --> 00:07:36,080 but it's a little bit funny because the Hamiltonian here 164 00:07:36,080 --> 00:07:39,180 depends on the coefficients that solve the equation. 165 00:07:39,180 --> 00:07:42,410 So this is sort of an eigenvalue equation. 166 00:07:45,990 --> 00:07:54,250 And so what we're seeing here is that H depends 167 00:07:54,250 --> 00:07:55,720 on the wave function somehow. 168 00:07:55,720 --> 00:07:58,280 It seems like, depending on what the wave function is, 169 00:07:58,280 --> 00:08:00,370 I have different Hamiltonians. 170 00:08:00,370 --> 00:08:02,470 And to understand why this is the case, 171 00:08:02,470 --> 00:08:08,630 I want us to think about the potential, v, 172 00:08:08,630 --> 00:08:13,747 that an electron feels in an atom or in a molecule. 173 00:08:13,747 --> 00:08:15,830 And there's generally going to be two pieces here. 174 00:08:15,830 --> 00:08:17,220 And they're actually-- 175 00:08:17,220 --> 00:08:19,250 I can color-code them with the terms 176 00:08:19,250 --> 00:08:21,410 that exist in the independent particle expression. 177 00:08:21,410 --> 00:08:23,720 So there is the one-electron terms, 178 00:08:23,720 --> 00:08:26,235 the terms in the energy that just come from one electron. 179 00:08:26,235 --> 00:08:28,610 And those are going to be things like, well, I might have 180 00:08:28,610 --> 00:08:31,490 a nucleus of charge plus z. 181 00:08:31,490 --> 00:08:37,020 And I might have my electron out here, the one 182 00:08:37,020 --> 00:08:38,580 that I'm interested in. 183 00:08:38,580 --> 00:08:42,162 And it's going to have some attraction to that nucleus. 184 00:08:42,162 --> 00:08:43,620 So that just involves one electron. 185 00:08:43,620 --> 00:08:46,330 It involves a nucleus, but just one electron. 186 00:08:46,330 --> 00:08:49,780 So there's some nuclear potential 187 00:08:49,780 --> 00:08:52,480 that each electron is going to feel. 188 00:08:52,480 --> 00:08:55,270 But then there's the term that correspond 189 00:08:55,270 --> 00:08:56,840 to these average repulsion terms. 190 00:08:56,840 --> 00:09:00,640 So there are other electrons in my system. 191 00:09:00,640 --> 00:09:04,780 And each of those electrons interacts with the electron 192 00:09:04,780 --> 00:09:07,490 that I'm looking at. 193 00:09:07,490 --> 00:09:10,900 So there's going to be some electron-electron repulsion 194 00:09:10,900 --> 00:09:12,680 term here. 195 00:09:12,680 --> 00:09:14,860 And that electron-electron repulsion 196 00:09:14,860 --> 00:09:17,077 depends on where the other electrons are. 197 00:09:17,077 --> 00:09:19,160 And those other electrons have some wave function. 198 00:09:19,160 --> 00:09:22,890 They're kind of spread out here, all over the place. 199 00:09:22,890 --> 00:09:25,900 And depending on how spread out those other electrons are, 200 00:09:25,900 --> 00:09:28,460 I might have more or less electron repulsion. 201 00:09:28,460 --> 00:09:30,820 So if they're very, very concentrated in the region 202 00:09:30,820 --> 00:09:32,617 that I'm looking at, the electron repulsion 203 00:09:32,617 --> 00:09:33,700 is going to be very large. 204 00:09:33,700 --> 00:09:35,574 If they're very spread out, and very diffuse, 205 00:09:35,574 --> 00:09:38,020 they're not going to have very much electron repulsion. 206 00:09:38,020 --> 00:09:43,470 But overall, the electron-electron repulsion 207 00:09:43,470 --> 00:09:46,710 depends on the wave functions that I've chosen. 208 00:09:46,710 --> 00:09:56,354 So the potential here is called a mean field, or average field. 209 00:09:56,354 --> 00:09:58,270 And it's because it's the potential that comes 210 00:09:58,270 --> 00:09:59,510 from the average repulsion. 211 00:09:59,510 --> 00:10:02,560 It's the effective potential due to the averaging, the smearing 212 00:10:02,560 --> 00:10:05,700 out, over all the other electrons. 213 00:10:05,700 --> 00:10:09,290 So what we do have, then, is that the Hamiltonian 214 00:10:09,290 --> 00:10:12,530 does into indeed depend on the wave functions, 215 00:10:12,530 --> 00:10:14,330 because the wave functions determine 216 00:10:14,330 --> 00:10:15,890 that average potential. 217 00:10:15,890 --> 00:10:17,930 But then if I know the Hamiltonian, 218 00:10:17,930 --> 00:10:19,520 I can solve for the wave functions, 219 00:10:19,520 --> 00:10:21,920 because the wave functions are the eigenfunctions 220 00:10:21,920 --> 00:10:23,250 of the Hamiltonian. 221 00:10:23,250 --> 00:10:28,231 And so I have this sort of chicken and egg situation, 222 00:10:28,231 --> 00:10:30,230 that the Hamiltonian defines the wave functions, 223 00:10:30,230 --> 00:10:32,390 and the wave functions define the Hamiltonian. 224 00:10:32,390 --> 00:10:34,970 And the solution to this is to do 225 00:10:34,970 --> 00:10:37,730 what are called self-consistent field iterations. 226 00:10:42,160 --> 00:10:43,910 And you can more or less think about these 227 00:10:43,910 --> 00:10:47,420 as the simple process of, guess some wave functions, 228 00:10:47,420 --> 00:10:49,790 build the Hamiltonian, get some new wave functions, 229 00:10:49,790 --> 00:10:52,016 build a new Hamiltonian, get some new wave functions, 230 00:10:52,016 --> 00:10:53,390 build a new Hamiltonian, and keep 231 00:10:53,390 --> 00:10:58,040 going until everything becomes consistent. 232 00:10:58,040 --> 00:11:00,290 And the idea, then, is that the wave 233 00:11:00,290 --> 00:11:03,600 function we're looking for in Hartree-Fock 234 00:11:03,600 --> 00:11:15,620 are the eigenfunctions of H of psi Hf, which we just defined 235 00:11:15,620 --> 00:11:19,060 to be equal to H Hartree-Fock. 236 00:11:19,060 --> 00:11:22,410 So there's some Hartree-Fock Hamiltonian. 237 00:11:22,410 --> 00:11:23,940 And the Hartree-Fock wave functions 238 00:11:23,940 --> 00:11:26,485 are the eigenfunctions of that Hamiltonian. 239 00:11:26,485 --> 00:11:28,789 Does that makes sense? 240 00:11:28,789 --> 00:11:29,830 Any questions about that? 241 00:11:37,500 --> 00:11:38,050 All right. 242 00:11:38,050 --> 00:11:40,930 So Hartree-Fock is the starting point. 243 00:11:40,930 --> 00:11:43,400 So we have that as an approximation. 244 00:11:43,400 --> 00:11:44,890 It's a useful starting point. 245 00:11:44,890 --> 00:11:47,480 In many cases, Hartree-Fock is not good enough. 246 00:11:47,480 --> 00:11:50,410 And so the real thing that we're trying to figure out 247 00:11:50,410 --> 00:11:51,850 is, how do we make something that 248 00:11:51,850 --> 00:11:54,640 uses what we've learned from Hartree-Fock about orbitals, 249 00:11:54,640 --> 00:11:56,470 and energies, and average repulsion-- how 250 00:11:56,470 --> 00:12:00,110 do we make something that's like Hartree-Fock, but better? 251 00:12:00,110 --> 00:12:02,369 So one way to do this is something 252 00:12:02,369 --> 00:12:04,660 that you've spent a lot of time on this semester, which 253 00:12:04,660 --> 00:12:06,364 is perturbation theory. 254 00:12:06,364 --> 00:12:08,780 You have something that's pretty good as a starting point, 255 00:12:08,780 --> 00:12:11,050 and you want to make it better, perturbation theory 256 00:12:11,050 --> 00:12:14,260 is the natural thing to turn to do that. 257 00:12:14,260 --> 00:12:16,150 And so the way we apply perturbation theory 258 00:12:16,150 --> 00:12:19,350 in Hartree-Fock is we say, well, we've 259 00:12:19,350 --> 00:12:22,020 got this Hartree-Fock Hamiltonian, which 260 00:12:22,020 --> 00:12:29,485 we can solve, and then we've got the actual Hamiltonian, H, 261 00:12:29,485 --> 00:12:30,360 which we can't solve. 262 00:12:34,740 --> 00:12:37,490 And so what we can do is then we say, aha, well 263 00:12:37,490 --> 00:12:40,580 that means that if I just take the difference 264 00:12:40,580 --> 00:12:42,200 between the Hamiltonian I can solve 265 00:12:42,200 --> 00:12:43,740 and the Hamiltonian I can't solve, 266 00:12:43,740 --> 00:12:45,980 and I treat that as a perturbation, 267 00:12:45,980 --> 00:12:49,340 then I can do a well-defined perturbation theory where 268 00:12:49,340 --> 00:12:52,160 I know the solutions to H0, and then I 269 00:12:52,160 --> 00:12:55,400 treat that additional term as a perturbation. 270 00:12:55,400 --> 00:12:57,392 And so then if I do that, I end up 271 00:12:57,392 --> 00:12:58,850 with a perturbation expansion where 272 00:12:58,850 --> 00:13:01,940 I can go to first, second, third, fourth, fifth order, 273 00:13:01,940 --> 00:13:06,630 and that additional term that's the difference between the two. 274 00:13:06,630 --> 00:13:10,010 And if I look at the first two terms added together, 275 00:13:10,010 --> 00:13:11,690 they give me the Hartree-Fock energy. 