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 00:00:21,910 --> 00:00:26,240 ROBERT FIELD: This is the first of two lectures on spectroscopy 9 00:00:26,240 --> 00:00:27,590 and dynamics. 10 00:00:27,590 --> 00:00:31,130 Now, I'm a spectroscopist, and so this is 11 00:00:31,130 --> 00:00:34,160 the core of what I really love. 12 00:00:34,160 --> 00:00:36,910 And there are a lot of questions about, 13 00:00:36,910 --> 00:00:38,990 well, what are we trying to do? 14 00:00:38,990 --> 00:00:43,430 And you heard two lectures from Professor Van Voorhis, 15 00:00:43,430 --> 00:00:48,270 and he talked about ab initio calculations-- 16 00:00:48,270 --> 00:00:50,480 electronic structure calculations-- 17 00:00:50,480 --> 00:00:55,210 where you can get really close to the exact answer. 18 00:00:55,210 --> 00:00:57,510 And it's really a powerful tool. 19 00:01:01,710 --> 00:01:04,140 And it gets you the truth. 20 00:01:04,140 --> 00:01:06,840 But it gets you so much truth that you 21 00:01:06,840 --> 00:01:09,280 don't know what to do with it. 22 00:01:09,280 --> 00:01:12,150 And the same thing is true for spectroscopy. 23 00:01:12,150 --> 00:01:13,150 You get a spectrum. 24 00:01:13,150 --> 00:01:16,450 It contains a huge amount of information. 25 00:01:16,450 --> 00:01:18,640 And you can take lots and lots of spectra. 26 00:01:18,640 --> 00:01:20,050 But what are we trying to do? 27 00:01:20,050 --> 00:01:24,880 When I was a graduate student, we 28 00:01:24,880 --> 00:01:29,290 were doing a unique super high resolution spectroscopy. 29 00:01:29,290 --> 00:01:33,970 And so we thought what we were generating 30 00:01:33,970 --> 00:01:39,630 was excellent tests for quantitative theory. 31 00:01:39,630 --> 00:01:44,880 And if, in those days, I went to a lecture by a theorist, 32 00:01:44,880 --> 00:01:48,420 they were saying, we're generating stuff 33 00:01:48,420 --> 00:01:52,690 to check against experiment. 34 00:01:52,690 --> 00:01:57,750 And it's a circle, and that's not what we're trying to do. 35 00:01:57,750 --> 00:02:02,370 The theory and the experiment are just the beginning. 36 00:02:02,370 --> 00:02:05,400 And you can start from either extreme. 37 00:02:05,400 --> 00:02:08,070 What we want is, how does it all work? 38 00:02:08,070 --> 00:02:10,710 What is going on? 39 00:02:10,710 --> 00:02:19,080 Can we build a picture, which is intuitive and checkable and 40 00:02:19,080 --> 00:02:22,680 predictive, so that we can say, oh, yeah. 41 00:02:22,680 --> 00:02:28,090 If we want to know something, we can get to it this way. 42 00:02:28,090 --> 00:02:29,890 And the purpose of this lecture is 43 00:02:29,890 --> 00:02:34,860 to not provide a textbook view of spectra, 44 00:02:34,860 --> 00:02:39,710 but to give you a sense of how you can get to stuff 45 00:02:39,710 --> 00:02:41,200 that challenge your intuition. 46 00:02:46,620 --> 00:02:53,440 What we want to do as scientists is to be surprised. 47 00:02:53,440 --> 00:02:58,590 We want to do a good experiment or do a good calculation. 48 00:02:58,590 --> 00:03:03,190 And we want to find that the result is not what we expected. 49 00:03:03,190 --> 00:03:07,210 And we can figure out why it's not what we expected. 50 00:03:07,210 --> 00:03:11,820 And that's never conveyed in any textbooks. 51 00:03:11,820 --> 00:03:17,520 Now, this lecture is based on my little book of lecture notes. 52 00:03:17,520 --> 00:03:20,760 And I have a number of copies of it. 53 00:03:20,760 --> 00:03:24,240 And if people have a strong wish to have a copy, 54 00:03:24,240 --> 00:03:24,960 you can have one. 55 00:03:24,960 --> 00:03:28,020 I can give it to you. 56 00:03:28,020 --> 00:03:32,430 And a lot of this lecture is based on the first chapter 57 00:03:32,430 --> 00:03:34,020 of this book. 58 00:03:34,020 --> 00:03:38,870 But many of the topics are developed throughout the book. 59 00:03:38,870 --> 00:03:39,370 OK. 60 00:03:39,370 --> 00:03:46,650 So this is a two lecture sequence, 61 00:03:46,650 --> 00:03:52,062 and the first half will cover this stuff on this board. 