1 00:00:00,090 --> 00:00:02,430 The following content is provided under a Creative 2 00:00:02,430 --> 00:00:03,820 Commons license. 3 00:00:03,820 --> 00:00:06,030 Your support will help MIT OpenCourseWare 4 00:00:06,030 --> 00:00:10,120 continue to offer high-quality educational resources for free. 5 00:00:10,120 --> 00:00:12,660 To make a donation or to view additional materials 6 00:00:12,660 --> 00:00:16,620 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:16,620 --> 00:00:17,850 at ocw.mit.edu. 8 00:00:21,510 --> 00:00:25,530 ROBERT FIELD: Last time I talked about LCAO-MO 9 00:00:25,530 --> 00:00:27,690 for diatomic molecules. 10 00:00:27,690 --> 00:00:31,140 And I didn't finish, but the important point 11 00:00:31,140 --> 00:00:33,690 was that this is a toy model. 12 00:00:33,690 --> 00:00:36,840 This is based on a little bit of extension 13 00:00:36,840 --> 00:00:41,980 from something which is not really a toy model, H2 plus, 14 00:00:41,980 --> 00:00:44,830 to, basically, an interpretive framework 15 00:00:44,830 --> 00:00:47,140 that can be applied to, basically, 16 00:00:47,140 --> 00:00:50,200 all diatomic molecules. 17 00:00:50,200 --> 00:00:54,710 And the logic is relatively simple. 18 00:00:54,710 --> 00:00:56,470 We know this one. 19 00:00:56,470 --> 00:01:04,200 We get the molecular orbitals for H2 plus. 20 00:01:04,200 --> 00:01:05,470 There are basically two. 21 00:01:05,470 --> 00:01:08,230 There's a binding one and the anti-binding one. 22 00:01:08,230 --> 00:01:12,110 Of course, there's many more, but we don't care about them. 23 00:01:12,110 --> 00:01:14,200 And then we go to H2. 24 00:01:14,200 --> 00:01:17,560 And again, there's basically only two orbitals 25 00:01:17,560 --> 00:01:22,240 that we care about because the next higher principle quantum 26 00:01:22,240 --> 00:01:28,290 number has such high energy that we can just forget about them, 27 00:01:28,290 --> 00:01:31,620 because those states that derive from the higher principle 28 00:01:31,620 --> 00:01:36,300 quantum number are Rydberg states or complicated things, 29 00:01:36,300 --> 00:01:39,410 because they're at such high energy. 30 00:01:39,410 --> 00:01:46,200 And then it's a very small step to go from hydrogen to the AH 31 00:01:46,200 --> 00:01:49,530 molecules, because, well, we've got 32 00:01:49,530 --> 00:01:56,940 the electronic structure for the A atom, which is complicated-- 33 00:01:56,940 --> 00:02:01,350 more complicated than hydrogen. But because hydrogen only 34 00:02:01,350 --> 00:02:05,610 can make sigma bonds, because the p orbitals in hydrogen 35 00:02:05,610 --> 00:02:11,520 are so high that it's a simpler picture 36 00:02:11,520 --> 00:02:14,460 and can easily be understood. 37 00:02:14,460 --> 00:02:17,850 This is in all the textbooks, and it's 38 00:02:17,850 --> 00:02:22,150 more or less given to you as something to memorize. 39 00:02:22,150 --> 00:02:25,720 But there is a lot more to it than just memorization. 40 00:02:25,720 --> 00:02:29,890 And the important thing is that everything 41 00:02:29,890 --> 00:02:35,780 in LCAO-MO for diatomics is based on the periodic table. 42 00:02:35,780 --> 00:02:40,060 And the periodic table tells you about the ionization 43 00:02:40,060 --> 00:02:44,080 energies-- the periodicites of ionization energies. 44 00:02:44,080 --> 00:02:48,920 And so we can say for any non-integer-- 45 00:02:48,920 --> 00:02:51,190 Well, if we know what the ionization 46 00:02:51,190 --> 00:02:54,730 energy from a particular state is, 47 00:02:54,730 --> 00:02:57,310 we can use that ionization energy 48 00:02:57,310 --> 00:03:02,021 to derive a non-integer principle quantum number. 49 00:03:02,021 --> 00:03:05,370 This is all empirical. 50 00:03:05,370 --> 00:03:09,960 And then use that empirical principle quantum number 51 00:03:09,960 --> 00:03:12,500 to get the size of the orbital. 52 00:03:12,500 --> 00:03:16,290 And basically, the size is everything. 53 00:03:16,290 --> 00:03:18,450 Because everything is based on overlap, 54 00:03:18,450 --> 00:03:21,450 And the internuclear distance molecule 55 00:03:21,450 --> 00:03:26,790 is based on the relative sizes of the different atoms. 56 00:03:26,790 --> 00:03:31,826 And for different electronic states of the atoms, 57 00:03:31,826 --> 00:03:33,450 these sizes are different, because they 58 00:03:33,450 --> 00:03:40,050 have the ionization energy from that level implicitly 59 00:03:40,050 --> 00:03:41,640 expressed. 60 00:03:41,640 --> 00:03:45,390 So because you know about orbital sizes, 61 00:03:45,390 --> 00:03:49,830 and that the different atomic orbitals have different sizes, 62 00:03:49,830 --> 00:03:52,470 you can do an enormous amount as far 63 00:03:52,470 --> 00:03:54,870 as understanding the electronic structure 64 00:03:54,870 --> 00:03:57,920 of all diatomic molecules. 65 00:03:57,920 --> 00:04:06,060 Now, this-- when you have two states which have different 66 00:04:06,060 --> 00:04:07,170 energies-- 67 00:04:07,170 --> 00:04:15,204 We do something like this to describe the molecular orbitals 68 00:04:15,204 --> 00:04:16,079 that arise from them. 69 00:04:16,079 --> 00:04:17,450 That that's perturbation theory. 70 00:04:20,810 --> 00:04:24,490 So the solid line is the dominant character. 71 00:04:24,490 --> 00:04:29,260 The dotted line is the admix character of the other orbital. 72 00:04:29,260 --> 00:04:31,630 And this is all qualitative, but you 73 00:04:31,630 --> 00:04:35,500 have these different atomic orbital energies. 74 00:04:35,500 --> 00:04:38,600 You know them from the periodic table, basically. 75 00:04:38,600 --> 00:04:41,701 And so you can say, yes. 76 00:04:41,701 --> 00:04:43,450 There's going to be two orbitals arriving, 77 00:04:43,450 --> 00:04:46,720 derived from these two states. 78 00:04:46,720 --> 00:04:49,050 And one is polarized towards this atom. 79 00:04:49,050 --> 00:04:51,550 The other is polarized towards that atom. 80 00:04:51,550 --> 00:04:58,050 And so you get the shape of the orbital. 81 00:04:58,050 --> 00:05:03,360 And it's actually useful for saying, well, 82 00:05:03,360 --> 00:05:07,920 if I were to do chemistry, which end of this is electronegative, 83 00:05:07,920 --> 00:05:09,540 and which end is positive? 84 00:05:09,540 --> 00:05:15,370 Or which end of this would attach to a metal surface? 85 00:05:15,370 --> 00:05:21,080 Would it attach pointing into the surface or lying down? 86 00:05:21,080 --> 00:05:22,510 And there's all sorts of insights, 87 00:05:22,510 --> 00:05:25,240 if you can draw these sorts of pictures. 88 00:05:25,240 --> 00:05:28,330 And this is what we do as physical chemists. 89 00:05:28,330 --> 00:05:33,380 We take the crudest model, and we say, OK. 90 00:05:33,380 --> 00:05:36,140 We understand the important features. 91 00:05:36,140 --> 00:05:39,190 And as we discover new, important features, 92 00:05:39,190 --> 00:05:42,230 we build them in, or we forget about them 93 00:05:42,230 --> 00:05:45,440 because they're too subtle. 94 00:05:45,440 --> 00:05:49,550 And we're always looking for something where 95 00:05:49,550 --> 00:05:51,950 the crude picture doesn't work. 96 00:05:51,950 --> 00:05:58,610 And we then find the important thing that is needed. 