276 00:13:11,690 --> 00:13:16,460 So if I'm the one making up a perturbation expansion, 277 00:13:16,460 --> 00:13:18,470 I will always choose it so that the first order 278 00:13:18,470 --> 00:13:20,000 correction is 0. 279 00:13:20,000 --> 00:13:22,730 So that if otherwise I have chosen, 280 00:13:22,730 --> 00:13:25,374 I will just re-engineer my perturbation so that's true. 281 00:13:25,374 --> 00:13:27,790 So up through first order, I just get what I started with. 282 00:13:27,790 --> 00:13:29,930 I get Hartree-Fock. 283 00:13:29,930 --> 00:13:34,030 But then I have these additional terms. 284 00:13:34,030 --> 00:13:35,470 And so I can go to second order. 285 00:13:35,470 --> 00:13:37,950 And I will get a method that I call-- 286 00:13:37,950 --> 00:13:40,330 it's called MP2. 287 00:13:40,330 --> 00:13:44,490 So Hartree-Fock is named after Hartree and Fock. 288 00:13:44,490 --> 00:13:46,730 MP stands for Moller and Plesset, 289 00:13:46,730 --> 00:13:50,580 the names of the two people who wrote the paper in 1938, 290 00:13:50,580 --> 00:13:53,190 about this particular thing. 291 00:13:53,190 --> 00:13:55,030 And the 2 is second order. 292 00:13:55,030 --> 00:14:01,140 And then I could go on, and I could do MP3, MP4, dot dot dot 293 00:14:01,140 --> 00:14:02,550 dot dot. 294 00:14:02,550 --> 00:14:06,570 So in general, this is MP, and some number, n, 295 00:14:06,570 --> 00:14:09,690 telling me the order I go to in perturbation theory. 296 00:14:09,690 --> 00:14:11,810 So you could do this. 297 00:14:11,810 --> 00:14:15,660 In principle, this gets better and better answers. 298 00:14:15,660 --> 00:14:18,700 However, in practice, about 20 years ago, 299 00:14:18,700 --> 00:14:21,510 people discovered something that was rather disturbing. 300 00:14:21,510 --> 00:14:25,030 So if I take a look at the energy minus the [INAUDIBLE].. 301 00:14:25,030 --> 00:14:28,919 So let's say I have a case where I know the exact energy. 302 00:14:28,919 --> 00:14:31,210 And I look at the energy of these different approximate 303 00:14:31,210 --> 00:14:35,260 methods for, say, one particular simple thing, like a neon atom. 304 00:14:35,260 --> 00:14:37,180 So I take a neon atom, and I look 305 00:14:37,180 --> 00:14:41,140 at the energies of these MPn approximations, 306 00:14:41,140 --> 00:14:47,091 and I look at them as a function of n, so 1, 2, 3, 4, 5, 6, 7, 307 00:14:47,091 --> 00:14:47,590 8. 308 00:14:52,880 --> 00:14:54,675 And I'll say that for neon-- 309 00:14:54,675 --> 00:14:56,800 neon is something for which Hartree-Fock is already 310 00:14:56,800 --> 00:14:57,760 pretty good. 311 00:14:57,760 --> 00:14:59,380 So the difference between Hartree-Fock 312 00:14:59,380 --> 00:15:01,750 and the exact answer is not too bad. 313 00:15:01,750 --> 00:15:04,090 So Hartree-Fock might be, let's say, 314 00:15:04,090 --> 00:15:06,490 there on this particular scale. 315 00:15:06,490 --> 00:15:10,023 And then if you do MP2, MP2 is better. 316 00:15:10,023 --> 00:15:13,480 So my MP2 is maybe there. 317 00:15:13,480 --> 00:15:15,280 MP3 is also a little bit better than that, 318 00:15:15,280 --> 00:15:17,510 but it overshoots a little bit. 319 00:15:17,510 --> 00:15:20,050 MP4 corrects back in the opposite direction. 320 00:15:20,050 --> 00:15:27,450 And then we've got 5, 6, 7, 8. 321 00:15:27,450 --> 00:15:29,440 And you can-- if I connect the dots, 322 00:15:29,440 --> 00:15:31,680 you can see what's starting to happen here. 323 00:15:36,850 --> 00:15:38,059 So that's a series. 324 00:15:38,059 --> 00:15:39,600 So mathematically, I've got something 325 00:15:39,600 --> 00:15:41,760 that there is a value for everything in n. 326 00:15:41,760 --> 00:15:42,760 That's a series. 327 00:15:42,760 --> 00:15:44,340 And the name for a series like this 328 00:15:44,340 --> 00:15:46,300 is one that is not converging. 329 00:15:46,300 --> 00:15:48,750 So there's an answer which is the infinite order answer, 330 00:15:48,750 --> 00:15:51,112 but these partial sums are not converging 331 00:15:51,112 --> 00:15:52,070 to the infinite answer. 332 00:15:56,890 --> 00:15:59,490 And very often this turns out to be the case with perturbation 333 00:15:59,490 --> 00:16:00,120 theory. 334 00:16:00,120 --> 00:16:01,536 There can be perturbation theories 335 00:16:01,536 --> 00:16:04,170 that are very useful at low orders for describing 336 00:16:04,170 --> 00:16:06,090 qualitatively how things shift. 337 00:16:06,090 --> 00:16:08,490 But if you go out to infinite order, 338 00:16:08,490 --> 00:16:12,000 they are series that, in principle, you can prove, 339 00:16:12,000 --> 00:16:14,640 give the correct, exact answer, because we've 340 00:16:14,640 --> 00:16:16,200 derived it as such. 341 00:16:16,200 --> 00:16:19,080 But just because a series in principle sums to something 342 00:16:19,080 --> 00:16:21,150 doesn't mean that any finite number of terms 343 00:16:21,150 --> 00:16:25,440 will give a convergent answer, or give smaller errors. 344 00:16:25,440 --> 00:16:28,470 And people find that for perturbation theory 345 00:16:28,470 --> 00:16:31,710 for electronic structure theory, this is-- 346 00:16:31,710 --> 00:16:33,240 not all of the time, but very often 347 00:16:33,240 --> 00:16:36,720 the case, that the perturbation expansion doesn't converge. 348 00:16:36,720 --> 00:16:39,827 And so this is one of those places where 349 00:16:39,827 --> 00:16:41,910 you make a nice empirical compromise, and you say, 350 00:16:41,910 --> 00:16:45,330 well, gee, every term-- 351 00:16:45,330 --> 00:16:47,820 it doesn't necessarily get better after MP2. 352 00:16:47,820 --> 00:16:50,070 MP2 usually improves on Hartree-Fock, 353 00:16:50,070 --> 00:16:52,320 but higher order might be better or worse. 354 00:16:52,320 --> 00:16:54,790 And certainly, higher order is more expensive. 355 00:16:54,790 --> 00:16:56,940 So MP2 is a good place to get off the elevator, 356 00:16:56,940 --> 00:16:59,190 because it's just going to get more expensive, and not 357 00:16:59,190 --> 00:17:00,777 necessarily more accurate. 358 00:17:00,777 --> 00:17:02,110 So this is a good place to stop. 359 00:17:09,109 --> 00:17:10,790 And so you'll see a lot of calculations 360 00:17:10,790 --> 00:17:13,609 out there that are variations on MP2, 361 00:17:13,609 --> 00:17:15,020 because it's the least expensive, 362 00:17:15,020 --> 00:17:17,780 and it's somewhat more accurate. 363 00:17:17,780 --> 00:17:20,480 OK so now we're going to take a pause, 364 00:17:20,480 --> 00:17:22,460 and we're going to talk-- 365 00:17:22,460 --> 00:17:25,589 and this is where I need audience participation. 366 00:17:25,589 --> 00:17:27,980 So we're going to discuss two different things 367 00:17:27,980 --> 00:17:29,146 that you might want to know. 368 00:17:29,146 --> 00:17:31,590 We've now got two methods, Hartree-Fock and MP2. 369 00:17:31,590 --> 00:17:33,798 And there are two things that you might want to know. 370 00:17:33,798 --> 00:17:36,260 One is, how accurate will one of these calculations be? 371 00:17:36,260 --> 00:17:38,135 If I do it, will it give me the right answer? 372 00:17:38,135 --> 00:17:40,070 And the other one is, how long will it take? 373 00:17:40,070 --> 00:17:41,736 In other words, will it finish by Friday 374 00:17:41,736 --> 00:17:43,000 when the P set is due? 375 00:17:43,000 --> 00:17:45,890 Will it finish by the end of the semester? 376 00:17:45,890 --> 00:17:49,010 Will it finish by the end of my time at MIT? 377 00:17:49,010 --> 00:17:51,000 What time scale are we looking at? 378 00:17:51,000 --> 00:17:53,840 So we'll start off talking about properties. 379 00:17:53,840 --> 00:17:56,090 So people have done many, many, many calculations 380 00:17:56,090 --> 00:17:58,750 using many different electronic structure methods, 381 00:17:58,750 --> 00:18:02,150 so that now there's sort of an empirical rule of thumb 382 00:18:02,150 --> 00:18:04,889 for just about any property you want to know about. 383 00:18:04,889 --> 00:18:07,430 There's a rule of thumb for how accurate one of these methods 384 00:18:07,430 --> 00:18:07,929 should be. 385 00:18:07,929 --> 00:18:10,910 So what's a property that you might want to know? 386 00:18:10,910 --> 00:18:14,089 This is where the audience participation comes in. 387 00:18:14,089 --> 00:18:16,130 I will only give you information about properties 388 00:18:16,130 --> 00:18:19,570 that you say are interesting. 389 00:18:19,570 --> 00:18:21,267 AUDIENCE: Time and space complexity. 