62 00:03:52,062 --> 00:03:53,520 And it's in the notes, so you don't 63 00:03:53,520 --> 00:03:54,970 have to copy all this stuff. 64 00:03:54,970 --> 00:03:57,700 I just want you to see where we're going. 65 00:03:57,700 --> 00:03:59,670 So I've already talked a little bit 66 00:03:59,670 --> 00:04:02,040 about experiment versus theory. 67 00:04:02,040 --> 00:04:04,090 They complement each other. 68 00:04:04,090 --> 00:04:07,410 We can use theory to devise an experiment, which 69 00:04:07,410 --> 00:04:10,200 is path breaking. 70 00:04:10,200 --> 00:04:13,410 Or we can use an experiment to challenge the theorist 71 00:04:13,410 --> 00:04:16,880 to calculate a new kind of thing. 72 00:04:16,880 --> 00:04:20,220 And I hope that some of these things, those ideas, 73 00:04:20,220 --> 00:04:23,360 present themselves in this lecture. 74 00:04:23,360 --> 00:04:28,070 And it's really important that if there's something 75 00:04:28,070 --> 00:04:32,300 that you don't understand or don't 76 00:04:32,300 --> 00:04:35,330 capture the importance of, you should ask me a question. 77 00:04:35,330 --> 00:04:39,560 I really want to talk about what it's all for. 78 00:04:39,560 --> 00:04:40,070 OK. 79 00:04:40,070 --> 00:04:42,470 So I'm going to talk about spectra, 80 00:04:42,470 --> 00:04:45,830 and it will be what kinds-- 81 00:04:45,830 --> 00:04:52,310 rotation, vibration, electronic, and other ramifications. 82 00:04:52,310 --> 00:04:55,730 And going from atom to diatomic to polyatomic 83 00:04:55,730 --> 00:04:56,690 to condensed phase. 84 00:04:59,370 --> 00:05:05,130 Each step along this path leads to new complexities 85 00:05:05,130 --> 00:05:08,410 and new insights. 86 00:05:08,410 --> 00:05:11,530 Then I'll talk about, OK, we got a spectrum. 87 00:05:11,530 --> 00:05:13,550 What do we expect to be in the spectrum? 88 00:05:13,550 --> 00:05:16,060 Well, one of the things that's important 89 00:05:16,060 --> 00:05:21,160 is the transition selection rules or transition rules. 90 00:05:21,160 --> 00:05:25,600 Selection rules correspond to an operator-- eigenvalues 91 00:05:25,600 --> 00:05:29,860 of an operator-- that commutes with the exact Hamiltonian. 92 00:05:29,860 --> 00:05:32,560 And those correspond to symmetries. 93 00:05:32,560 --> 00:05:35,740 And there are propensity rules like, OK, 94 00:05:35,740 --> 00:05:38,830 which transitions are going to be strong 95 00:05:38,830 --> 00:05:41,360 and which are going to be weak? 96 00:05:41,360 --> 00:05:44,960 And a beautiful example of propensity rules 97 00:05:44,960 --> 00:05:47,830 are based on the Franck-Condon principle. 98 00:05:47,830 --> 00:05:51,370 And the Franck-Condon principle is one of the first keys 99 00:05:51,370 --> 00:05:54,760 you use to unlock what's in a spectrum, 100 00:05:54,760 --> 00:05:56,500 or what a molecule is doing. 101 00:05:56,500 --> 00:06:02,350 Because it's the first level of complexity 102 00:06:02,350 --> 00:06:05,080 that is presented to you in the spectrum. 103 00:06:05,080 --> 00:06:08,690 What are the vibrational bands, and how do we assign them, 104 00:06:08,690 --> 00:06:09,940 and what are they telling us? 105 00:06:12,560 --> 00:06:16,700 There is a very different kind of information in an absorption 106 00:06:16,700 --> 00:06:21,770 spectrum, because it's always from the lowest 107 00:06:21,770 --> 00:06:25,150 electronic state, lowest vibrational level. 