97 00:05:58,610 --> 00:06:01,640 And for example, if you were to look 98 00:06:01,640 --> 00:06:06,670 at the molecular orbital diagram for C2, which 99 00:06:06,670 --> 00:06:09,689 is a perfectly legitimate gaseous molecule, 100 00:06:09,689 --> 00:06:11,980 you'll see that there's a little bit of ambiguity about 101 00:06:11,980 --> 00:06:14,330 which is the ground state. 102 00:06:14,330 --> 00:06:16,030 And this is an important thing. 103 00:06:16,030 --> 00:06:18,700 And there was a big controversy about this that 104 00:06:18,700 --> 00:06:20,690 was settled by spectroscopy. 105 00:06:20,690 --> 00:06:21,190 OK. 106 00:06:21,190 --> 00:06:25,750 So if you can build an intuitive picture 107 00:06:25,750 --> 00:06:30,850 for all diatomic molecules, you can also 108 00:06:30,850 --> 00:06:35,100 build an intuitive picture for what 109 00:06:35,100 --> 00:06:37,006 we could call chromophores. 110 00:06:40,270 --> 00:06:42,820 So there are a lot of molecules that are 111 00:06:42,820 --> 00:06:46,640 larger than diatomic molecules. 112 00:06:46,640 --> 00:06:51,650 But the electronic structure is mostly nothing, 113 00:06:51,650 --> 00:06:53,900 except a few atoms that are close to each other, where 114 00:06:53,900 --> 00:06:56,960 there's a double bond or there's something special. 115 00:06:56,960 --> 00:07:03,620 And so if you can do LCAO-MO for diatomic molecules, 116 00:07:03,620 --> 00:07:07,250 you can also address electronic structure 117 00:07:07,250 --> 00:07:09,740 in much larger molecules, because it's 118 00:07:09,740 --> 00:07:13,400 really due to a few important atoms 119 00:07:13,400 --> 00:07:17,210 or several different groups of important atoms. 120 00:07:17,210 --> 00:07:19,700 And so the idea is, in here, enable 121 00:07:19,700 --> 00:07:23,495 you to talk about the electronic structure of almost anything. 122 00:07:27,350 --> 00:07:31,010 And one of the things that we care about is spectroscopy. 123 00:07:35,270 --> 00:07:38,600 How do we learn about the structure of a molecule? 124 00:07:38,600 --> 00:07:42,760 And mostly, you learn about it from doing 125 00:07:42,760 --> 00:07:44,200 electronic spectroscopy. 126 00:07:44,200 --> 00:07:47,290 You do transitions between the ground state and some higher 127 00:07:47,290 --> 00:07:48,640 states. 128 00:07:48,640 --> 00:07:51,800 And you want to be able to predict 129 00:07:51,800 --> 00:07:53,670 what you are going to see. 130 00:07:53,670 --> 00:07:57,320 And for example-- and I think you may hear about this 131 00:07:57,320 --> 00:07:59,840 a little bit more-- 132 00:07:59,840 --> 00:08:03,280 if you know the spectrum of nitrogen, 133 00:08:03,280 --> 00:08:08,140 it doesn't start absorbing until far into the vacuum UV. 134 00:08:08,140 --> 00:08:10,480 And then go one atom over-- 135 00:08:10,480 --> 00:08:15,730 the spectrum of oxygen. Well, that defines the vacuum UV. 136 00:08:15,730 --> 00:08:18,640 It starts to absorb around 200 nanometers. 137 00:08:18,640 --> 00:08:25,520 And 2 and 1/2 billion years ago, oxygen 138 00:08:25,520 --> 00:08:29,060 started to enter into the atmosphere and profoundly 139 00:08:29,060 --> 00:08:30,890 changed life on Earth. 140 00:08:30,890 --> 00:08:33,890 Because with the vacuum UV radiation, 141 00:08:33,890 --> 00:08:36,950 nothing could live on the surface of land, 142 00:08:36,950 --> 00:08:39,820 because it would all be killed by this hard UV. 143 00:08:42,500 --> 00:08:49,110 And so everything was underwater at the bottom of the ocean. 144 00:08:49,110 --> 00:08:54,370 But all of a sudden, when oxygen appeared, there was protection. 145 00:08:54,370 --> 00:08:58,600 And then there's additional protection farther out 146 00:08:58,600 --> 00:09:02,310 towards the visible from ozone. 147 00:09:02,310 --> 00:09:09,210 It's not such a good shield, but it absorbs between 200 and 350 148 00:09:09,210 --> 00:09:10,380 nanometers. 149 00:09:10,380 --> 00:09:14,100 And it protects us, too. 150 00:09:14,100 --> 00:09:17,340 But the big thing is oxygen. And why should oxygen 151 00:09:17,340 --> 00:09:20,107 be so different from nitrogen? 152 00:09:20,107 --> 00:09:21,690 And maybe you should think about that. 153 00:09:24,260 --> 00:09:24,760 OK. 154 00:09:24,760 --> 00:09:28,870 I should have mentioned the schedule up there. 155 00:09:28,870 --> 00:09:31,630 I'm going to give another lecture on perturbation 156 00:09:31,630 --> 00:09:34,360 theory-- one of my favorites, which I thought I wasn't 157 00:09:34,360 --> 00:09:37,930 going to get to give on Friday. 158 00:09:37,930 --> 00:09:40,570 And then Troy is going to give two lectures 159 00:09:40,570 --> 00:09:41,710 on quantum chemistry. 160 00:09:41,710 --> 00:09:46,690 And there's going to be significant lab experience. 161 00:09:46,690 --> 00:09:49,270 And you'll hear about that from the TAs. 162 00:09:49,270 --> 00:09:53,380 And so you will do real calculations. 163 00:09:53,380 --> 00:10:01,300 So LCAO-MO could be the organization structure 164 00:10:01,300 --> 00:10:04,000 for quantitative calculations. 165 00:10:04,000 --> 00:10:08,740 But quantitative calculations are usually huge basis sets, 166 00:10:08,740 --> 00:10:12,130 and you lose the LCAO completely. 167 00:10:12,130 --> 00:10:15,820 One could take the results of such a calculation, 168 00:10:15,820 --> 00:10:20,637 and then project it onto a LCAO-MO picture. 169 00:10:20,637 --> 00:10:21,970 And that's generally what we do. 170 00:10:21,970 --> 00:10:26,410 We reduce experiments, we reduce theory, to something 171 00:10:26,410 --> 00:10:28,730 that we can intuit. 172 00:10:28,730 --> 00:10:31,601 And it's really important to have intuition. 173 00:10:31,601 --> 00:10:32,100 OK. 174 00:10:32,100 --> 00:10:42,860 Huckel theory-- Huckel theory is another kind 175 00:10:42,860 --> 00:10:49,570 of non-rigorous theory. 176 00:10:49,570 --> 00:10:54,070 In fact, it's laughable in its simplicity. 177 00:10:54,070 --> 00:10:59,770 And the idea is that even such a crude theory can make sense out 178 00:10:59,770 --> 00:11:03,790 of huge families of molecules and enable 179 00:11:03,790 --> 00:11:06,910 you to be quantitative about things without having 180 00:11:06,910 --> 00:11:09,550 to do a huge calculation. 181 00:11:09,550 --> 00:11:15,090 Now, you almost always have to diagonize a matrix. 182 00:11:15,090 --> 00:11:19,050 And because usually, in Huckel theory, 183 00:11:19,050 --> 00:11:25,420 you're dealing with more than two or three relevant orbitals, 184 00:11:25,420 --> 00:11:27,110 it's a big deal. 185 00:11:27,110 --> 00:11:30,070 I don't know whether you have experience 186 00:11:30,070 --> 00:11:32,200 with matrix diagonalization routines, 187 00:11:32,200 --> 00:11:33,670 but there are many of them. 188 00:11:33,670 --> 00:11:35,620 And you're going to have experience with them. 189 00:11:39,610 --> 00:11:43,300 But the things you put into them have absolutely nothing 190 00:11:43,300 --> 00:11:45,730 to do with a real Hamiltonian. 191 00:11:45,730 --> 00:11:48,130 It's a make up picture. 192 00:11:48,130 --> 00:11:51,610 It's a picture based on, basically, two parameters-- 193 00:11:51,610 --> 00:11:52,810 alpha and beta. 194 00:11:52,810 --> 00:11:55,000 And I'll talk about this. 195 00:11:55,000 --> 00:11:59,590 And these are parameters that are agreed upon 196 00:11:59,590 --> 00:12:01,450 by international committee-- 197 00:12:01,450 --> 00:12:02,650 It's not a committee. 