390 00:18:21,267 --> 00:18:23,100 TROY VAN VOORHIS: Time and space complexity. 391 00:18:23,100 --> 00:18:23,825 All right. 392 00:18:23,825 --> 00:18:24,950 Could you be more specific. 393 00:18:24,950 --> 00:18:27,617 What type of time and space complexity? 394 00:18:27,617 --> 00:18:28,450 AUDIENCE: How well-- 395 00:18:28,450 --> 00:18:31,779 I mean, for a size of molecule, how long it takes-- 396 00:18:31,779 --> 00:18:32,820 TROY VAN VOORHIS: Ah, OK. 397 00:18:32,820 --> 00:18:34,580 AUDIENCE: --and also how much space it takes. 398 00:18:34,580 --> 00:18:36,050 TROY VAN VOORHIS: Yes, so that will be the second thing 399 00:18:36,050 --> 00:18:37,165 that we will talk about. 400 00:18:37,165 --> 00:18:38,540 Those are the two variables we'll 401 00:18:38,540 --> 00:18:41,087 talk about, in terms of how long and how fast. 402 00:18:41,087 --> 00:18:42,920 I'm interested here, where I'm saying, like, 403 00:18:42,920 --> 00:18:44,330 molecular properties-- things you 404 00:18:44,330 --> 00:18:47,105 might want to know about a molecule, or a reaction, 405 00:18:47,105 --> 00:18:49,150 or something like that. 406 00:18:49,150 --> 00:18:52,220 AUDIENCE: Like the energy of a molecule in transition state. 407 00:18:52,220 --> 00:18:58,890 TROY VAN VOORHIS: OK, so that would be sort of activation 408 00:18:58,890 --> 00:19:04,390 energy, right/ OK, what's something else? 409 00:19:08,907 --> 00:19:10,490 We only want to know about activation? 410 00:19:10,490 --> 00:19:11,060 Yeah. 411 00:19:11,060 --> 00:19:12,130 AUDIENCE: Charge distribution. 412 00:19:12,130 --> 00:19:13,130 TROY VAN VOORHIS: Right. 413 00:19:13,130 --> 00:19:15,110 OK, so I'll give one variable that probes that, 414 00:19:15,110 --> 00:19:16,693 which is a dipole moment, for example. 415 00:19:16,693 --> 00:19:19,010 You might want to know a dipole moment. 416 00:19:19,010 --> 00:19:21,227 AUDIENCE: What the MOs look like [INAUDIBLE] 417 00:19:21,227 --> 00:19:22,310 TROY VAN VOORHIS: OK, yes. 418 00:19:22,310 --> 00:19:26,570 So that is something we want to know about, 419 00:19:26,570 --> 00:19:28,220 but it's a qualitative one. 420 00:19:28,220 --> 00:19:29,590 So what I'm going to give is I'm going to give you things where 421 00:19:29,590 --> 00:19:31,550 I can say, oh, for a dipole moment, 422 00:19:31,550 --> 00:19:33,810 it's within this many debye, or this much percent. 423 00:19:33,810 --> 00:19:36,710 So MOs are qualitative, and we can get that, but I can't say, 424 00:19:36,710 --> 00:19:40,840 well, this MO is 75% right. 425 00:19:40,840 --> 00:19:43,690 AUDIENCE: Average distance of an electron from an atom. 426 00:19:43,690 --> 00:19:46,865 TROY VAN VOORHIS: OK, so that would be like size. 427 00:19:46,865 --> 00:19:47,490 AUDIENCE: Yeah. 428 00:19:47,490 --> 00:19:49,878 TROY VAN VOORHIS: Yeah, yeah. 429 00:19:49,878 --> 00:19:51,197 AUDIENCE: [INAUDIBLE]. 430 00:19:51,197 --> 00:19:52,530 TROY VAN VOORHIS: What was that? 431 00:19:52,530 --> 00:19:53,880 AUDIENCE: Color. 432 00:19:53,880 --> 00:19:54,970 TROY VAN VOORHIS: Yeah. 433 00:19:54,970 --> 00:19:57,340 I mean, so yeah, that's about an absorption-- 434 00:19:57,340 --> 00:19:59,080 like an electronic absorption spectrum. 435 00:20:07,140 --> 00:20:09,844 Other things people might want to know? 436 00:20:09,844 --> 00:20:10,760 AUDIENCE: Bond length. 437 00:20:10,760 --> 00:20:11,380 TROY VAN VOORHIS: Bond length. 438 00:20:11,380 --> 00:20:12,030 There you go. 439 00:20:16,690 --> 00:20:17,310 Anything else? 440 00:20:17,310 --> 00:20:18,226 We've got a good list. 441 00:20:20,990 --> 00:20:23,870 OK, we'll use that as our list. 442 00:20:23,870 --> 00:20:25,862 OK, so I will start off with bond lengths, 443 00:20:25,862 --> 00:20:27,570 because that's the one where Hartree-Fock 444 00:20:27,570 --> 00:20:30,320 does the best of these things. 445 00:20:30,320 --> 00:20:32,970 So it makes you feel like it's encouraging. 446 00:20:32,970 --> 00:20:35,580 So bond lengths-- Hartree-Fock usually 447 00:20:35,580 --> 00:20:38,434 predicts bond lengths to be a little bit too short. 448 00:20:38,434 --> 00:20:40,350 So the molecules are a little bit too compact. 449 00:20:40,350 --> 00:20:42,000 But it's not too bad. 450 00:20:42,000 --> 00:20:46,660 They're usually too short by about 1%, which is not too bad. 451 00:20:46,660 --> 00:20:50,170 It gets 99% percent of the bond length right. 452 00:20:50,170 --> 00:20:53,980 MP2 does better, so that it doesn't actually 453 00:20:53,980 --> 00:20:55,590 have a systematic error. 454 00:20:55,590 --> 00:20:58,120 It typically gets bond lengths correct, 455 00:20:58,120 --> 00:21:00,930 plus or minus one picometer. 456 00:21:00,930 --> 00:21:04,960 So a picometer is 10 to the minus 12th meters. 457 00:21:04,960 --> 00:21:08,560 So 0.01 angstroms, since a bond length 458 00:21:08,560 --> 00:21:11,680 is usually about an angstrom. 459 00:21:11,680 --> 00:21:19,330 So we'll go from that to size. 460 00:21:19,330 --> 00:21:23,980 So for the size of an atom or a molecule, 461 00:21:23,980 --> 00:21:26,800 this has to do with the Van der Waal's radius, basically. 462 00:21:26,800 --> 00:21:30,550 And actually, Hartree-Fock does a fairly good job 463 00:21:30,550 --> 00:21:32,500 of describing Van der Waal's radii. 464 00:21:32,500 --> 00:21:37,900 I would say that it more or less goes with the bond length 465 00:21:37,900 --> 00:21:40,730 prescription, so that the size is a little bit too small. 466 00:21:40,730 --> 00:21:45,860 So it might be minus 2% to 3% to small. 467 00:21:45,860 --> 00:21:50,440 And then there's very few people that actually do sizes 468 00:21:50,440 --> 00:21:52,900 with MP2, for various reasons. 469 00:21:52,900 --> 00:21:55,060 But I would say that it's-- 470 00:21:59,327 --> 00:22:00,410 I'll put "accurate" there. 471 00:22:00,410 --> 00:22:04,070 Because basically the size of an atom or a molecule is fuzzy. 472 00:22:04,070 --> 00:22:06,650 So I can talk about the radius, and it's a little bit fuzzy. 473 00:22:06,650 --> 00:22:08,357 The errors in MP2 would be smaller 474 00:22:08,357 --> 00:22:10,940 than whatever fuzziness I could have in my definition of size. 475 00:22:14,110 --> 00:22:17,210 For activation energies, Hartree-Fock is pretty poor, 476 00:22:17,210 --> 00:22:23,820 so the activation barriers here are too high, by 30% to 50%. 477 00:22:23,820 --> 00:22:27,080 So the barrier heights are very high in Hartree-Fock. 478 00:22:27,080 --> 00:22:28,700 MP2 has barrier heights that are still 479 00:22:28,700 --> 00:22:34,400 too high, but only by about 10% on average. 480 00:22:34,400 --> 00:22:36,440 Dipole moments in Hartree-Fock-- there 481 00:22:36,440 --> 00:22:37,880 are no systematic studies on that. 482 00:22:37,880 --> 00:22:40,560 I will just say that they are bad. 483 00:22:40,560 --> 00:22:43,640 They are bad enough that nobody does a systematic study. 484 00:22:43,640 --> 00:22:49,940 But for MP2, they're quite a bit better, and typically 485 00:22:49,940 --> 00:22:51,950 accurate say, 0.1 debye. 486 00:22:54,830 --> 00:22:59,210 And then for excitation absorption energy-- 487 00:22:59,210 --> 00:23:01,580 so this is going to be the difference between the lowest 488 00:23:01,580 --> 00:23:05,330 electronic state and the next highest electronic state-- 489 00:23:05,330 --> 00:23:08,840 so you can get these things through various means. 490 00:23:08,840 --> 00:23:13,165 The typical absorption-- and also, I'll say that doing that 491 00:23:13,165 --> 00:23:15,290 involves an extension beyond what I've talked about 492 00:23:15,290 --> 00:23:17,540 so far, because I've been focused on the ground state. 493 00:23:17,540 --> 00:23:20,220 But if you use the logical extensions of these things, 494 00:23:20,220 --> 00:23:22,940 the absorption energies in Hartree-Fock 495 00:23:22,940 --> 00:23:34,560 tend to be too big, and too big by, say, half an eV or so. 496 00:23:34,560 --> 00:23:40,560 And then for MP2, they're more accurate, but not that much. 497 00:23:40,560 --> 00:23:45,140 So plus or minus, say, 0.3 eV. 498 00:23:45,140 --> 00:23:48,780 Electronic inside states tend to be fairly difficult to get. 499 00:23:48,780 --> 00:23:51,380 But we see that MP2 is doing better on all accounts, 500 00:23:51,380 --> 00:23:52,130 than Hartree-Fock. 501 00:23:52,130 --> 00:23:53,120 So it's improving. 502 00:23:53,120 --> 00:23:54,470 That's good. 503 00:23:54,470 --> 00:23:58,400 But then, all right, what is this going to cost me? 504 00:23:58,400 --> 00:23:59,930 How long is this going to take? 505 00:23:59,930 --> 00:24:02,720 Well, there's two different ways that we can measure 506 00:24:02,720 --> 00:24:03,890 computational complexity. 