108 00:06:25,150 --> 00:06:27,160 And so there's a simplicity, because 109 00:06:27,160 --> 00:06:29,540 of a kind of state selection. 110 00:06:29,540 --> 00:06:32,100 And in emission, it's a very different ballgame. 111 00:06:32,100 --> 00:06:35,390 Because in the gas phase, the emission 112 00:06:35,390 --> 00:06:38,060 is from many different levels. 113 00:06:38,060 --> 00:06:40,920 In the condensed phase, it's not. 114 00:06:40,920 --> 00:06:43,260 Why? 115 00:06:43,260 --> 00:06:43,900 OK. 116 00:06:43,900 --> 00:06:48,280 And then we get to dynamics. 117 00:06:48,280 --> 00:06:51,610 And the main thing I want to do in these two lectures 118 00:06:51,610 --> 00:06:54,250 is to whet your appetite for dynamics. 119 00:06:54,250 --> 00:06:56,560 And there are many kinds of dynamics ranging 120 00:06:56,560 --> 00:07:02,310 from a simple two level quantum beat to intramolecular 121 00:07:02,310 --> 00:07:04,720 vibrational redistribution. 122 00:07:04,720 --> 00:07:08,940 Which you can understand by perturbation theory 123 00:07:08,940 --> 00:07:16,590 to the strange behavior of electronically excited states 124 00:07:16,590 --> 00:07:18,960 losing their ability to fluoresce 125 00:07:18,960 --> 00:07:24,030 not because the molecule breaks, but because the bright state-- 126 00:07:24,030 --> 00:07:25,930 that's an important concept-- 127 00:07:25,930 --> 00:07:27,180 the bright state is something. 128 00:07:27,180 --> 00:07:29,100 It's not an eigenstate. 129 00:07:29,100 --> 00:07:32,370 It's a special state that we understand well. 130 00:07:32,370 --> 00:07:35,730 It's one of the things that we build perturbation theory 131 00:07:35,730 --> 00:07:36,840 around. 132 00:07:36,840 --> 00:07:41,430 The bright state mixes into an enormous number of dark states. 133 00:07:41,430 --> 00:07:45,360 And the molecule forgets that it knows 134 00:07:45,360 --> 00:07:48,630 how to fluoresce because the different components, 135 00:07:48,630 --> 00:07:51,740 different eigenstates dephase. 136 00:07:51,740 --> 00:07:54,000 And that is a beautiful theory. 137 00:07:54,000 --> 00:07:55,620 And when I was a graduate student, 138 00:07:55,620 --> 00:07:58,050 this theory of radiationless transitions-- 139 00:07:58,050 --> 00:08:01,620 Bixon-Jortner theory-- was just created 140 00:08:01,620 --> 00:08:04,260 and many people didn't believe it. 141 00:08:04,260 --> 00:08:07,680 They thought molecules, big molecules, 142 00:08:07,680 --> 00:08:10,440 they're really easy to be quenched by collision 143 00:08:10,440 --> 00:08:12,900 and the loss of the ability to fluoresce 144 00:08:12,900 --> 00:08:15,270 or the absence of fluorescence was somehow 145 00:08:15,270 --> 00:08:20,340 collision related as opposed to a physical process. 146 00:08:20,340 --> 00:08:22,140 So there's lots of good stuff. 147 00:08:22,140 --> 00:08:27,710 And some of the good stuff is Ahmed Zewail's Nobel Prize 148 00:08:27,710 --> 00:08:30,350 where he claims-- 149 00:08:30,350 --> 00:08:32,659 and that's why he got the Nobel Prize, because people 150 00:08:32,659 --> 00:08:34,610 believed that claim. 151 00:08:34,610 --> 00:08:36,470 Now, I'm not saying it's wrong. 152 00:08:36,470 --> 00:08:40,760 But part of getting famous is to have 153 00:08:40,760 --> 00:08:44,070 a package, which you can sell. 154 00:08:44,070 --> 00:08:46,730 And he sold the daylights out of it. 155 00:08:46,730 --> 00:08:55,290 And he calls it clocking real dynamics in real time. 156 00:08:55,290 --> 00:08:58,380 And it's basically wave packets. 157 00:08:58,380 --> 00:09:01,770 But they're wave packets doing neat stuff. 