198 00:12:02,650 --> 00:12:05,020 But people say, OK. 199 00:12:05,020 --> 00:12:06,850 This is the value of alpha and beta 200 00:12:06,850 --> 00:12:09,805 that we're going to use to describe many problems. 201 00:12:12,830 --> 00:12:15,700 But I just want to make sure you understand 202 00:12:15,700 --> 00:12:19,270 that Huckel theory is only a little bit more 203 00:12:19,270 --> 00:12:25,250 ridiculous than LCAO-MO theory. 204 00:12:25,250 --> 00:12:32,120 Because in LCAO-MO, you really have some semi-empirical sense 205 00:12:32,120 --> 00:12:33,360 of how big orbitals are. 206 00:12:33,360 --> 00:12:34,984 And once you know how big orbitals are, 207 00:12:34,984 --> 00:12:36,900 you know what's going on. 208 00:12:36,900 --> 00:12:40,840 In this, you just have a couple of parameters that says, OK. 209 00:12:40,840 --> 00:12:41,910 These are the rules. 210 00:12:41,910 --> 00:12:45,360 Let's now apply that. 211 00:12:45,360 --> 00:12:45,860 OK. 212 00:12:48,788 --> 00:12:50,730 I'm carrying around too much stuff. 213 00:12:53,580 --> 00:13:00,420 So organic chemists are really wonderful, 214 00:13:00,420 --> 00:13:04,740 because they give you an abbreviated way of drawing 215 00:13:04,740 --> 00:13:06,660 a molecular structure. 216 00:13:06,660 --> 00:13:09,570 And almost everyone in this room is 217 00:13:09,570 --> 00:13:11,880 gifted in being able to visualize 218 00:13:11,880 --> 00:13:13,860 three-dimensional structures. 219 00:13:13,860 --> 00:13:16,800 Physical chemists tend not to be so gifted. 220 00:13:16,800 --> 00:13:20,100 But here is something where we don't have any hydrogens-- 221 00:13:20,100 --> 00:13:22,590 I mean, there are hydrogens, but they're implied. 222 00:13:22,590 --> 00:13:24,720 And we don't put carbons here. 223 00:13:24,720 --> 00:13:28,780 We just say, at every vertex, there is a carbon atom. 224 00:13:28,780 --> 00:13:31,140 And so we consider these things. 225 00:13:31,140 --> 00:13:36,070 And every organic chemist knows, if I draw something like this, 226 00:13:36,070 --> 00:13:39,390 that either I was stupid and I drew 227 00:13:39,390 --> 00:13:44,110 something that was impossible or what the structure is. 228 00:13:44,110 --> 00:13:44,610 OK. 229 00:13:44,610 --> 00:13:46,770 Now, this is a conjugated system. 230 00:13:46,770 --> 00:13:49,630 We have double bonds and single bonds. 231 00:13:49,630 --> 00:13:55,980 And that kind of thing is known to be unusually stable. 232 00:13:55,980 --> 00:14:00,870 And in order for it to work, it has to be planar. 233 00:14:00,870 --> 00:14:04,890 If you have a conjugated system which is not planar, 234 00:14:04,890 --> 00:14:09,180 then it's not as stable. 235 00:14:09,180 --> 00:14:17,280 So Huckel theory is based on the existence of unusual stability 236 00:14:17,280 --> 00:14:20,600 of conjugated systems. 237 00:14:20,600 --> 00:14:28,730 And it can be extended to non-conjugated systems-- 238 00:14:28,730 --> 00:14:32,870 to non-planar systems-- and we have rules for how to do that. 239 00:14:32,870 --> 00:14:41,220 But the simple rules are, you start with a picture, 240 00:14:41,220 --> 00:14:44,660 and you can write down a Hamiltonian. 241 00:14:44,660 --> 00:14:48,130 And it's a toy model, but it's still something 242 00:14:48,130 --> 00:14:49,990 that the computer has to diagonalize. 243 00:14:58,440 --> 00:15:02,990 So when we're doing a variational calculation 244 00:15:02,990 --> 00:15:04,790 expressed in matrix language-- 245 00:15:04,790 --> 00:15:06,740 We have a Hamiltonian matrix. 246 00:15:06,740 --> 00:15:15,210 We have eigenvectors, and we have eigenvalues. 247 00:15:15,210 --> 00:15:18,600 And we have the overlap matrix. 248 00:15:18,600 --> 00:15:26,580 And we have something where this is the alpha-- 249 00:15:26,580 --> 00:15:32,580 where, here, we can have non-orthonormal basis 250 00:15:32,580 --> 00:15:34,050 functions. 251 00:15:34,050 --> 00:15:37,640 And this makes them normalized. 252 00:15:37,640 --> 00:15:43,290 So these are the orthonormalized basis functions or basis 253 00:15:43,290 --> 00:15:44,470 vectors. 254 00:15:44,470 --> 00:15:47,010 And this is the kind of equation we have to solve. 255 00:15:47,010 --> 00:15:50,340 It's the generalized eigenvalue equation. 256 00:15:50,340 --> 00:15:54,420 And we don't like this, because it 257 00:15:54,420 --> 00:15:57,930 doesn't have the simplicity of just a Hamiltonian, where 258 00:15:57,930 --> 00:16:02,040 we have matrix elements along the diagonal and somewhere 259 00:16:02,040 --> 00:16:03,090 else. 260 00:16:03,090 --> 00:16:08,790 And the idea is, we want to take the secular determinate 261 00:16:08,790 --> 00:16:14,400 and make it 0 by adjusting values of the energy 262 00:16:14,400 --> 00:16:16,830 differences along the diagonal. 263 00:16:16,830 --> 00:16:20,770 But when we have this overlap matrix, it's complicated. 264 00:16:20,770 --> 00:16:24,360 Now, there's a way of dealing with the generalized eigenvalue 265 00:16:24,360 --> 00:16:31,890 equations, but one way to deal with it is to say s 266 00:16:31,890 --> 00:16:34,065 is equal to the unit matrix. 267 00:16:39,910 --> 00:16:41,900 You can do anything you want. 268 00:16:41,900 --> 00:16:47,620 And it's basically saying, there is no overlap between orbitals 269 00:16:47,620 --> 00:16:49,902 on adjacent atoms. 270 00:16:49,902 --> 00:16:51,610 We're going to neglect it, and then we're 271 00:16:51,610 --> 00:16:54,520 going to bring it back if we need it. 272 00:16:54,520 --> 00:16:56,500 But it's a wonderful simplification, 273 00:16:56,500 --> 00:17:01,840 because it enables you to write a simple, effective 274 00:17:01,840 --> 00:17:09,230 Hamiltonian, which looks just like h c alpha is 275 00:17:09,230 --> 00:17:13,800 equal to e alpha c alpha. 276 00:17:13,800 --> 00:17:14,300 OK. 277 00:17:14,300 --> 00:17:17,530 This, we know how to solve. 278 00:17:17,530 --> 00:17:19,890 And we can use the same procedure. 279 00:17:26,420 --> 00:17:27,940 OK. 280 00:17:27,940 --> 00:17:28,960 So these are the rules. 281 00:17:39,550 --> 00:17:45,760 If we have a planar molecule, we can say there are p orbitals-- 282 00:17:45,760 --> 00:17:49,030 one on each carbon atom or one on each atom that's 283 00:17:49,030 --> 00:17:54,980 not hydrogen, which are perpendicular to the plane 284 00:17:54,980 --> 00:17:57,260 of the molecule. 285 00:17:57,260 --> 00:18:00,730 So easy orbitals. 286 00:18:00,730 --> 00:18:04,330 And those are special. 287 00:18:04,330 --> 00:18:07,510 They give rise to pi bonds. 288 00:18:07,510 --> 00:18:14,470 Pi bonds are bonds where there is one plane of symmetry 289 00:18:14,470 --> 00:18:25,510 containing the bond, And then there are p x, and p y, and s. 290 00:18:25,510 --> 00:18:27,255 And these give rise to sigma bonds. 291 00:18:29,850 --> 00:18:35,210 So we have pi bonds and sigma bonds. 292 00:18:35,210 --> 00:18:38,554 Never the twain shall meet. 293 00:18:38,554 --> 00:18:40,220 And we don't care about the sigma bonds, 294 00:18:40,220 --> 00:18:42,530 because anybody can make sigma bonds. 295 00:18:42,530 --> 00:18:48,950 But only the special, perpendicular-to-the-plane 296 00:18:48,950 --> 00:18:52,280 guys, which are responsible for the fact that the molecule 297 00:18:52,280 --> 00:18:53,690 likes to be planar. 298 00:18:53,690 --> 00:18:58,280 And so we're only going to consider these p z orbitals-- 299 00:18:58,280 --> 00:18:59,530 one on each atom. 300 00:19:04,580 --> 00:19:12,080 So we don't care about no pi-sigma interactions. 