507 00:24:03,890 --> 00:24:06,725 One is by the amount of storage that's required. 508 00:24:06,725 --> 00:24:09,710 It's the amount of information you have to store in order 509 00:24:09,710 --> 00:24:11,340 to do the calculation. 510 00:24:11,340 --> 00:24:13,670 The other way is by how long it's going to take, 511 00:24:13,670 --> 00:24:17,440 how many operations the computer has to do to solve the problem. 512 00:24:17,440 --> 00:24:24,760 And so for Hartree-Fock, the main memory piece that we need 513 00:24:24,760 --> 00:24:26,200 is actually the Hamiltonian. 514 00:24:26,200 --> 00:24:28,210 So we need that Hamiltonian matrix. 515 00:24:28,210 --> 00:24:32,770 We've got to store that in order to compute its eigenvalues. 516 00:24:32,770 --> 00:24:38,050 And it is an end-- it's a number of basis functions by number 517 00:24:38,050 --> 00:24:39,511 of basis functions. 518 00:24:45,640 --> 00:24:52,060 And so then now we have to say, all right-- 519 00:24:52,060 --> 00:24:54,660 well, first of all, we have to figure out how much RAM. 520 00:24:54,660 --> 00:24:57,470 So when I'm figuring out how much RAM is required-- 521 00:24:57,470 --> 00:25:02,170 does anybody know how much RAM they have on their laptop 522 00:25:02,170 --> 00:25:04,020 or desktop computer? 523 00:25:04,020 --> 00:25:05,800 Nobody knows. 524 00:25:05,800 --> 00:25:06,670 16 gigs. 525 00:25:06,670 --> 00:25:10,270 All right, that's what's on my mine, too. 526 00:25:14,320 --> 00:25:19,570 16 gigabytes, which is 16 times 10 to the 9th bytes, roughly. 527 00:25:25,670 --> 00:25:28,340 So now we have to say, all right, 528 00:25:28,340 --> 00:25:33,020 for storing n basis times n basis numbers, 529 00:25:33,020 --> 00:25:35,370 how much does that take? 530 00:25:35,370 --> 00:25:46,260 Well, let's say that we have a atoms. 531 00:25:46,260 --> 00:25:49,539 So we've got-- the number of atoms we have is a. 532 00:25:49,539 --> 00:25:51,830 Then the number of basis functions we have-- last time, 533 00:25:51,830 --> 00:25:53,810 we figured out that a DZP basis, which 534 00:25:53,810 --> 00:25:57,660 is a decent-size basis for carbon had around 15. 535 00:25:57,660 --> 00:25:59,270 It was 14 basis functions per carbon. 536 00:25:59,270 --> 00:26:01,880 I'm just going round to 15 because it's easier math. 537 00:26:01,880 --> 00:26:05,060 So that means that the number of basis functions is 15, 538 00:26:05,060 --> 00:26:07,320 roughly, times the number of atoms. 539 00:26:07,320 --> 00:26:14,570 So there's 15a times 15a things that I need 540 00:26:14,570 --> 00:26:17,000 to store to get this matrix. 541 00:26:17,000 --> 00:26:23,460 And that means I need about 225 a squared numbers. 542 00:26:23,460 --> 00:26:26,360 So these are how many numbers I have to store. 543 00:26:26,360 --> 00:26:30,434 And then I have to know that each number-- 544 00:26:30,434 --> 00:26:32,350 I'm usually storing these in double precision. 545 00:26:32,350 --> 00:26:36,010 So each number requires eight bytes in double precision. 546 00:26:36,010 --> 00:26:45,010 So that means that I have something like 1,700 a squared 547 00:26:45,010 --> 00:26:47,170 bytes to store that object. 548 00:26:51,192 --> 00:26:52,900 So now that says, all right, if I've only 549 00:26:52,900 --> 00:26:56,680 got 16 times 10 to the 9 bytes of storage space 550 00:26:56,680 --> 00:27:01,750 on my computer, and I need 1,700 a squared bytes for a molecule 551 00:27:01,750 --> 00:27:04,210 with a atoms, that just gives me a natural limit 552 00:27:04,210 --> 00:27:05,680 on the number of atoms. 553 00:27:05,680 --> 00:27:10,030 If you back it out, that implies that a is less than or about 554 00:27:10,030 --> 00:27:13,700 equal to 3,000, which is a big number. 555 00:27:13,700 --> 00:27:17,200 So your molecule could have up to about 3,000 atoms in it, 556 00:27:17,200 --> 00:27:19,069 and this calculation would run. 557 00:27:19,069 --> 00:27:21,360 And that rough calculation turns out to be about right. 558 00:27:23,940 --> 00:27:28,420 Now let's take a look at the CPU time required. 559 00:27:28,420 --> 00:27:32,190 So the CPU time, we're going to measure this in hours, 560 00:27:32,190 --> 00:27:36,060 because that's my human unit of time. 561 00:27:36,060 --> 00:27:41,950 So one hour is 3,600 seconds. 562 00:27:41,950 --> 00:27:45,100 And then what I need to know is well, 563 00:27:45,100 --> 00:27:46,840 what actually takes the computer time? 564 00:27:46,840 --> 00:27:49,810 What takes the computer time is doing mathematical operations-- 565 00:27:49,810 --> 00:27:55,060 so add, subtract, multiply, divide, if, things like that. 566 00:27:55,060 --> 00:27:59,410 And usually, we measure the speed of a CPU 567 00:27:59,410 --> 00:28:00,910 in terms of the number of operations 568 00:28:00,910 --> 00:28:02,030 you can do per second. 569 00:28:02,030 --> 00:28:04,690 And does anybody know the order of magnitude 570 00:28:04,690 --> 00:28:08,650 of operations per second the CPU on your computer can do? 571 00:28:08,650 --> 00:28:10,230 It's about a billion-- 572 00:28:10,230 --> 00:28:13,020 some number of billion operations per second, 573 00:28:13,020 --> 00:28:14,930 depending on how recent a model you have, 574 00:28:14,930 --> 00:28:19,190 and whether you play video games or not, 575 00:28:19,190 --> 00:28:22,710 you will do more or less than that. 576 00:28:22,710 --> 00:28:26,822 So that means that our computer, in one hour, can do about one-- 577 00:28:26,822 --> 00:28:28,530 and I'm going to round up, and say you've 578 00:28:28,530 --> 00:28:29,210 got a really good one. 579 00:28:29,210 --> 00:28:31,170 So we're going to say that your thing can do 10 580 00:28:31,170 --> 00:28:32,860 billion operations per second. 581 00:28:32,860 --> 00:28:36,180 If you've got multiple cores, they can each go independently. 582 00:28:36,180 --> 00:28:42,130 So 1 times 10 to the 10 operations per second. 583 00:28:45,020 --> 00:28:48,710 And that means that you can do about 3 times 10 584 00:28:48,710 --> 00:28:51,974 to 13 operations in an hour. 585 00:28:51,974 --> 00:28:54,140 I'm just going to set the hour as my patience limit. 586 00:28:54,140 --> 00:28:56,930 I don't want to wait longer than an hour. 587 00:28:56,930 --> 00:29:02,370 And so, then, for Hartree-Fock, the rate-determining step 588 00:29:02,370 --> 00:29:05,788 is computing the eigenvalues of the Hamiltonian. 589 00:29:10,090 --> 00:29:18,956 And that requires n cubed operations, 590 00:29:18,956 --> 00:29:20,580 where n is the dimension of the matrix. 591 00:29:20,580 --> 00:29:24,500 So if our matrix n basis on a side, it requires n cubed-- 592 00:29:24,500 --> 00:29:28,440 n basis functions cubed operations. 593 00:29:28,440 --> 00:29:30,860 And so then, again, using our translation of that 594 00:29:30,860 --> 00:29:35,360 into atoms, that's 15 times the number of atoms cubed. 595 00:29:35,360 --> 00:29:38,060 Which, if I did my math right back in my office, 596 00:29:38,060 --> 00:29:44,300 that's about 3,000 a cubed. 597 00:29:44,300 --> 00:29:46,220 And then if I back that out-- 598 00:29:46,220 --> 00:29:50,580 again, if I compare that to the number of operations, 599 00:29:50,580 --> 00:29:52,170 that gives me a limit on the number 600 00:29:52,170 --> 00:29:54,360 of atoms that I can handle in an hour, 601 00:29:54,360 --> 00:29:57,160 and that turns out to be about 1,000. 602 00:29:57,160 --> 00:29:59,530 So similar sizes. 603 00:29:59,530 --> 00:30:01,680 So I run out of storage about the same time 604 00:30:01,680 --> 00:30:03,839 as I run out of patience. 605 00:30:03,839 --> 00:30:05,880 If I was willing to be a little bit more patient, 606 00:30:05,880 --> 00:30:08,400 I might be able to do a few more. 607 00:30:08,400 --> 00:30:14,490 But order of magnitude, we're at 1,000. 608 00:30:14,490 --> 00:30:15,420 So I'll summarize. 609 00:30:15,420 --> 00:30:20,110 What we found here is that the storage requirements 610 00:30:20,110 --> 00:30:23,620 were something like a squared of order a squared. 611 00:30:23,620 --> 00:30:27,620 CPU time was of order a cubed. 612 00:30:27,620 --> 00:30:30,060 The maximum feasible number of basis functions 613 00:30:30,060 --> 00:30:34,740 that I could do here was something 614 00:30:34,740 --> 00:30:37,500 like-- so if I have 1,000 atoms, that was my limit. 