158 00:09:01,770 --> 00:09:06,730 And for example, one way a molecule 159 00:09:06,730 --> 00:09:09,760 can lose the ability to fluoresce 160 00:09:09,760 --> 00:09:13,000 is because the molecule breaks. 161 00:09:13,000 --> 00:09:15,790 And what is the mechanism by which a molecule breaks? 162 00:09:15,790 --> 00:09:20,080 Does the bond just simply break or is there some motion 163 00:09:20,080 --> 00:09:22,200 that precedes that? 164 00:09:22,200 --> 00:09:26,310 And what Zewail did was to show what 165 00:09:26,310 --> 00:09:28,830 are the motions that lead the molecule 166 00:09:28,830 --> 00:09:34,740 into the region of state space where the bond breaks. 167 00:09:34,740 --> 00:09:37,200 And, of course, if you want to manipulate molecules, 168 00:09:37,200 --> 00:09:41,060 you either want to get to those regions or avoid them. 169 00:09:41,060 --> 00:09:45,268 And so there's all sorts of insight there. 170 00:09:45,268 --> 00:09:47,610 OK. 171 00:09:47,610 --> 00:09:52,390 Now this is what I believe. 172 00:09:52,390 --> 00:09:56,360 That if you understand small molecules, 173 00:09:56,360 --> 00:09:58,250 you will see examples of everything 174 00:09:58,250 --> 00:10:02,390 you need to know to deal with almost any dynamical process 175 00:10:02,390 --> 00:10:03,980 in chemistry. 176 00:10:03,980 --> 00:10:06,390 Now, this is certainly an exaggeration, 177 00:10:06,390 --> 00:10:10,490 but this has been my motto for years. 178 00:10:10,490 --> 00:10:15,680 And so I really stress the small molecules. 179 00:10:15,680 --> 00:10:20,210 And it's not that small molecules are really hard. 180 00:10:20,210 --> 00:10:22,400 They're really beautiful, and they 181 00:10:22,400 --> 00:10:26,480 do enough so that you can anticipate what you need 182 00:10:26,480 --> 00:10:29,840 to deal with bigger molecules. 183 00:10:29,840 --> 00:10:32,630 So let's begin. 184 00:10:38,380 --> 00:10:40,780 OK. 185 00:10:40,780 --> 00:10:41,980 So what is a molecule? 186 00:10:41,980 --> 00:10:45,170 As chemists, we would never think 187 00:10:45,170 --> 00:10:48,830 of a molecule as a bag of nuclei and electrons. 188 00:10:53,550 --> 00:10:57,680 We wouldn't think of it as a bag of atoms either. 189 00:10:57,680 --> 00:11:00,250 We believe in chemical bonds. 190 00:11:00,250 --> 00:11:02,160 This is an important thing. 191 00:11:02,160 --> 00:11:04,490 It's not a conserved quantity. 192 00:11:04,490 --> 00:11:06,260 Bonds can break. 193 00:11:06,260 --> 00:11:10,370 But we believe that bonds tell an important story. 194 00:11:16,520 --> 00:11:23,480 And so almost all of our pictures for complicated 195 00:11:23,480 --> 00:11:27,200 phenomena are based on the-- 196 00:11:27,200 --> 00:11:29,420 I hesitate to use the word sanctity-- 197 00:11:29,420 --> 00:11:34,990 but the importance of bonds. 198 00:11:34,990 --> 00:11:36,790 OK. 199 00:11:36,790 --> 00:11:39,160 We start-- I'm going to erase this, 200 00:11:39,160 --> 00:11:42,340 because I've made my point, and it's 201 00:11:42,340 --> 00:11:48,520 embarrassing to keep emphasizing my secret motto. 202 00:11:48,520 --> 00:11:49,460 But it is true. 203 00:11:49,460 --> 00:11:49,960 OK. 204 00:11:49,960 --> 00:11:51,970 We have the Born-Oppenheimer approximation. 205 00:11:56,550 --> 00:12:02,110 And this is very important, because we can't 206 00:12:02,110 --> 00:12:04,680 solve a three body problem. 207 00:12:04,680 --> 00:12:06,840 We can solve a two body problem. 208 00:12:06,840 --> 00:12:10,560 But we have molecules, which are consisting 209 00:12:10,560 --> 00:12:13,350 of nuclei and electrons. 210 00:12:13,350 --> 00:12:16,380 And this Born-Oppenheimer approximation 211 00:12:16,380 --> 00:12:20,310 enables us to separate the nuclear part of the problem 212 00:12:20,310 --> 00:12:22,560 from the electronic part of the problem. 