301 00:19:15,540 --> 00:19:18,160 And we're going to neglect the sigma orbitals, because they 302 00:19:18,160 --> 00:19:19,571 take care of themselves. 303 00:19:22,277 --> 00:19:25,610 And we can do anything we want. 304 00:19:25,610 --> 00:19:28,130 It's just a question of, if we make 305 00:19:28,130 --> 00:19:31,700 too many ridiculous assumptions, we'll get ridiculous results. 306 00:19:31,700 --> 00:19:34,880 And this has been time tested, and it 307 00:19:34,880 --> 00:19:36,830 gives pretty useful stuff. 308 00:19:36,830 --> 00:19:39,980 And it provides a framework for making arguments 309 00:19:39,980 --> 00:19:44,030 about molecular structure and molecular reactivity. 310 00:19:44,030 --> 00:19:48,060 In organic chemistry you learn about resonance forms. 311 00:19:48,060 --> 00:19:52,340 And this is compatible with generating the resonance forms 312 00:19:52,340 --> 00:19:54,650 and saying, what is the relative importance? 313 00:19:54,650 --> 00:19:56,930 And what is the charge distribution, 314 00:19:56,930 --> 00:19:59,360 and bond strength, and everything like that? 315 00:19:59,360 --> 00:20:04,080 So it's really useful using the most primitive tools 316 00:20:04,080 --> 00:20:08,240 that organic chemists introduce at an early stage 317 00:20:08,240 --> 00:20:09,829 in your education. 318 00:20:09,829 --> 00:20:11,870 And that's one of the reasons why a lot of people 319 00:20:11,870 --> 00:20:14,395 become organic chemists, because it's so beautiful. 320 00:20:17,310 --> 00:20:18,990 OK. 321 00:20:18,990 --> 00:20:25,270 So our h i j matrix-- 322 00:20:25,270 --> 00:20:29,550 So we have a bunch of matrix elements, and we say, OK. 323 00:20:29,550 --> 00:20:34,080 h i i is equal to alpha. 324 00:20:34,080 --> 00:20:41,950 So every carbon atom has its alpha value. 325 00:20:41,950 --> 00:20:44,760 It's the same for all carbon atoms, 326 00:20:44,760 --> 00:20:46,395 regardless of who is nearby. 327 00:20:49,180 --> 00:20:56,970 And we have h I i plus or minus 1 adjacent atoms, 328 00:20:56,970 --> 00:20:59,240 and it's beta. 329 00:20:59,240 --> 00:20:59,850 That's it. 330 00:20:59,850 --> 00:21:03,380 That's the whole ball game. 331 00:21:03,380 --> 00:21:06,670 And it's really a simple-- 332 00:21:06,670 --> 00:21:10,750 There's far less here than in LCAO-MO. 333 00:21:13,830 --> 00:21:15,390 But it's still a toy. 334 00:21:15,390 --> 00:21:19,380 Both are toy models, and they're both very useful 335 00:21:19,380 --> 00:21:21,880 OK. 336 00:21:21,880 --> 00:21:31,875 And everything else which is not diagonal or off diagonal by 1-- 337 00:21:31,875 --> 00:21:32,375 0. 338 00:21:36,070 --> 00:21:38,080 That's really convenient. 339 00:21:38,080 --> 00:21:41,830 And so you can draw the h matrix, 340 00:21:41,830 --> 00:21:46,210 regardless of what it looks like, as alpha, alpha, alpha, 341 00:21:46,210 --> 00:21:49,120 alpha, et cetera, along the diagonal, 342 00:21:49,120 --> 00:21:56,430 and beta, beta, beta, et cetera, beta, along the near diagonal. 343 00:21:56,430 --> 00:22:00,240 And if it's a ring, you have a beta here, here, and here. 344 00:22:03,740 --> 00:22:05,060 So that's pretty simple. 345 00:22:05,060 --> 00:22:08,200 So we have 0s, 0s-- 346 00:22:08,200 --> 00:22:12,410 so that it has a tri-diagonal structure with something 347 00:22:12,410 --> 00:22:14,360 up here and there. 348 00:22:14,360 --> 00:22:17,880 Never forget this thing here and there, 349 00:22:17,880 --> 00:22:20,280 which is present when you have a ring, 350 00:22:20,280 --> 00:22:23,440 and it's not present when you don't. 351 00:22:23,440 --> 00:22:23,940 That's it. 352 00:22:23,940 --> 00:22:25,680 That's Huckel theory. 353 00:22:25,680 --> 00:22:27,710 It's just there. 354 00:22:27,710 --> 00:22:28,290 OK. 355 00:22:28,290 --> 00:22:34,960 And so now, things can be more complicated. 356 00:22:41,420 --> 00:22:43,150 So if we're not content with what 357 00:22:43,150 --> 00:22:45,370 we get from the really primitive theory, 358 00:22:45,370 --> 00:22:48,370 we can do something like saying, well, we 359 00:22:48,370 --> 00:22:53,450 can make beta be dependent on internuclear distance. 360 00:22:53,450 --> 00:22:55,240 If the molecule, for some reason, 361 00:22:55,240 --> 00:22:58,550 is constrained to have not equal bonds lengths. 362 00:22:58,550 --> 00:23:02,690 So we can add an additional parameter-- some kind of reason 363 00:23:02,690 --> 00:23:06,500 for this beta to be dependent on r. 364 00:23:06,500 --> 00:23:10,230 But that's already getting sophisticated. 365 00:23:10,230 --> 00:23:16,770 And heteroatoms-- In other words, 366 00:23:16,770 --> 00:23:19,970 if you have a nitrogen instead of a carbon in a benzene-type 367 00:23:19,970 --> 00:23:22,700 ring, you can have-- 368 00:23:22,700 --> 00:23:27,550 So, well, nitrogen is different from carbon. 369 00:23:27,550 --> 00:23:29,465 It has a different-- 370 00:23:29,465 --> 00:23:32,740 In LCAO theory, the distance-- 371 00:23:32,740 --> 00:23:35,110 The ionization energy for nitrogen 372 00:23:35,110 --> 00:23:36,520 is different from carbon. 373 00:23:36,520 --> 00:23:38,550 It's larger. 374 00:23:38,550 --> 00:23:42,290 And so heteroatoms can be included if you 375 00:23:42,290 --> 00:23:44,990 use a different value of alpha. 376 00:23:44,990 --> 00:23:50,160 And now, alpha and beta are both negative. 377 00:23:50,160 --> 00:23:52,821 Now, this is a little bit fraudulent. 378 00:23:52,821 --> 00:23:53,320 Yeah. 379 00:23:53,320 --> 00:23:55,540 AUDIENCE: Do you also have to pick a new beta? 380 00:23:55,540 --> 00:23:56,360 Or-- 381 00:23:56,360 --> 00:23:57,190 ROBERT FIELD: Yes. 382 00:23:57,190 --> 00:23:58,900 But the main thing is alpha. 383 00:23:58,900 --> 00:24:00,660 AUDIENCE: I see. 384 00:24:00,660 --> 00:24:01,870 Why is that? 385 00:24:01,870 --> 00:24:02,980 ROBERT FIELD: Why is that? 386 00:24:02,980 --> 00:24:11,410 Because beta comes from overlap, even though we're neglecting 387 00:24:11,410 --> 00:24:12,950 overlap. 388 00:24:12,950 --> 00:24:19,150 And so the bond distances between normal and heteroatoms 389 00:24:19,150 --> 00:24:22,880 are not usually that different. 390 00:24:22,880 --> 00:24:26,320 But the thing is, you put what you need into the model. 391 00:24:26,320 --> 00:24:28,870 And the first thing you do is, you 392 00:24:28,870 --> 00:24:30,400 solve the most simple model. 393 00:24:30,400 --> 00:24:33,160 And you say, this is not quite what I wanted. 394 00:24:33,160 --> 00:24:36,920 And so I allow a couple of extra degrees of freedom. 395 00:24:36,920 --> 00:24:40,760 And it's really instructive how these things work. 396 00:24:40,760 --> 00:24:44,950 But the thing is, you're not calculating the matrix element 397 00:24:44,950 --> 00:24:46,930 of a real operator. 398 00:24:46,930 --> 00:24:48,920 It's all make believe. 399 00:24:48,920 --> 00:24:52,810 But it's really powerful, because what you're comparing 400 00:24:52,810 --> 00:24:55,090 is families of molecules. 401 00:24:55,090 --> 00:24:58,400 And the reality might be really complicated, 402 00:24:58,400 --> 00:25:00,940 but the complexity in each member of the family 403 00:25:00,940 --> 00:25:02,320 is pretty much the same. 404 00:25:02,320 --> 00:25:04,960 And what it's allowing you to see 405 00:25:04,960 --> 00:25:07,042 is, what are the differences? 