615 00:30:37,500 --> 00:30:40,950 Something like 15,000 would be the maximum feasible 616 00:30:40,950 --> 00:30:43,110 number of basis functions. 617 00:30:43,110 --> 00:30:48,540 And maximum feasible number of atoms was about 1,000. 618 00:30:48,540 --> 00:30:50,640 So we won't go through the same exercise for MP2 619 00:30:50,640 --> 00:30:51,480 in such detail. 620 00:30:51,480 --> 00:30:55,492 I will just tell you that everything is worse for MP2. 621 00:30:55,492 --> 00:30:57,450 Everything takes longer, it takes more storage, 622 00:30:57,450 --> 00:30:58,590 everything's worse. 623 00:30:58,590 --> 00:31:04,090 So it requires order a cubed storage. 624 00:31:04,090 --> 00:31:09,000 It requires order a to the fifth CPU time. 625 00:31:09,000 --> 00:31:11,160 The maximum feasible number of basis functions 626 00:31:11,160 --> 00:31:14,520 is therefore smaller, about 2,000. 627 00:31:14,520 --> 00:31:18,270 And in MP2, you actually require more basis functions per atom 628 00:31:18,270 --> 00:31:21,570 to get an accurate answer, so that the number 629 00:31:21,570 --> 00:31:27,190 of feasible atoms is more like 50 rather than 1,000. 630 00:31:27,190 --> 00:31:30,390 So that's the cumulative effect of larger basis sets and worse 631 00:31:30,390 --> 00:31:35,650 scaling with number of atoms that makes MP2 deal with much 632 00:31:35,650 --> 00:31:36,400 smaller systems. 633 00:31:39,330 --> 00:31:40,800 So questions about that. 634 00:31:44,981 --> 00:31:45,480 Yes. 635 00:31:45,480 --> 00:31:49,790 AUDIENCE: So what are some examples of [INAUDIBLE]?? 636 00:31:49,790 --> 00:31:53,360 TROY VAN VOORHIS: So C60 is 60 atoms. 637 00:31:53,360 --> 00:31:55,710 That's an easy one. 638 00:31:55,710 --> 00:32:00,330 There are a number of small dyes, 639 00:32:00,330 --> 00:32:03,210 for example, that we worked with in our group, where you might 640 00:32:03,210 --> 00:32:05,293 have seen absorption spectra, or emission spectra, 641 00:32:05,293 --> 00:32:07,320 or HOMOs, or LUMOs, or hole transport 642 00:32:07,320 --> 00:32:10,050 properties, or electron transport properties, 643 00:32:10,050 --> 00:32:12,750 that are around 50 atoms in size. 644 00:32:12,750 --> 00:32:15,330 And then the main thing that people get interested in that 645 00:32:15,330 --> 00:32:21,360 are on the 1,000-atom regimes are enzymes and peptides. 646 00:32:21,360 --> 00:32:22,480 Those are the sort of-- 647 00:32:22,480 --> 00:32:24,480 when you start saying, I want to do 1,000 atoms, 648 00:32:24,480 --> 00:32:29,105 it's usually because it's really some polymer or heteropolymer. 649 00:32:31,777 --> 00:32:34,360 Another case where you also are sometimes interested in things 650 00:32:34,360 --> 00:32:36,220 where you do simulations with 1,000 atoms, 651 00:32:36,220 --> 00:32:37,700 if you're interested in a surface. 652 00:32:37,700 --> 00:32:41,140 Because if you have a chunk of gold, for example, 653 00:32:41,140 --> 00:32:43,450 1,000 atoms is just 10 gold atoms, by 10 gold atoms, 654 00:32:43,450 --> 00:32:46,210 by 10 gold atoms, which is just a little chunk-- a very, very 655 00:32:46,210 --> 00:32:48,040 small chunk of gold, but still a chunk. 656 00:32:55,220 --> 00:32:56,408 Other questions. 657 00:33:00,160 --> 00:33:07,840 OK, so I'm going to spend seven minutes talking about density 658 00:33:07,840 --> 00:33:08,890 functional theory. 659 00:33:08,890 --> 00:33:12,320 And then we're going to go over and do some examples. 660 00:33:12,320 --> 00:33:15,160 So the idea of density functional theory 661 00:33:15,160 --> 00:33:19,420 is that it'd be really nice if you had some magical way 662 00:33:19,420 --> 00:33:20,800 to do a Hartree-Fock calculation, 663 00:33:20,800 --> 00:33:23,980 but have a give you exactly the right answer. 664 00:33:23,980 --> 00:33:26,650 That basically would be the dream because we saw, well, 665 00:33:26,650 --> 00:33:29,080 for Hartree-Fock, we can do big systems, it's cheap, 666 00:33:29,080 --> 00:33:31,480 it has good scaling, all these kinds of things. 667 00:33:31,480 --> 00:33:33,450 It just doesn't give us very good answers. 668 00:33:33,450 --> 00:33:35,025 The results are pretty poor. 669 00:33:35,025 --> 00:33:36,650 So what if we had something that scaled 670 00:33:36,650 --> 00:33:38,774 like it had the computational cost of Hartree-Fock, 671 00:33:38,774 --> 00:33:42,980 but was, say, the accuracy of MP2, or even better than that? 672 00:33:42,980 --> 00:33:46,030 And so in density functional theory, what we use 673 00:33:46,030 --> 00:33:51,700 is the idea of looking at the electron density rho of r 674 00:33:51,700 --> 00:33:53,090 as the fundamental variable. 675 00:33:53,090 --> 00:33:56,350 So you can actually work out, for a determinant, 676 00:33:56,350 --> 00:33:58,001 what the electron density is. 677 00:33:58,001 --> 00:33:59,500 And it's just the sum of the squares 678 00:33:59,500 --> 00:34:01,321 of the orbital densities-- 679 00:34:01,321 --> 00:34:02,820 or the sum of the orbital densities. 680 00:34:02,820 --> 00:34:05,190 So you square each orbital, then you add that up, 681 00:34:05,190 --> 00:34:08,530 and that gives you the total electron density. 682 00:34:08,530 --> 00:34:10,360 It's basically just the probability 683 00:34:10,360 --> 00:34:13,330 of finding an electron at a point, r. 684 00:34:27,190 --> 00:34:29,400 OK, so that's a nice observable. 685 00:34:29,400 --> 00:34:31,139 The thing that's kind of amazing is 686 00:34:31,139 --> 00:34:33,472 that there is a theorem, a mathematical theorem that you 687 00:34:33,472 --> 00:34:36,389 can prove in about two pages using just sort of proof 688 00:34:36,389 --> 00:34:37,620 by contradiction. 689 00:34:37,620 --> 00:34:42,630 You can prove that there exists a functional, which is always 690 00:34:42,630 --> 00:34:46,650 given then in e sub v of rho, such 691 00:34:46,650 --> 00:34:49,060 that when you're given the ground state density, rho 692 00:34:49,060 --> 00:34:52,409 0, if you plug that into this magical functional, 693 00:34:52,409 --> 00:34:54,219 it will give you the ground state energy. 694 00:34:54,219 --> 00:34:56,280 So if you found the ground state density lying around 695 00:34:56,280 --> 00:34:57,390 in the gutter, and you picked it up, 696 00:34:57,390 --> 00:34:58,889 and you put it into this functional, 697 00:34:58,889 --> 00:35:01,642 it would give you the ground state energy. 698 00:35:01,642 --> 00:35:03,600 And e0 is not just the approximate ground state 699 00:35:03,600 --> 00:35:04,100 energy. 700 00:35:04,100 --> 00:35:09,330 It is the exact ground state energy, the exact thing. 701 00:35:09,330 --> 00:35:11,910 Further, for any density, rho prime, 702 00:35:11,910 --> 00:35:13,950 that is not the ground state density, 703 00:35:13,950 --> 00:35:18,740 if you plug that density into eV, you get a higher energy. 704 00:35:18,740 --> 00:35:21,890 So what you could do is you could say, well, let me 705 00:35:21,890 --> 00:35:23,150 just search. 706 00:35:23,150 --> 00:35:25,680 Let me try density, see what energy it gives. 707 00:35:25,680 --> 00:35:26,930 Then I'll try another density. 708 00:35:26,930 --> 00:35:29,200 If it gives a lower energy, that's a better density. 709 00:35:29,200 --> 00:35:31,741 And I'll keep going, and going, and going, and going, until I 710 00:35:31,741 --> 00:35:34,470 find that ground state density. 711 00:35:34,470 --> 00:35:36,480 So the idea is that then you would say, aha, 712 00:35:36,480 --> 00:35:44,384 if I solve that equation-- so I search over densities, 713 00:35:44,384 --> 00:35:46,300 and that's not going to be a very hard search, 714 00:35:46,300 --> 00:35:48,790 because the density is just a three-dimensional object. 715 00:35:48,790 --> 00:35:50,200 It's not like the wave function that depends 716 00:35:50,200 --> 00:35:51,283 on a bunch of coordinates. 717 00:35:51,283 --> 00:35:53,120 It's just three dimensional. 718 00:35:53,120 --> 00:35:56,910 So I solve that equation, that will give me rho 0. 719 00:35:56,910 --> 00:36:02,100 And then I can take that rho 0, plug it back in there, 720 00:36:02,100 --> 00:36:05,090 and I will get the ground state energy. 721 00:36:05,090 --> 00:36:08,161 That gives me a closed set of conditions 722 00:36:08,161 --> 00:36:09,910 that lets me find the ground state energy, 723 00:36:09,910 --> 00:36:13,176 and then report back the energy of that density. 724 00:36:13,176 --> 00:36:15,050 AUDIENCE: Where do we get this [? EV ?] from? 725 00:36:15,050 --> 00:36:16,800 TROY VAN VOORHIS: That's a great question. 726 00:36:16,800 --> 00:36:18,480 The proof is a proof by contradiction. 