213 00:12:22,560 --> 00:12:26,730 Because these two things move at very different velocities. 214 00:12:26,730 --> 00:12:29,710 And so it's a profound simplification. 215 00:12:29,710 --> 00:12:35,250 We get potential energy curves or potential energy surfaces. 216 00:12:35,250 --> 00:12:39,180 And that is the repository of essentially everything 217 00:12:39,180 --> 00:12:40,410 we want to know. 218 00:12:40,410 --> 00:12:43,140 If we know the potential surface, 219 00:12:43,140 --> 00:12:46,540 we can begin to do almost anything. 220 00:12:46,540 --> 00:12:49,450 And certainly for a big molecule, 221 00:12:49,450 --> 00:12:53,590 it's not just a simple curve like this. 222 00:12:53,590 --> 00:12:59,580 If you have N atoms, there's 3N minus 6 vibrational modes. 223 00:12:59,580 --> 00:13:03,210 And well, that sounds terrible. 224 00:13:03,210 --> 00:13:07,140 But even for this, you have essentially an infinite number 225 00:13:07,140 --> 00:13:10,580 of vibrational levels and an infinite number 226 00:13:10,580 --> 00:13:12,390 of rotational levels. 227 00:13:12,390 --> 00:13:15,830 And so if you have a polyatomic molecule, 228 00:13:15,830 --> 00:13:20,480 you have 3N minus 6 infinities of infinities. 229 00:13:20,480 --> 00:13:24,470 So you're not wanting to get everything. 230 00:13:24,470 --> 00:13:26,960 You want to generate enough information 231 00:13:26,960 --> 00:13:31,310 to be able to calculate anything you want. 232 00:13:31,310 --> 00:13:33,070 And sometimes, you make approximations 233 00:13:33,070 --> 00:13:36,176 and you're not sure that those approximations are good. 234 00:13:36,176 --> 00:13:37,300 And you want them to break. 235 00:13:37,300 --> 00:13:39,520 You want to discover something new. 236 00:13:39,520 --> 00:13:41,890 So the Born-Oppenheimer approximation, 237 00:13:41,890 --> 00:13:44,890 we go from clamped nuclei calculation 238 00:13:44,890 --> 00:13:47,560 where the-- since the nuclei moves slow 239 00:13:47,560 --> 00:13:51,600 compared to the electrons, well, let's not let them move at all. 240 00:13:51,600 --> 00:13:54,930 And then we build a perturbation theory picture 241 00:13:54,930 --> 00:13:56,550 where we let them move. 242 00:13:56,550 --> 00:13:58,530 And we can deal with that because we understand 243 00:13:58,530 --> 00:14:00,090 vibrations and rotations. 244 00:14:03,480 --> 00:14:04,150 OK. 245 00:14:04,150 --> 00:14:05,920 So we have a potential energy surface. 246 00:14:09,790 --> 00:14:13,440 And there are things that we can anticipate 247 00:14:13,440 --> 00:14:15,680 about a potential energy surface. 248 00:14:15,680 --> 00:14:21,840 And LCAO-MO theory enables you to say 249 00:14:21,840 --> 00:14:27,840 a lot of important things about the potential energy surface. 250 00:14:27,840 --> 00:14:33,450 So it provides a qualitative framework. 251 00:14:33,450 --> 00:14:37,880 And so from molecular orbital theory-- 252 00:14:37,880 --> 00:14:41,160 and this is not what Professor Van Voorhis talked about. 253 00:14:41,160 --> 00:14:44,590 This is the baby stuff. 254 00:14:44,590 --> 00:14:48,490 And we don't expect to get the exact answer. 255 00:14:48,490 --> 00:14:52,410 But we do expect to be able to explain trends. 256 00:14:52,410 --> 00:14:55,800 Trends within a molecule and between related molecules. 257 00:14:59,130 --> 00:15:04,760 So this provides a framework for expectations. 258 00:15:04,760 --> 00:15:08,270 And there are things that we get like bond order. 259 00:15:12,680 --> 00:15:23,270 And we talk about orbitals that are bonding, non-bonding, 260 00:15:23,270 --> 00:15:24,180 and antibonding. 261 00:15:27,170 --> 00:15:31,170 And this comes directly out of the simple ideas. 