406 00:25:07,042 --> 00:25:10,890 It allows you to see the big picture. 407 00:25:10,890 --> 00:25:12,420 I love this. 408 00:25:12,420 --> 00:25:15,900 And in fact, at an early stage of my education, 409 00:25:15,900 --> 00:25:18,250 I thought Huckel theory was wonderful. 410 00:25:18,250 --> 00:25:21,130 And it was what got me interested in quantum 411 00:25:21,130 --> 00:25:21,630 mechanics. 412 00:25:21,630 --> 00:25:24,327 Because you normally see Huckel theory before you 413 00:25:24,327 --> 00:25:25,910 know anything about quantum mechanics, 414 00:25:25,910 --> 00:25:28,190 because it's just a game. 415 00:25:28,190 --> 00:25:30,120 OK So heteroatoms-- 416 00:25:30,120 --> 00:25:33,570 You can fiddle with alpha. 417 00:25:33,570 --> 00:25:40,180 Now, the ionization energies for carbon and nitrogen 418 00:25:40,180 --> 00:25:42,371 are not that different. 419 00:25:42,371 --> 00:25:42,870 I'm sorry. 420 00:25:42,870 --> 00:25:46,410 They're very different, but the effect 421 00:25:46,410 --> 00:25:50,910 on the alpha value in Huckel theory is very small. 422 00:25:50,910 --> 00:25:53,000 Well, so it is. 423 00:25:53,000 --> 00:26:00,170 But the more electronegative or the higher the ionization 424 00:26:00,170 --> 00:26:05,660 energy, the alpha value becomes increasingly negative. 425 00:26:05,660 --> 00:26:07,790 Now, I was starting to say something is fraudulent, 426 00:26:07,790 --> 00:26:12,590 and I was distracted by a really good question. 427 00:26:12,590 --> 00:26:14,840 Alpha is on the diagonal. 428 00:26:14,840 --> 00:26:21,840 And we know we can determine the sign of a diagonal element. 429 00:26:21,840 --> 00:26:25,400 And we know we can't determine the sign 430 00:26:25,400 --> 00:26:28,070 of an off-diagonal element. 431 00:26:28,070 --> 00:26:32,300 But we say that alpha and beta are both less than 0. 432 00:26:34,890 --> 00:26:39,350 And the reason for this is that, when you saw the secular 433 00:26:39,350 --> 00:26:41,600 equation, especially-- 434 00:26:45,130 --> 00:26:48,820 You get two eigenvalues-- one where 435 00:26:48,820 --> 00:26:52,360 you have minus beta and one where you have plus beta. 436 00:26:52,360 --> 00:26:54,640 And so in a sense, beta is present, 437 00:26:54,640 --> 00:26:56,680 but it's just a sign choice. 438 00:26:56,680 --> 00:27:00,520 And since alpha and beta have to do with stability, 439 00:27:00,520 --> 00:27:03,200 we just say alpha is negative. 440 00:27:03,200 --> 00:27:04,000 We know that. 441 00:27:04,000 --> 00:27:08,900 And beta is chosen to be always negative. 442 00:27:08,900 --> 00:27:12,310 There's no, you could have had beta be positive. 443 00:27:12,310 --> 00:27:13,840 And you could do all the theory. 444 00:27:13,840 --> 00:27:18,961 It's just a lot more complicated explaining all the cases. 445 00:27:18,961 --> 00:27:19,460 All right. 446 00:27:19,460 --> 00:27:23,738 So let's continue with this. 447 00:27:23,738 --> 00:27:25,490 And so we can do heteroatoms. 448 00:27:32,840 --> 00:27:37,150 So if we have a non-planar system, 449 00:27:37,150 --> 00:27:42,190 we can say that beta is a function of the dihedral angle. 450 00:27:42,190 --> 00:27:43,160 We can put that in. 451 00:27:43,160 --> 00:27:45,290 We can do anything we want. 452 00:27:45,290 --> 00:27:47,840 We have a molecule. 453 00:27:47,840 --> 00:27:49,840 We're trying to describe its properties 454 00:27:49,840 --> 00:27:53,020 relative to the normal members of the group, which 455 00:27:53,020 --> 00:27:55,570 are planar and no heteroatoms. 456 00:27:55,570 --> 00:27:58,120 And we can do stuff that will accommodate 457 00:27:58,120 --> 00:28:02,020 these interesting differences, which you can impose 458 00:28:02,020 --> 00:28:06,310 by you putting the molecule in a constrained environment 459 00:28:06,310 --> 00:28:11,970 or doing stuff that distorts the geometry. 460 00:28:11,970 --> 00:28:13,855 So we do this. 461 00:28:17,700 --> 00:28:19,700 So when we do this, we get a Hamiltonian. 462 00:28:19,700 --> 00:28:25,960 We get the energies of the orbitals, 463 00:28:25,960 --> 00:28:35,370 and we get the eigenvector that corresponds to each energy. 464 00:28:35,370 --> 00:28:37,570 And the total energy involves the sum 465 00:28:37,570 --> 00:28:41,500 over the energies of each of the occupied orbitals-- the number 466 00:28:41,500 --> 00:28:43,240 of electrons in that. 467 00:28:43,240 --> 00:28:50,485 And we can also get bond order and charge. 468 00:28:53,608 --> 00:28:58,660 So sometimes, we want to know, what 469 00:28:58,660 --> 00:29:01,990 is the charge on each atom if the charge is 470 00:29:01,990 --> 00:29:04,880 going to be different from 0? 471 00:29:04,880 --> 00:29:08,570 Because that also controls chemistry. 472 00:29:08,570 --> 00:29:12,110 Negatively charged atoms are sought out 473 00:29:12,110 --> 00:29:14,300 by certain classes of reactants. 474 00:29:14,300 --> 00:29:18,370 Anyway, so you get all this stuff. 475 00:29:18,370 --> 00:29:21,510 And of course, some of it will require 476 00:29:21,510 --> 00:29:27,000 a little bit of patchwork, but you do this for your career. 477 00:29:27,000 --> 00:29:29,160 And you discover that there are certain things 478 00:29:29,160 --> 00:29:31,210 that I know how to handle. 479 00:29:31,210 --> 00:29:33,510 And I use my favorite parameters for it. 480 00:29:33,510 --> 00:29:36,520 And you get closer to the truth. 481 00:29:36,520 --> 00:29:39,400 And you always want to be surprised, 482 00:29:39,400 --> 00:29:46,150 because when the crude theory cannot be made consistent with 483 00:29:46,150 --> 00:29:49,630 observations, you know you did something special. 484 00:29:49,630 --> 00:29:54,040 You have a molecule, which has a property which is unexpected, 485 00:29:54,040 --> 00:29:54,880 and we like that. 486 00:29:58,360 --> 00:29:59,930 OK. 487 00:29:59,930 --> 00:30:05,820 So when you're doing quantum chemistry, 488 00:30:05,820 --> 00:30:06,900 there are five steps-- 489 00:30:06,900 --> 00:30:11,060 or when you're doing molecular orbital theory. 490 00:30:11,060 --> 00:30:15,200 And this is one of Troy's rules. 491 00:30:15,200 --> 00:30:17,210 We have a five-step procedure. 492 00:30:17,210 --> 00:30:25,580 And we define the atomic orbital basis set. 493 00:30:28,940 --> 00:30:31,550 And the basis set is one p z orbital per atom. 494 00:30:35,160 --> 00:30:45,280 And so we can have a molecular orbital, which is the sum i 495 00:30:45,280 --> 00:30:52,260 equals 1 to n c i u p z i. 496 00:30:52,260 --> 00:30:56,160 So this is the p z orbital on the i-th atom. 497 00:30:56,160 --> 00:31:00,116 This is the coefficient for this particular linear combination. 498 00:31:04,141 --> 00:31:04,640 OK. 499 00:31:04,640 --> 00:31:09,190 So we have a bunch of molecular orbitals. 500 00:31:09,190 --> 00:31:16,140 Next step-- compute h and s. 501 00:31:16,140 --> 00:31:21,860 But s is 0 because we made this ridiculous assumption. 502 00:31:21,860 --> 00:31:24,510 It's convenient. 503 00:31:24,510 --> 00:31:28,090 This is called the complete neglect of overlap. 504 00:31:28,090 --> 00:31:30,430 And we can do that. 505 00:31:30,430 --> 00:31:32,730 It's wrong. 506 00:31:32,730 --> 00:31:36,900 But when you include overlap, it leads 507 00:31:36,900 --> 00:31:39,300 to greater complexity of the calculation, 508 00:31:39,300 --> 00:31:42,550 and not much improvement in the results. 