727 00:36:18,480 --> 00:36:19,470 It's not constructive. 728 00:36:19,470 --> 00:36:21,290 It proves that it exists, but gives you 729 00:36:21,290 --> 00:36:23,880 no way of constructing it. 730 00:36:23,880 --> 00:36:27,932 So kind of the frustrating thing is like, oh, cookies exist, 731 00:36:27,932 --> 00:36:29,390 but we don't know how to make them. 732 00:36:33,370 --> 00:36:35,930 But the idea that such a thing exists 733 00:36:35,930 --> 00:36:37,737 gives you hope to say, well, maybe 734 00:36:37,737 --> 00:36:39,820 we can construct-- if we can't find the exact one, 735 00:36:39,820 --> 00:36:42,920 maybe we can find one that's very, very good. 736 00:36:42,920 --> 00:36:46,010 And that one that's very good, we would just use. 737 00:36:46,010 --> 00:36:48,319 We would pretend it was exact, minimize the energy, 738 00:36:48,319 --> 00:36:50,360 find the density, and then report back the ground 739 00:36:50,360 --> 00:36:52,070 state for that approximating. 740 00:36:52,070 --> 00:36:55,220 And that's actually what you do in density functional theory. 741 00:36:55,220 --> 00:36:57,458 You have approximate functionals. 742 00:37:04,430 --> 00:37:06,740 And they're all-- just like with basis sets-- 743 00:37:06,740 --> 00:37:09,650 named after the authors of the people who wrote the papers. 744 00:37:09,650 --> 00:37:11,480 Almost all of them are. 745 00:37:11,480 --> 00:37:17,420 So there's the Local Spin Density Approximation, LSDA. 746 00:37:17,420 --> 00:37:20,840 There's the Perdew-Burke-Ernzerhof 747 00:37:20,840 --> 00:37:22,940 functional. 748 00:37:22,940 --> 00:37:30,170 There's the Becke-Lee-Yang-Parr functional, 749 00:37:30,170 --> 00:37:34,910 the Perdew-Burke-Ernzerhof Zero functional, 750 00:37:34,910 --> 00:37:39,840 the Becke 3 Lee-Yang-Parr functional, 751 00:37:39,840 --> 00:37:41,791 and then on and on and on. 752 00:37:41,791 --> 00:37:43,290 Now these things have been sort of-- 753 00:37:43,290 --> 00:37:45,330 there's a mishmash of different exact conditions 754 00:37:45,330 --> 00:37:48,270 that were used to derive these functionals. 755 00:37:48,270 --> 00:37:51,570 And then, empirically, people have gone and shown 756 00:37:51,570 --> 00:37:55,110 that PBE is better than LSDA. 757 00:37:55,110 --> 00:37:57,300 It's about as good as BLYP. 758 00:37:57,300 --> 00:38:01,190 BLYP is not as good as PBE0, but it was just about as good 759 00:38:01,190 --> 00:38:01,690 as B3LYP. 760 00:38:04,760 --> 00:38:06,150 So we have, then, sort of an idea 761 00:38:06,150 --> 00:38:08,970 that, OK, if we go over towards the right-hand side of this, 762 00:38:08,970 --> 00:38:11,730 we're going to get better results out of DFT. 763 00:38:11,730 --> 00:38:14,760 But DFT, because it's based off of a Slater determinant, 764 00:38:14,760 --> 00:38:16,920 has exactly the same computational scaling, 765 00:38:16,920 --> 00:38:19,090 and storage, and everything, as Hartree-Fock. 766 00:38:19,090 --> 00:38:20,760 It's the same kind of expense. 767 00:38:20,760 --> 00:38:22,740 But if I go back to this accuracy thing, 768 00:38:22,740 --> 00:38:26,530 and I say, well, what would I get from that best 769 00:38:26,530 --> 00:38:34,060 density functional B3LYP, well what 770 00:38:34,060 --> 00:38:36,880 I find is that activation energies are still a little bit 771 00:38:36,880 --> 00:38:40,700 difficult. Now I underpredict activation energies. 772 00:38:40,700 --> 00:38:44,400 But my dipole moments are good. 773 00:38:44,400 --> 00:38:48,170 My sizes are again accurate. 774 00:38:48,170 --> 00:38:55,010 My absorption energies are just as accurate as MP2, 775 00:38:55,010 --> 00:38:57,300 and my bond lengths are just as accurate as MP2. 776 00:39:00,660 --> 00:39:02,550 There are other things, but basically B3LYP 777 00:39:02,550 --> 00:39:05,010 is as accurate or more accurate than MP2 778 00:39:05,010 --> 00:39:07,470 for virtually every property you would want, but has 779 00:39:07,470 --> 00:39:09,399 Hartree-Fock-like cost. 780 00:39:09,399 --> 00:39:11,190 And so that makes density functional theory 781 00:39:11,190 --> 00:39:14,310 the workhorse of computational chemistry these days. 782 00:39:14,310 --> 00:39:17,250 Because it's inexpensive, you can do lots of calculations 783 00:39:17,250 --> 00:39:25,014 with it, but it's accurate enough for grunt work. 784 00:39:25,014 --> 00:39:27,430 And then there's a blank column there that, if I had time, 785 00:39:27,430 --> 00:39:28,590 I would talk about. 786 00:39:28,590 --> 00:39:29,830 There's a whole other category of things 787 00:39:29,830 --> 00:39:31,829 where you say, well, perturbation theory doesn't 788 00:39:31,829 --> 00:39:35,220 work, but are there more constructive ways 789 00:39:35,220 --> 00:39:38,250 that I could use wave functions to approximate the correlation 790 00:39:38,250 --> 00:39:39,280 energy? 791 00:39:39,280 --> 00:39:41,370 And the answer is yes, you can get good energies 792 00:39:41,370 --> 00:39:42,547 and good answers out. 793 00:39:42,547 --> 00:39:44,880 But then it comes at the cost of the calculation getting 794 00:39:44,880 --> 00:39:46,480 much more expensive. 795 00:39:46,480 --> 00:39:49,530 So there are methods that improve all these properties 796 00:39:49,530 --> 00:39:52,154 on the wave function side, that just require more. 797 00:39:52,154 --> 00:39:53,820 So if you want to know more about those, 798 00:39:53,820 --> 00:39:56,040 they're in the notes. 799 00:39:56,040 --> 00:40:02,980 And so now I think I will switch over and show you how we 800 00:40:02,980 --> 00:40:04,650 do some calculations with this. 801 00:40:04,650 --> 00:40:08,831 So now I will switch. 802 00:40:08,831 --> 00:40:10,330 There's two different tools that you 803 00:40:10,330 --> 00:40:12,820 have available to you for free on Athena 804 00:40:12,820 --> 00:40:15,819 that you can use to do electronic structure 805 00:40:15,819 --> 00:40:16,360 calculations. 806 00:40:16,360 --> 00:40:19,384 One is Gaussian, and the other one is Q-Chem. 807 00:40:19,384 --> 00:40:20,800 There may be even some other ones, 808 00:40:20,800 --> 00:40:22,758 but those are the two that I've collected notes 809 00:40:22,758 --> 00:40:24,470 about how to use. 810 00:40:24,470 --> 00:40:26,290 So I'm going to show you how we use Q-Chem. 811 00:40:26,290 --> 00:40:27,664 I'm not going to do it on Athena. 812 00:40:27,664 --> 00:40:30,290 I'm going to do it on my laptop, but it's basically the same. 813 00:40:30,290 --> 00:40:32,065 So there's this GUI called IQmol. 814 00:40:35,410 --> 00:40:38,920 And when you open it up, it gives you 815 00:40:38,920 --> 00:40:43,020 a window that looks like this. 816 00:40:43,020 --> 00:40:44,970 And there's a few things here. 817 00:40:44,970 --> 00:40:48,660 So there's, like, an open file, you can create something, 818 00:40:48,660 --> 00:40:51,870 you can open a file, you can save something, 819 00:40:51,870 --> 00:40:54,960 you can move something around after you've built it. 820 00:40:54,960 --> 00:40:58,260 And then these are the build tools over here. 821 00:40:58,260 --> 00:41:01,290 So that just turns it into build mode. 822 00:41:01,290 --> 00:41:04,290 If I click on this, I can choose any item in the periodic table. 823 00:41:04,290 --> 00:41:06,600 Right now, it's on carbon. 824 00:41:06,600 --> 00:41:11,180 This lets me select various fragments, 825 00:41:11,180 --> 00:41:13,230 and also what the attach point of the fragment's 826 00:41:13,230 --> 00:41:16,452 going to be in some cases. 827 00:41:16,452 --> 00:41:18,660 And I don't want to do it, so I'll select it for now, 828 00:41:18,660 --> 00:41:21,000 but then I'm going to go back to this one. 829 00:41:21,000 --> 00:41:23,260 This is the Add Hydrogens button. 830 00:41:23,260 --> 00:41:24,640 So if you have something that-- 831 00:41:24,640 --> 00:41:26,140 you just wrote your Lewis structure, 832 00:41:26,140 --> 00:41:28,230 it had no hydrogens in it, you click that button, 833 00:41:28,230 --> 00:41:30,510 it'll put all the hydrogens where they should be, 834 00:41:30,510 --> 00:41:32,170 or where it thinks they should be. 835 00:41:32,170 --> 00:41:34,920 This is a Minimize Energy thing, which 836 00:41:34,920 --> 00:41:37,110 basically if your structure looks really crazy, 837 00:41:37,110 --> 00:41:40,140 it'll kind of make it look less crazy. 838 00:41:40,140 --> 00:41:44,370 And then this is the Select button, which 839 00:41:44,370 --> 00:41:45,550 lets you pick certain atoms. 