262 00:15:31,170 --> 00:15:35,970 Recall when we had atom with a hydrogen, 263 00:15:35,970 --> 00:15:38,330 the hydrogen doesn't make pi bonds. 264 00:15:38,330 --> 00:15:44,300 And so they're pi orbitals for the atom A, which have nothing 265 00:15:44,300 --> 00:15:45,620 to interact with. 266 00:15:45,620 --> 00:15:47,780 And they're usually non-bonding. 267 00:15:47,780 --> 00:15:50,660 So we have these sorts of things. 268 00:15:50,660 --> 00:15:56,030 We have spN hybridization. 269 00:16:01,270 --> 00:16:05,210 And this is just telling you if a molecule wants 270 00:16:05,210 --> 00:16:09,240 to make the maximum number of bonds, you do something. 271 00:16:09,240 --> 00:16:14,130 And if it's sp cubed, you have four 272 00:16:14,130 --> 00:16:17,490 tetrahedrally arranged bonds. 273 00:16:17,490 --> 00:16:23,370 And if it's sp cubed, sp squared is planar with 120 degrees. 274 00:16:23,370 --> 00:16:27,540 These things tell you something about geometric expectations. 275 00:16:27,540 --> 00:16:29,850 Now, molecules don't follow the rules exactly, 276 00:16:29,850 --> 00:16:31,900 but they come pretty close. 277 00:16:31,900 --> 00:16:34,410 And so if you have some reason to believe 278 00:16:34,410 --> 00:16:37,500 that a particular hybridization is appropriate, 279 00:16:37,500 --> 00:16:41,889 then you have certain expectations for the geometry 280 00:16:41,889 --> 00:16:44,180 and how that's going to present itself in the spectrum. 281 00:16:50,300 --> 00:16:52,990 So bond order's related to internuclear distance 282 00:16:52,990 --> 00:16:55,270 and vibrational frequencies. 283 00:16:55,270 --> 00:16:58,930 Sp hybridization has to do with geometry, 284 00:16:58,930 --> 00:17:01,190 and all of these things are really important. 285 00:17:01,190 --> 00:17:02,740 So we have a potential surface. 286 00:17:05,810 --> 00:17:08,710 And let's say this is a boldface thing, 287 00:17:08,710 --> 00:17:12,130 implying that there are 3N minus 6 different displacement 288 00:17:12,130 --> 00:17:13,149 coordinates. 289 00:17:17,020 --> 00:17:23,859 This potential encodes the normal modes. 290 00:17:23,859 --> 00:17:27,130 What's a normal mode? 291 00:17:27,130 --> 00:17:29,980 Well, it's a classical mechanical concept. 292 00:17:29,980 --> 00:17:35,290 And it basically corresponds to situations 293 00:17:35,290 --> 00:17:39,820 where all of the atoms move at the same frequency 294 00:17:39,820 --> 00:17:42,620 in each normal mode. 295 00:17:42,620 --> 00:17:47,200 And these normal modes each has an expected frequency 296 00:17:47,200 --> 00:17:50,890 and expected geometry. 297 00:17:50,890 --> 00:17:52,870 Because if it's a polyatomic molecule, 298 00:17:52,870 --> 00:17:55,870 you have not just two things moving. 299 00:17:55,870 --> 00:17:59,090 They'll always be moving at the same frequency. 300 00:17:59,090 --> 00:18:02,380 But you have three or four or 100. 301 00:18:02,380 --> 00:18:03,630 And OK. 302 00:18:03,630 --> 00:18:08,200 So you learn about the shape of the potential and the force 303 00:18:08,200 --> 00:18:09,240 constants and so on. 304 00:18:11,940 --> 00:18:14,976 Now, we have the rotational structure. 305 00:18:18,550 --> 00:18:22,060 Now, molecules are not rigid rotors. 306 00:18:22,060 --> 00:18:26,590 But it's useful to think about molecules as rigid rotors 307 00:18:26,590 --> 00:18:30,310 to develop a basis set for describing rotation. 308 00:18:30,310 --> 00:18:32,110 And perturbation theory enables us 309 00:18:32,110 --> 00:18:40,970 to describe the energy levels of a non-rigid vibrating rotor. 310 00:18:40,970 --> 00:18:42,770 And it's straightforward. 