509 00:31:42,550 --> 00:31:47,160 And so we normally don't worry about the overlap. 510 00:31:47,160 --> 00:31:49,770 So all we care about is this. 511 00:31:49,770 --> 00:31:56,040 And as I said, h has this structure 512 00:31:56,040 --> 00:32:01,360 of alpha, beta, beta-- 513 00:32:01,360 --> 00:32:04,301 tri-diagonal structure and maybe something up here. 514 00:32:04,301 --> 00:32:04,800 That's it. 515 00:32:08,700 --> 00:32:15,360 Then we diagonalize the Hamiltonian. 516 00:32:18,420 --> 00:32:24,680 And so this gives us the eigenvectors or eigenvalues-- 517 00:32:24,680 --> 00:32:28,010 the energies and eigenvectors associated 518 00:32:28,010 --> 00:32:31,070 with each orbital energy. 519 00:32:31,070 --> 00:32:34,870 So we fill electrons into orbitals. 520 00:32:34,870 --> 00:32:40,710 Now, for benzene, what you would get-- 521 00:32:40,710 --> 00:32:43,150 Now, this is important. 522 00:32:43,150 --> 00:32:47,340 How many carbon atoms in benzene? 523 00:32:47,340 --> 00:32:48,310 Right. 524 00:32:48,310 --> 00:32:50,890 And how many orbitals will you get 525 00:32:50,890 --> 00:32:54,240 from six primitive orbitals? 526 00:32:54,240 --> 00:32:55,440 Right. 527 00:32:55,440 --> 00:32:58,800 And so there are six energy levels. 528 00:32:58,800 --> 00:33:04,530 And it happens that when you solve the secular equation, 529 00:33:04,530 --> 00:33:05,940 you get this pattern. 530 00:33:16,450 --> 00:33:19,090 This is a problem where the number of nodal planes 531 00:33:19,090 --> 00:33:21,160 determines the order of energy. 532 00:33:21,160 --> 00:33:24,070 And so if you have benzene, you can 533 00:33:24,070 --> 00:33:29,740 imagine that the orbitals that you can have will be no nodes-- 534 00:33:29,740 --> 00:33:32,830 nodal planes-- one nodal plane. 535 00:33:32,830 --> 00:33:36,130 And you could have the nodal planes between the atoms 536 00:33:36,130 --> 00:33:38,410 or through atoms-- 537 00:33:38,410 --> 00:33:42,860 and two nodal planes or three. 538 00:33:42,860 --> 00:33:46,680 You can't have any more with six atoms. 539 00:33:46,680 --> 00:33:52,490 And so you almost don't have to solve this secular equation 540 00:33:52,490 --> 00:33:53,210 at all. 541 00:33:53,210 --> 00:33:56,480 You can anticipate what the structure 542 00:33:56,480 --> 00:34:00,630 is going to be just by saying, no nodes, one node, two 543 00:34:00,630 --> 00:34:02,240 node, three node. 544 00:34:02,240 --> 00:34:07,140 And you can even anticipate double degeneracies, 545 00:34:07,140 --> 00:34:09,989 because if you have two nodal planes, 546 00:34:09,989 --> 00:34:17,489 you can put them through opposite bonds 547 00:34:17,489 --> 00:34:19,540 or through opposite atoms. 548 00:34:19,540 --> 00:34:25,060 And those are examples of the forms that you would deal with. 549 00:34:25,060 --> 00:34:28,949 The only thing you can't do is to know what the order 550 00:34:28,949 --> 00:34:31,389 the energies are. 551 00:34:31,389 --> 00:34:32,860 But there are tricks for that, too. 552 00:34:35,949 --> 00:34:38,380 The tricks for that include-- 553 00:34:38,380 --> 00:34:42,244 The sum of the eigenvalues is equal to sum 554 00:34:42,244 --> 00:34:43,160 of the diagonal nodes. 555 00:34:46,210 --> 00:34:50,080 And so we know that the six orbital energies 556 00:34:50,080 --> 00:34:52,530 will sum to 0-- 557 00:34:52,530 --> 00:34:55,557 will sum to 6 alpha. 558 00:34:55,557 --> 00:34:58,100 The betas go away. 559 00:34:58,100 --> 00:35:00,620 And there are other tricks that you can do, 560 00:35:00,620 --> 00:35:02,780 but generally, you solve the equation. 561 00:35:02,780 --> 00:35:09,110 You don't want to push your requirements for symmetry 562 00:35:09,110 --> 00:35:10,010 too far. 563 00:35:10,010 --> 00:35:13,040 So we end up doing that. 564 00:35:13,040 --> 00:35:18,550 And then stick diagrams-- 565 00:35:22,700 --> 00:35:27,500 We fill electrons into orbitals in energy order. 566 00:35:27,500 --> 00:35:29,970 And for benzene, there are six of them. 567 00:35:29,970 --> 00:35:35,675 And so this is, then, the lowest energy state of benzene. 568 00:35:38,740 --> 00:35:43,570 This is called the independent electron approximation. 569 00:35:43,570 --> 00:35:46,010 They don't know about each other. 570 00:35:46,010 --> 00:35:48,860 This is illegal. 571 00:35:48,860 --> 00:35:50,360 But it's legal in a sense, if you 572 00:35:50,360 --> 00:35:52,730 have two electrons in every orbital, 573 00:35:52,730 --> 00:35:54,950 you have nothing but singlet states. 574 00:35:54,950 --> 00:35:58,040 The ground state is always going to be a singlet state 575 00:35:58,040 --> 00:36:01,130 unless you have something really weird going on-- 576 00:36:01,130 --> 00:36:03,450 like in O2-- 577 00:36:03,450 --> 00:36:05,690 But that's not Huckel theory. 578 00:36:05,690 --> 00:36:09,190 But it's examinable. 579 00:36:09,190 --> 00:36:11,180 OK. 580 00:36:11,180 --> 00:36:15,660 Why does oxygen have a triplet ground state? 581 00:36:15,660 --> 00:36:18,465 That's something that every textbook says. 582 00:36:18,465 --> 00:36:19,340 You got to know that. 583 00:36:19,340 --> 00:36:21,830 And of course, they don't really tell you anything more 584 00:36:21,830 --> 00:36:24,301 except something to memorize. 585 00:36:24,301 --> 00:36:24,800 OK. 586 00:36:24,800 --> 00:36:27,950 So generally, you have two electrons 587 00:36:27,950 --> 00:36:30,940 in each orbital in singlets. 588 00:36:30,940 --> 00:36:33,190 Now, if you're going to do spectroscopy 589 00:36:33,190 --> 00:36:36,850 you would perhaps promote one of these guys to a higher state. 590 00:36:39,400 --> 00:36:43,270 And so you'll have singlets and triplets. 591 00:36:43,270 --> 00:36:45,400 But of course the only transition you would see 592 00:36:45,400 --> 00:36:47,590 would be a singlet-to-singlet transition. 593 00:36:47,590 --> 00:36:50,040 So you might as well forget about triplets. 594 00:36:50,040 --> 00:36:52,570 And you might as well forget about having 595 00:36:52,570 --> 00:36:53,650 to add [INAUDIBLE]. 596 00:36:56,270 --> 00:37:00,750 You can get away with murder, especially with Huckel theory. 597 00:37:00,750 --> 00:37:03,230 So we have a stick diagram. 598 00:37:03,230 --> 00:37:05,750 We fill electrons into the thing. 599 00:37:05,750 --> 00:37:17,440 And then we compute the energy of the many electron problem. 600 00:37:17,440 --> 00:37:20,990 And so let's just do this. 601 00:37:20,990 --> 00:37:22,490 And you've all seen this. 602 00:37:35,560 --> 00:37:38,980 So we number the atoms. 603 00:37:38,980 --> 00:37:42,580 We have our symmetric structure. 604 00:37:42,580 --> 00:37:52,375 And psi mu is going to be sum from i equals 1 to 6. 605 00:37:52,375 --> 00:37:59,106 C i mu p z i. 606 00:37:59,106 --> 00:38:12,140 And c mu is c 1 mu c 2 mu to c 6 mu. 607 00:38:12,140 --> 00:38:13,690 These are the mixing coefficients. 608 00:38:16,520 --> 00:38:18,430 Well, for benzene, it's pretty simple 609 00:38:18,430 --> 00:38:24,370 because you know that if you have no nodes and symmetry, 610 00:38:24,370 --> 00:38:27,470 all of these are going to be the same. 611 00:38:27,470 --> 00:38:30,540 And so if you put 1s here, you put a 1 over square root of 6 612 00:38:30,540 --> 00:38:34,410 out in front for normalization. 613 00:38:34,410 --> 00:38:38,160 And you can figure out the eigenvectors for benzene-- 614 00:38:38,160 --> 00:38:41,490 all of them-- just by counting the number of nodes. 