840 00:41:45,550 --> 00:41:48,900 This lets you delete certain atoms, take a picture, 841 00:41:48,900 --> 00:41:50,702 do a movie. 842 00:41:50,702 --> 00:41:53,055 And then that changes it over to fullscreen mode. 843 00:41:53,055 --> 00:41:57,170 And that's the life preserver for Help. 844 00:41:57,170 --> 00:41:58,860 But we'll do building a molecule. 845 00:41:58,860 --> 00:42:01,440 So does somebody want to tell me a molecule that 846 00:42:01,440 --> 00:42:04,410 has less than 15 atoms in it that they 847 00:42:04,410 --> 00:42:09,590 would like me to draw here, that I can actually draw? 848 00:42:09,590 --> 00:42:10,940 AUDIENCE: Diethyl ether. 849 00:42:10,940 --> 00:42:12,969 TROY VAN VOORHIS: Diethyl ether. 850 00:42:12,969 --> 00:42:13,760 Yes, I can do that. 851 00:42:13,760 --> 00:42:15,660 OK. 852 00:42:15,660 --> 00:42:20,340 All right, so put a carbon, carbon, 853 00:42:20,340 --> 00:42:31,940 then I got to switch to an oxygen, oxygen, carbon, carbon. 854 00:42:31,940 --> 00:42:36,426 And then I will click the Add Hydrogen button. 855 00:42:36,426 --> 00:42:38,790 [LAUGHTER] 856 00:42:38,790 --> 00:42:39,750 There we go. 857 00:42:39,750 --> 00:42:43,410 I've got my diethyl ether now. 858 00:42:43,410 --> 00:42:46,400 And while my geometry here is not perfect, 859 00:42:46,400 --> 00:42:47,720 it doesn't look totally crazy. 860 00:42:47,720 --> 00:42:49,190 So I'll just go with that. 861 00:42:49,190 --> 00:42:50,690 And then the thing we're going to be 862 00:42:50,690 --> 00:42:52,640 most interested in is this little tab here, 863 00:42:52,640 --> 00:42:54,290 called Calculation. 864 00:42:54,290 --> 00:42:58,040 So calculation here has a little button that says Q-Chem Setup. 865 00:42:58,040 --> 00:43:01,520 So I'll do Q-Chem Setup. 866 00:43:01,520 --> 00:43:04,745 And now it's got a bunch of things that I can specify. 867 00:43:07,302 --> 00:43:09,260 There are a few things that are most important. 868 00:43:09,260 --> 00:43:13,361 So the first thing is, what kind of calculation I want it to do. 869 00:43:13,361 --> 00:43:15,110 So I could have it just compute the energy 870 00:43:15,110 --> 00:43:17,960 of the molecule for the configuration I specified. 871 00:43:17,960 --> 00:43:20,390 That would be one useful thing. 872 00:43:20,390 --> 00:43:21,870 I could compute forces. 873 00:43:21,870 --> 00:43:23,540 In other words, the forces on the atoms, 874 00:43:23,540 --> 00:43:25,640 where the atoms want to move. 875 00:43:25,640 --> 00:43:28,850 I could optimize the geometry, which would then 876 00:43:28,850 --> 00:43:33,230 relax the thing down, and figure out what the best geometry is. 877 00:43:33,230 --> 00:43:34,820 I could do various things that scan 878 00:43:34,820 --> 00:43:36,800 across the potential energy surface. 879 00:43:36,800 --> 00:43:39,340 I can compute vibrational frequencies. 880 00:43:39,340 --> 00:43:41,810 I can compute NMR chemical shifts. 881 00:43:41,810 --> 00:43:45,110 I can do molecular dynamics. 882 00:43:45,110 --> 00:43:47,780 I can compute properties-- 883 00:43:47,780 --> 00:43:48,594 so lots of things. 884 00:43:48,594 --> 00:43:50,260 I'm just going to optimize the geometry. 885 00:43:50,260 --> 00:43:52,860 That's going to be what I'm going to choose for now. 886 00:43:52,860 --> 00:43:55,140 And then I get to choose a method. 887 00:43:55,140 --> 00:43:57,170 So I can do Hartree-Fock. 888 00:43:57,170 --> 00:43:59,680 I can do B3LYP. 889 00:43:59,680 --> 00:44:03,260 I can go down-- so all the things above this dotted line 890 00:44:03,260 --> 00:44:06,560 here are different density functions. 891 00:44:06,560 --> 00:44:11,450 And then, down here, we've got MP2, MP2 bracket v-- 892 00:44:11,450 --> 00:44:13,760 I'm not actually sure what that is. 893 00:44:13,760 --> 00:44:18,020 I've got various versions of MP2 that try to make it faster. 894 00:44:18,020 --> 00:44:20,630 And then I've got a couple of cluster methods 895 00:44:20,630 --> 00:44:23,090 that we didn't talk about yet, but that are in the notes. 896 00:44:23,090 --> 00:44:24,590 And then various methods for getting 897 00:44:24,590 --> 00:44:27,540 electronic excited states, down here at the bottom. 898 00:44:27,540 --> 00:44:30,920 So for now, let's just do B3LYP, and then I 899 00:44:30,920 --> 00:44:32,420 get to choose a basis. 900 00:44:32,420 --> 00:44:34,900 Because this is on my laptop, and we're in class, 901 00:44:34,900 --> 00:44:36,600 I'm going to choose a small basis. 902 00:44:36,600 --> 00:44:41,420 So I'm going to leave it at 321g, which is a small basis. 903 00:44:41,420 --> 00:44:44,840 And then I can also specify the charge, which 904 00:44:44,840 --> 00:44:46,940 is like the total charge of the molecule, 905 00:44:46,940 --> 00:44:48,210 and the spin multiplicity. 906 00:44:48,210 --> 00:44:52,190 So I want it to be a neutral molecule, spin 1-- 907 00:44:52,190 --> 00:44:55,910 or spin multiplicity 1, which is spin 0. 908 00:44:55,910 --> 00:45:00,920 Because it's 2s plus 1 is the spin multiplicity. 909 00:45:00,920 --> 00:45:03,460 And then there's various things about convergence control 910 00:45:03,460 --> 00:45:05,293 that you probably won't need to worry about. 911 00:45:05,293 --> 00:45:07,610 And then there's advanced things that you also probably 912 00:45:07,610 --> 00:45:09,170 won't need to worry about. 913 00:45:09,170 --> 00:45:11,630 So now I'm happy with that. 914 00:45:11,630 --> 00:45:14,280 I can submit my job. 915 00:45:14,280 --> 00:45:23,420 And I will call this diethyl ether, hit OK. 916 00:45:23,420 --> 00:45:26,650 And now it doesn't appear to be doing anything. 917 00:45:26,650 --> 00:45:30,102 In the background, it is doing something. 918 00:45:30,102 --> 00:45:32,060 I think this will finish in just a few seconds, 919 00:45:32,060 --> 00:45:35,890 but if you get concerned that the job might not be running 920 00:45:35,890 --> 00:45:37,340 or something might have happened, 921 00:45:37,340 --> 00:45:39,607 you can go over to the Job Monitor. 922 00:45:39,607 --> 00:45:41,690 Methane-- that was the job that I ran this morning 923 00:45:41,690 --> 00:45:42,815 to make sure that it works. 924 00:45:42,815 --> 00:45:47,455 Diethyl ether, we just submitted it at 10:50 and 21 seconds. 925 00:45:47,455 --> 00:45:48,300 Ah, there we go. 926 00:45:48,300 --> 00:45:49,924 Now it's been running for four seconds. 927 00:45:52,174 --> 00:45:53,340 Let's see how long it takes. 928 00:45:59,610 --> 00:46:02,200 OK, it's taking a lot longer than I think. 929 00:46:02,200 --> 00:46:02,967 All right. 930 00:46:02,967 --> 00:46:06,240 AUDIENCE: They close [? in 15 ?] [INAUDIBLE].. 931 00:46:06,240 --> 00:46:08,426 TROY VAN VOORHIS: That's all right. 932 00:46:08,426 --> 00:46:09,800 We still we've got a few minutes. 933 00:46:09,800 --> 00:46:13,210 So what will happen here in a minute, when we-- 934 00:46:13,210 --> 00:46:15,872 AUDIENCE: [INAUDIBLE]. 935 00:46:15,872 --> 00:46:17,330 TROY VAN VOORHIS: That's all right. 936 00:46:20,960 --> 00:46:22,660 It's going along. 937 00:46:22,660 --> 00:46:24,150 So when it's done, what will happen 938 00:46:24,150 --> 00:46:26,790 is, over here, it'll let us know. 939 00:46:26,790 --> 00:46:29,070 And over here will appear a thing 940 00:46:29,070 --> 00:46:31,189 where we can start looking at the results. 941 00:46:31,189 --> 00:46:33,480 It might have gone faster if I had minimized the energy 942 00:46:33,480 --> 00:46:35,980 to begin with, because then it would do less geometry steps. 943 00:46:38,775 --> 00:46:40,879 It's going along. 944 00:46:40,879 --> 00:46:41,420 I don't know. 945 00:46:41,420 --> 00:46:43,503 We might run out of class time before it finishes. 946 00:46:43,503 --> 00:46:44,270 It's 10:51. 947 00:46:44,270 --> 00:46:45,260 We've got four minutes. 948 00:46:45,260 --> 00:46:47,390 Let's see. 949 00:46:47,390 --> 00:46:48,290 Come on, laptop. 950 00:46:48,290 --> 00:46:50,950 I've got the MacBook Pro. 951 00:46:50,950 --> 00:46:54,534 Maybe I should have upgraded the processor, this kind of thing. 952 00:46:54,534 --> 00:46:55,980 AUDIENCE: How long do you usually 953 00:46:55,980 --> 00:46:57,910 let your group calculations go for? 954 00:46:57,910 --> 00:46:59,784 TROY VAN VOORHIS: Well, in my research group, 955 00:46:59,784 --> 00:47:03,907 we have 1,000 cores that we-- 956 00:47:03,907 --> 00:47:05,990 maybe around 1,200 cores, even, I think, actually. 957 00:47:05,990 --> 00:47:06,470 Oh, there we go. 958 00:47:06,470 --> 00:47:06,970 It finished. 959 00:47:06,970 --> 00:47:09,720 But we have around 1,200 cores for around 10 people. 960 00:47:09,720 --> 00:47:13,700 So that means that you can let a job run for a very, very, very 961 00:47:13,700 --> 00:47:15,860 long time, on multiple cores, and it 962 00:47:15,860 --> 00:47:17,630 doesn't get in anyone's way. 963 00:47:17,630 --> 00:47:22,520 So it's not unusual for us to put something in there that 964 00:47:22,520 --> 00:47:25,070 lasts a week. 965 00:47:25,070 --> 00:47:26,090 So now it's finished. 966 00:47:26,090 --> 00:47:27,923 We want to copy the results from the server. 967 00:47:27,923 --> 00:47:31,580 This is an arcane way of saying things. 968 00:47:31,580 --> 00:47:34,580 Because it's designed for supercomputers, where 969 00:47:34,580 --> 00:47:37,520 you are sitting at a terminal in your office, 970 00:47:37,520 --> 00:47:39,504 and then it sends all the results over 971 00:47:39,504 --> 00:47:41,420 to the supercomputer, it runs the calculation, 972 00:47:41,420 --> 00:47:42,770 and then you have to copy the results back 973 00:47:42,770 --> 00:47:44,032 to your little computer. 974 00:47:44,032 --> 00:47:45,740 I was actually running it on my computer. 975 00:47:45,740 --> 00:47:49,180 It's just the computer is both-- 976 00:47:49,180 --> 00:47:50,230 I'll just put it-- 977 00:47:53,674 --> 00:47:54,658 New Folder. 978 00:48:03,014 --> 00:48:03,514 OK. 979 00:48:08,861 --> 00:48:09,360 There we go. 980 00:48:09,360 --> 00:48:11,516 Now it has a little star next to it, which lets 981 00:48:11,516 --> 00:48:12,640 me know that it's finished. 982 00:48:12,640 --> 00:48:14,130 So let's see, Info-- 983 00:48:14,130 --> 00:48:16,450 so I can show the dipole moment. 984 00:48:16,450 --> 00:48:18,720 So there we go, that little blue arrow-- 985 00:48:18,720 --> 00:48:20,880 which, magnitude and direction is the dipole moment 986 00:48:20,880 --> 00:48:21,990 of diethyl ether. 987 00:48:21,990 --> 00:48:25,936 It does have a dipole moment, with its positive end here, 988 00:48:25,936 --> 00:48:27,060 pointing in that direction. 989 00:48:27,060 --> 00:48:28,860 That seems about right. 990 00:48:28,860 --> 00:48:30,300 Let's see. 991 00:48:30,300 --> 00:48:31,617 I can look at the files. 992 00:48:31,617 --> 00:48:34,200 These are only if you're really, really interested in details. 993 00:48:34,200 --> 00:48:35,460 You see these are very verbose. 994 00:48:35,460 --> 00:48:36,876 They have every bit of information 995 00:48:36,876 --> 00:48:38,600 you could possibly want in them. 996 00:48:38,600 --> 00:48:40,725 So if there's ever something that you can't get out 997 00:48:40,725 --> 00:48:42,750 of the GUI, if you just kind of go-- 998 00:48:42,750 --> 00:48:43,890 you can search. 999 00:48:43,890 --> 00:48:47,122 And I can look for converge. 1000 00:48:47,122 --> 00:48:49,240 Oh, look, optimization converged, and then it 1001 00:48:49,240 --> 00:48:52,460 tells me everything after that. 1002 00:48:52,460 --> 00:48:54,680 I can look at the different atoms in the molecule, 1003 00:48:54,680 --> 00:48:56,480 and it will highlight them for me. 1004 00:48:59,420 --> 00:49:02,532 And then I can look at bonds, and it will tell me-- 1005 00:49:02,532 --> 00:49:04,490 I don't if you guys can read, but down here, it 1006 00:49:04,490 --> 00:49:06,110 has the bond length in the corner. 1007 00:49:06,110 --> 00:49:08,330 So that C-H bond is 1.093. 1008 00:49:08,330 --> 00:49:13,260 All these C-H bonds are going to be 1.09, plus or minus a bit. 1009 00:49:13,260 --> 00:49:17,940 The carbon-oxygen bond there is 1.46, rounding. 1010 00:49:17,940 --> 00:49:20,870 That one is also 1.46. 1011 00:49:20,870 --> 00:49:26,940 And then this here-- so as it did the geometry optimization, 1012 00:49:26,940 --> 00:49:30,150 it was adjusting the geometry trying to find the minimum. 1013 00:49:30,150 --> 00:49:32,190 And these are the different total energies 1014 00:49:32,190 --> 00:49:34,320 that it had as it went through the optimization. 1015 00:49:34,320 --> 00:49:36,430 You'll see that eventually it sort of slows down. 1016 00:49:36,430 --> 00:49:37,980 The energy is going down every time, 1017 00:49:37,980 --> 00:49:39,450 and eventually it sort of slows down to where 1018 00:49:39,450 --> 00:49:40,574 it's not changing any more. 1019 00:49:40,574 --> 00:49:42,270 It says, all right, that's the minimum. 1020 00:49:42,270 --> 00:49:46,120 You also notice that these are in atomic units. 1021 00:49:46,120 --> 00:49:49,590 So that's minus 232 Hartrees. 1022 00:49:49,590 --> 00:49:52,620 So that's like 4,000 kcals. 1023 00:49:52,620 --> 00:49:54,390 No, that's more than 4,000 kcals. 1024 00:49:54,390 --> 00:49:56,976 Anyway, it's tens of thousands of kcals. 1025 00:49:56,976 --> 00:49:58,350 The reason it's such a low number 1026 00:49:58,350 --> 00:50:01,200 is this is the energy it would take to rip every electron off 1027 00:50:01,200 --> 00:50:04,620 of the molecule, including all of the 1s electrons. 1028 00:50:04,620 --> 00:50:06,630 So the energies are typically going 1029 00:50:06,630 --> 00:50:08,522 to be very, very large negative numbers. 1030 00:50:08,522 --> 00:50:09,480 Don't worry about that. 1031 00:50:09,480 --> 00:50:12,660 It's energy differences that matter. 1032 00:50:12,660 --> 00:50:16,840 So you notice, even my bad geometry was already 232.1. 1033 00:50:16,840 --> 00:50:18,780 The correct answer is 232.3. 1034 00:50:18,780 --> 00:50:21,132 So the total energy change wasn't that much. 1035 00:50:21,132 --> 00:50:22,590 And then the last cool thing that I 1036 00:50:22,590 --> 00:50:25,740 want to show you is that we can plot orbitals. 1037 00:50:29,435 --> 00:50:35,360 So I'm going to count some orbitals. 1038 00:50:35,360 --> 00:50:35,880 There we go. 1039 00:50:35,880 --> 00:50:38,136 So I just had to calculate the HOMO and the LUMO. 1040 00:50:38,136 --> 00:50:39,760 I didn't go through all of it with you. 1041 00:50:39,760 --> 00:50:42,425 But if you go there, it gives you some things-- 1042 00:50:42,425 --> 00:50:44,300 I should actually go back and look, show you. 1043 00:50:44,300 --> 00:50:47,180 So you get to choose a range of orbitals. 1044 00:50:47,180 --> 00:50:50,100 The iso value is basically where it draws the surface. 1045 00:50:50,100 --> 00:50:51,670 So it's inverse of what you expect. 1046 00:50:51,670 --> 00:50:54,709 A larger iso value is-- it's an isosurface, 1047 00:50:54,709 --> 00:50:56,750 so if you make it larger, the place where the iso 1048 00:50:56,750 --> 00:50:58,750 value is larger is closer in. 1049 00:50:58,750 --> 00:51:00,290 So it makes a smaller bubble. 1050 00:51:00,290 --> 00:51:02,626 And a smaller iso value makes a bigger bubble. 1051 00:51:02,626 --> 00:51:04,250 And you can change the number of points 1052 00:51:04,250 --> 00:51:07,400 that it samples to make the surface, change the colors, 1053 00:51:07,400 --> 00:51:09,530 change the opacity of it in terms 1054 00:51:09,530 --> 00:51:12,232 of how you want it to look. 1055 00:51:12,232 --> 00:51:13,190 But I already did this. 1056 00:51:13,190 --> 00:51:17,100 So there is the HOMO. 1057 00:51:17,100 --> 00:51:23,230 So the HOMO is something in the pi manifold. 1058 00:51:23,230 --> 00:51:27,850 It's pi star here. 1059 00:51:27,850 --> 00:51:30,944 And then I can take out the HOMO and show the LUMO. 1060 00:51:30,944 --> 00:51:32,860 And the LUMO is, in this particular basis set, 1061 00:51:32,860 --> 00:51:36,320 not very useful, because I don't have a lot of basis functions. 1062 00:51:36,320 --> 00:51:40,000 So my basis functions, just there's not enough of them 1063 00:51:40,000 --> 00:51:42,844 to really describe the LUMO, or the LUMO plus 1. 1064 00:51:42,844 --> 00:51:44,260 If I want to get a LUMO, we'd need 1065 00:51:44,260 --> 00:51:47,209 a bigger basis set than this. 1066 00:51:47,209 --> 00:51:49,000 So that's the kind of thing that you can do 1067 00:51:49,000 --> 00:51:51,340 with Q-Chem or with Gaussian. 1068 00:51:51,340 --> 00:51:54,220 Gaussian has a GUI called GaussView. 1069 00:51:54,220 --> 00:51:55,526 You can use either one. 1070 00:51:55,526 --> 00:51:57,900 But they both work pretty well, and are pretty intuitive. 1071 00:51:57,900 --> 00:51:59,590 And so I think, on the next problem set, 1072 00:51:59,590 --> 00:52:02,290 you will have some homework problems 1073 00:52:02,290 --> 00:52:05,420 that involve running some calculations with these things. 1074 00:52:05,420 --> 00:52:06,640 And I hope you enjoy it. 1075 00:52:06,640 --> 00:52:09,140 Well, I don't know if homework is ever really enjoyable. 1076 00:52:09,140 --> 00:52:12,428 But I hope it's marginally enjoyable anyway.