615 00:38:41,490 --> 00:38:45,660 And that's useful, because if you know the eigenvectors, then 616 00:38:45,660 --> 00:38:48,480 you can show what the eigenvalue is 617 00:38:48,480 --> 00:38:51,993 by multiplying the original Hamiltonian by an eigenvalue. 618 00:38:55,470 --> 00:39:01,380 So anyway, we have this. 619 00:39:01,380 --> 00:39:05,580 And then the Hamiltonian-- 620 00:39:05,580 --> 00:39:12,700 we have alpha, beta, and then 0s. 621 00:39:12,700 --> 00:39:17,610 And beta, alpha, beta, and then 0s, et cetera. 622 00:39:17,610 --> 00:39:22,710 And so we have this tri-diagonal structure. 623 00:39:22,710 --> 00:39:29,420 And because it's a ring, we have a beta here and here. 624 00:39:32,885 --> 00:39:35,360 OK. 625 00:39:35,360 --> 00:39:40,270 So then, we ask our computer to diagonalize this, 626 00:39:40,270 --> 00:39:45,320 or we use clever tricks from linear algebra 627 00:39:45,320 --> 00:39:49,340 to find the eigenvalues, but I don't recommend it. 628 00:39:49,340 --> 00:39:52,700 I mean, the general problem is going 629 00:39:52,700 --> 00:39:56,070 to be something where you have to use a computer. 630 00:39:56,070 --> 00:39:58,730 So don't develop tricks unless you 631 00:39:58,730 --> 00:40:01,040 want to check to see whether you programmed 632 00:40:01,040 --> 00:40:02,602 the computer correctly. 633 00:40:05,320 --> 00:40:13,380 And so when you do this, you get-- 634 00:40:13,380 --> 00:40:19,090 E 1, the lowest energy, is alpha plus 2 beta. 635 00:40:19,090 --> 00:40:23,575 E 2 is alpha plus beta. 636 00:40:23,575 --> 00:40:27,280 And E 3 is alpha minus beta. 637 00:40:32,940 --> 00:40:34,110 I'm sorry. 638 00:40:34,110 --> 00:40:37,080 E 2 and e 3 are both this. 639 00:40:37,080 --> 00:40:40,820 And E 4 and e 5 are this. 640 00:40:40,820 --> 00:40:46,482 And e 6 is alpha minus 2 beta. 641 00:40:46,482 --> 00:40:48,440 Now, we don't really care about these orbitals, 642 00:40:48,440 --> 00:40:50,148 because they don't put electrons in them. 643 00:40:52,810 --> 00:40:56,200 And so when you put the electrons in the orbitals, 644 00:40:56,200 --> 00:40:59,661 you get 2 for this, 2 for this, 2 for that. 645 00:40:59,661 --> 00:41:00,160 OK? 646 00:41:00,160 --> 00:41:03,580 And so we just calculate the sum of the energies. 647 00:41:03,580 --> 00:41:12,520 And we end up getting the energy levels for benzene. 648 00:41:12,520 --> 00:41:21,790 The ground state is going to be 2 alpha, 649 00:41:21,790 --> 00:41:36,580 plus 2 beta, plus 2 alpha, plus beta, plus 2 alpha, plus beta. 650 00:41:36,580 --> 00:41:38,370 Right? 651 00:41:38,370 --> 00:41:44,330 So this gives you 6 alpha plus 8 beta. 652 00:41:54,290 --> 00:41:58,620 Now, this-- You might have guessed it, 653 00:41:58,620 --> 00:41:59,620 but you didn't guess it. 654 00:41:59,620 --> 00:42:02,236 Your computer told you that. 655 00:42:02,236 --> 00:42:07,590 And the computer actually likes numbers rather than symbols. 656 00:42:07,590 --> 00:42:14,120 And so you actually obtain this simple structure 657 00:42:14,120 --> 00:42:18,000 from the computer-- requires a little bit of manipulation. 658 00:42:18,000 --> 00:42:21,960 But you still get 6 alpha plus 8 beta. 659 00:42:21,960 --> 00:42:29,750 And you also get the eigenvectors. 660 00:42:29,750 --> 00:42:32,530 And that's the stuff that you know. 661 00:42:32,530 --> 00:42:38,330 So c 1 is 1 over square root of 6. 662 00:42:38,330 --> 00:42:38,830 1. 663 00:42:38,830 --> 00:42:39,330 1. 664 00:42:39,330 --> 00:42:39,920 All of those. 665 00:42:39,920 --> 00:42:48,020 And c 2 is going to be something a little bit more complicated. 666 00:42:48,020 --> 00:42:53,050 We can have the nodal plane going through atoms. 667 00:42:53,050 --> 00:42:55,780 And so if it goes through atom one, 668 00:42:55,780 --> 00:42:58,030 it's also going to go through atom four, 669 00:42:58,030 --> 00:43:03,370 and so we have 0 and 0, 1, 1, 1, 1. 670 00:43:03,370 --> 00:43:09,900 And now, to figure out how to normalize that, we just-- 671 00:43:09,900 --> 00:43:13,040 1 over square root of 4, and so on. 672 00:43:13,040 --> 00:43:14,970 We can figure these things out. 673 00:43:14,970 --> 00:43:20,020 C 3 is going to be an eigenvector. 674 00:43:20,020 --> 00:43:24,880 Now, instead of having the nodal plane going through atoms, 675 00:43:24,880 --> 00:43:26,710 it's going between atoms. 676 00:43:26,710 --> 00:43:29,770 And so instead of having any 0s, you're 677 00:43:29,770 --> 00:43:33,760 going to have something more complicated. 678 00:43:33,760 --> 00:43:41,200 And now I can see that there is something in my notes 679 00:43:41,200 --> 00:43:51,170 which is subject to how you'd actually impose the symmetry. 680 00:43:51,170 --> 00:43:53,630 But suppose you have something like this-- 681 00:43:53,630 --> 00:44:02,830 2, 1, minus 1, minus 2, minus 1, 1. 682 00:44:02,830 --> 00:44:05,380 So why did I use these numbers? 683 00:44:05,380 --> 00:44:14,080 Well, I had to have a nodal plane here between atoms two 684 00:44:14,080 --> 00:44:15,310 and three. 685 00:44:15,310 --> 00:44:19,467 And the corresponding guy will be-- 686 00:44:19,467 --> 00:44:21,300 Well, that should have been 2 at the bottom. 687 00:44:24,830 --> 00:44:27,481 Let me just make sure I'm doing this right. 688 00:44:27,481 --> 00:44:27,980 No. 689 00:44:27,980 --> 00:44:28,621 I had a 2. 690 00:44:28,621 --> 00:44:29,120 OK. 691 00:44:29,120 --> 00:44:30,280 It's a 1. 692 00:44:30,280 --> 00:44:30,860 OK. 693 00:44:30,860 --> 00:44:37,790 And so the last 1 is here. 694 00:44:37,790 --> 00:44:42,530 So the sign change occurs twice. 695 00:44:42,530 --> 00:44:46,160 And those correspond to opposite bonds. 696 00:44:49,760 --> 00:44:51,590 Why were there 2s? 697 00:44:51,590 --> 00:44:59,180 Well, the 2s are between two atoms that have 1s, 698 00:44:59,180 --> 00:45:02,070 and so it's going to have a larger eigenvector. 699 00:45:02,070 --> 00:45:03,830 And you can figure it out lots of ways. 700 00:45:03,830 --> 00:45:07,610 You can also say, well, every atom has to be used up. 701 00:45:07,610 --> 00:45:08,840 Yes. 702 00:45:08,840 --> 00:45:10,870 AUDIENCE: So should there also be a sign change? 703 00:45:10,870 --> 00:45:11,828 ROBERT FIELD: I can't-- 704 00:45:11,828 --> 00:45:14,276 AUDIENCE: Should there also be a sign change in c 2? 705 00:45:14,276 --> 00:45:15,942 AUDIENCE: Yeah. 706 00:45:15,942 --> 00:45:22,182 In c 2 eigen c 2 6 should be [INAUDIBLE].. 707 00:45:22,182 --> 00:45:23,390 ROBERT FIELD: Did I screw up? 708 00:45:23,390 --> 00:45:24,015 AUDIENCE: Yeah. 709 00:45:27,770 --> 00:45:29,110 No, not-- in c 2. 710 00:45:29,110 --> 00:45:30,030 Not c 3. 711 00:45:30,030 --> 00:45:30,600 ROBERT FIELD: I'm sorry? 712 00:45:30,600 --> 00:45:31,725 AUDIENCE: The previous one. 713 00:45:33,992 --> 00:45:34,950 ROBERT FIELD: Oh, yeah. 714 00:45:34,950 --> 00:45:35,449 Yeah. 715 00:45:38,040 --> 00:45:41,540 So opposite sides. 716 00:45:41,540 --> 00:45:42,040 OK? 717 00:45:45,440 --> 00:45:45,940 All right. 718 00:45:45,940 --> 00:45:46,990 It doesn't matter. 719 00:45:46,990 --> 00:45:48,280 The computer tells you. 720 00:45:48,280 --> 00:45:52,360 But sometimes you can approach the problem very quickly, 721 00:45:52,360 --> 00:45:55,042 and just say, I know what the eigenvalues are going to be, 722 00:45:55,042 --> 00:45:56,250 and I have to normalize them. 723 00:46:01,580 --> 00:46:03,270 But one of the things that you often 724 00:46:03,270 --> 00:46:08,970 do if you're trying to be smart and skip steps is to say, OK. 725 00:46:08,970 --> 00:46:10,950 We have six eigenvectors. 726 00:46:10,950 --> 00:46:17,070 And each of the atomic orbitals gets used up completely 727 00:46:17,070 --> 00:46:19,560 among the six eigenvectors. 728 00:46:19,560 --> 00:46:22,170 And I do that all the time, because it's 729 00:46:22,170 --> 00:46:26,410 a very useful way of making sure I haven't screwed up. 730 00:46:26,410 --> 00:46:27,870 OK. 731 00:46:27,870 --> 00:46:37,510 So we get this, and that's perfectly OK. 732 00:46:37,510 --> 00:46:39,120 We don't know how significant it is, 733 00:46:39,120 --> 00:46:42,405 but say we had three of these-- 734 00:46:45,300 --> 00:46:48,900 three ethylenes-- I guess an organic chemist 735 00:46:48,900 --> 00:46:50,060 would just do that, right? 736 00:46:53,070 --> 00:46:56,580 But anyway, if we had three of these, when you do this 737 00:46:56,580 --> 00:47:03,464 you get 2 alpha plus 2 theta. 738 00:47:03,464 --> 00:47:04,350 OK? 739 00:47:04,350 --> 00:47:08,330 So we get 6 alpha plus 6 beta. 740 00:47:08,330 --> 00:47:10,670 And alpha and beta are both negative. 741 00:47:10,670 --> 00:47:14,660 And so the energy for three ethylenes-- 742 00:47:14,660 --> 00:47:15,890 did I screw up again? 743 00:47:15,890 --> 00:47:17,265 AUDIENCE: How are those ethylenes 744 00:47:17,265 --> 00:47:18,950 arranged with one another? 745 00:47:18,950 --> 00:47:20,306 ROBERT FIELD: They're separate. 746 00:47:25,584 --> 00:47:26,500 We could draw benzene. 747 00:47:29,130 --> 00:47:31,720 And so we could say, we have three isolated bonds. 748 00:47:31,720 --> 00:47:34,210 And since we don't care about the sigma structure, 749 00:47:34,210 --> 00:47:39,880 and no bonds are adjacent, you know that they're additive. 750 00:47:39,880 --> 00:47:42,730 And so we represent benzene. 751 00:47:42,730 --> 00:47:46,300 The primitive structure for benzene is three ethylenes. 752 00:47:46,300 --> 00:47:49,280 And benzene is 2 beta better. 753 00:47:49,280 --> 00:47:50,270 AUDIENCE: Yeah. 754 00:47:50,270 --> 00:47:52,270 ROBERT FIELD: That's the resonant stabilization. 755 00:47:52,270 --> 00:47:54,910 That's a great thing. 756 00:47:54,910 --> 00:47:57,110 OK. 757 00:47:57,110 --> 00:48:02,760 So we get a resonant stabilization. 758 00:48:02,760 --> 00:48:09,250 And now, the more subtle and wonderful stuff 759 00:48:09,250 --> 00:48:16,550 is, we have these eigenvectors. 760 00:48:16,550 --> 00:48:18,120 And we have the energies. 761 00:48:18,120 --> 00:48:19,370 And we can do stuff with them. 762 00:48:19,370 --> 00:48:20,930 And one thing is bond order. 763 00:48:27,550 --> 00:48:29,410 And so we have a formula. 764 00:48:29,410 --> 00:48:37,925 The bond order between atoms i and j is equal to the sum 765 00:48:37,925 --> 00:48:50,160 for mu equals 1, including only the occupied orbitals, of c i 766 00:48:50,160 --> 00:48:52,650 mu c j mu. 767 00:48:52,650 --> 00:48:56,320 So we have the new molecular orbitals, 768 00:48:56,320 --> 00:48:59,040 and we have the adjacent atoms. 769 00:48:59,040 --> 00:49:01,900 And this gives you the bond order. 770 00:49:01,900 --> 00:49:04,180 And so we can calculate the bond order. 771 00:49:04,180 --> 00:49:09,340 And we find that every bond is a pi bond 772 00:49:09,340 --> 00:49:20,510 order of 2/3, which is neat, because the non-resonant 773 00:49:20,510 --> 00:49:24,050 structure says, three of the bonds 774 00:49:24,050 --> 00:49:31,900 have a pi bond order of 0, and 3 have a bond order of 1. 775 00:49:31,900 --> 00:49:36,340 And so the average is 1/2. 776 00:49:36,340 --> 00:49:40,477 And this is bigger than 1/2. 777 00:49:40,477 --> 00:49:41,185 And it's uniform. 778 00:49:46,470 --> 00:49:48,900 So you can do this, and you can say, OK. 779 00:49:48,900 --> 00:49:50,250 The 1, 2 bond order-- 780 00:49:50,250 --> 00:49:53,580 the 2, 3 bond order-- you do the laborious calculation-- 781 00:49:53,580 --> 00:49:54,360 always get 2/3. 782 00:50:02,010 --> 00:50:07,890 Now, you could also calculate the charge on atom i. 783 00:50:07,890 --> 00:50:13,800 And that is, again, mu for the occupied orbitals. 784 00:50:13,800 --> 00:50:20,580 And that would be c i mu c i mu. 785 00:50:20,580 --> 00:50:23,610 And so for benzene, you would expect 786 00:50:23,610 --> 00:50:24,780 this to come out to be 0. 787 00:50:24,780 --> 00:50:27,680 And it does. 788 00:50:27,680 --> 00:50:29,095 So there is no charge. 789 00:50:37,160 --> 00:50:38,740 Let me just make sure that that-- 790 00:50:43,921 --> 00:50:47,640 Well, there is an equal charge on each atom, 791 00:50:47,640 --> 00:50:53,080 whether it's 0 or 1/6. 792 00:50:53,080 --> 00:50:54,600 That I haven't done. 793 00:50:54,600 --> 00:50:57,225 And I am a little uncertain about what this 794 00:50:57,225 --> 00:50:58,350 is going to come out to be. 795 00:50:58,350 --> 00:51:03,250 But they're going to be equal on every atom in benzene. 796 00:51:03,250 --> 00:51:10,770 And so now, suppose, instead of benzene, you have amylin. 797 00:51:13,820 --> 00:51:15,630 Well, you can do stuff. 798 00:51:15,630 --> 00:51:17,840 And you could say, well, the amylin out here 799 00:51:17,840 --> 00:51:21,290 is going to affect the alpha value here a lot, 800 00:51:21,290 --> 00:51:23,840 and here less and less. 801 00:51:23,840 --> 00:51:25,260 And then you do the calculation. 802 00:51:25,260 --> 00:51:27,880 And you find when you do this calculation, 803 00:51:27,880 --> 00:51:30,710 you get unequal charges. 804 00:51:30,710 --> 00:51:36,950 And you get the normal rules for ortho versus meta. 805 00:51:36,950 --> 00:51:39,170 And everything is great. 806 00:51:39,170 --> 00:51:41,530 And you also, when you do this, you 807 00:51:41,530 --> 00:51:44,240 can actually write resonance structures 808 00:51:44,240 --> 00:51:46,250 and you could calculate, well, what 809 00:51:46,250 --> 00:51:48,260 is the energy of that resonance structure-- 810 00:51:48,260 --> 00:51:50,010 in a Huckel-like theory. 811 00:51:50,010 --> 00:51:58,640 So there are lots of things you can do in order to say, OK. 812 00:51:58,640 --> 00:52:01,760 We do Huckel theory to get this 0 in our picture. 813 00:52:01,760 --> 00:52:05,120 And then-- I'm way over time. 814 00:52:05,120 --> 00:52:07,460 And then we can add special effects, 815 00:52:07,460 --> 00:52:10,600 and we know how to parameterize them. 816 00:52:10,600 --> 00:52:13,750 And if we're close to being OK, we'll 817 00:52:13,750 --> 00:52:17,140 get results that correspond to experiment. 818 00:52:17,140 --> 00:52:21,300 And this is so much better than you deserve, 819 00:52:21,300 --> 00:52:23,430 because it's all garbage. 820 00:52:23,430 --> 00:52:26,510 But it's empirically-calibrated garbage. 821 00:52:26,510 --> 00:52:30,950 And it it's calibrated over an enormous number of molecules 822 00:52:30,950 --> 00:52:33,850 and a huge amount of experience. 823 00:52:33,850 --> 00:52:35,360 And that's good. 824 00:52:35,360 --> 00:52:36,860 That's what we do. 825 00:52:36,860 --> 00:52:37,730 OK. 826 00:52:37,730 --> 00:52:42,230 So I'll see you on Friday with some perturbation theory. 827 00:52:42,230 --> 00:52:44,570 One of my favorite problems, too. 828 00:52:44,570 --> 00:52:46,120 OK.