311 00:18:42,770 --> 00:18:47,960 It may be ugly, but it tells you how you take information 312 00:18:47,960 --> 00:18:50,780 from the spectrum and learn about, say, 313 00:18:50,780 --> 00:18:53,150 the internuclear distance dependence 314 00:18:53,150 --> 00:18:56,900 of molecular constants like a spin orbit constant. 315 00:18:56,900 --> 00:18:58,970 Or some hyperfine constant. 316 00:18:58,970 --> 00:19:02,130 Or just the rotational constant. 317 00:19:02,130 --> 00:19:03,890 And it's a simple thing. 318 00:19:03,890 --> 00:19:09,260 And I'm toying with the idea of using this sort of problem 319 00:19:09,260 --> 00:19:11,720 on the exam. 320 00:19:11,720 --> 00:19:15,120 And I'm not sure whether I did on the second exam. 321 00:19:15,120 --> 00:19:19,680 But if I did, you'll have a chance to redeem yourself. 322 00:19:19,680 --> 00:19:20,180 OK. 323 00:19:25,041 --> 00:19:25,540 OK. 324 00:19:25,540 --> 00:19:33,620 Then in the gas phase, nothing much happens. 325 00:19:33,620 --> 00:19:36,710 You can have collisions, but the time between collisions 326 00:19:36,710 --> 00:19:40,800 can be controlled by what the pressure you use. 327 00:19:40,800 --> 00:19:43,160 And so you can sort of think about the gas 328 00:19:43,160 --> 00:19:45,950 phase as something where the molecules are isolated. 329 00:19:48,990 --> 00:19:51,725 And another way of saying that is the expectation 330 00:19:51,725 --> 00:20:01,845 value of the Hamiltonian in any state is time independent. 331 00:20:01,845 --> 00:20:04,950 The Hamiltonian is energy. 332 00:20:04,950 --> 00:20:06,650 Energy is conserved. 333 00:20:06,650 --> 00:20:11,350 And unless there are collisions, energy will be conserved. 334 00:20:11,350 --> 00:20:12,010 And so. 335 00:20:16,930 --> 00:20:20,930 But in the condensed phase, you have lot of collisions. 336 00:20:20,930 --> 00:20:24,210 They're very fast. 337 00:20:24,210 --> 00:20:28,190 And so one big difference between the gas phase 338 00:20:28,190 --> 00:20:32,440 and the condensed phase is energy is not conserved. 339 00:20:32,440 --> 00:20:35,230 And it's not conserved at different rates 340 00:20:35,230 --> 00:20:37,660 for different kinds of motions. 341 00:20:37,660 --> 00:20:41,350 And you want to understand that. 342 00:20:41,350 --> 00:20:41,890 OK. 343 00:20:41,890 --> 00:20:43,764 So now let's talk about the kinds of spectra. 344 00:20:50,700 --> 00:20:52,110 We have rotational spectra. 345 00:20:55,920 --> 00:21:00,870 And that usually is in the micro region of the spectrum. 346 00:21:00,870 --> 00:21:05,440 And it requires that the electric dipole 347 00:21:05,440 --> 00:21:07,510 moment be not equal to zero. 348 00:21:12,830 --> 00:21:14,790 I'll talk about this some more in a minute. 349 00:21:14,790 --> 00:21:15,720 We have vibration. 350 00:21:18,640 --> 00:21:22,010 And vibrational spectrum is in the infrared. 351 00:21:22,010 --> 00:21:30,350 And the requirement for vibration 352 00:21:30,350 --> 00:21:33,240 is that the dipole moment-- 353 00:21:33,240 --> 00:21:34,970 which is a vector quantity if you don't 354 00:21:34,970 --> 00:21:38,420 have a diatomic molecule-- 355 00:21:38,420 --> 00:21:43,580 changes with displacements, each of the normal modes. 356 00:21:43,580 --> 00:21:48,020 And so we have a molecule like CO2. 357 00:21:48,020 --> 00:21:51,590 CO2 does not have a dipole moment at equilibrium. 358 00:21:51,590 --> 00:21:54,340 It's a linear molecule, symmetric. 359 00:21:54,340 --> 00:22:02,760 But it can do this and that is a change in dipole moment. 360 00:22:02,760 --> 00:22:07,780 It can do this and that produces a new dipole moment. 361 00:22:07,780 --> 00:22:09,620 Or produces a dipole moment. 362 00:22:09,620 --> 00:22:11,510 And it can do this, which does not. 363 00:22:11,510 --> 00:22:14,120 So you have different modes, which are infrared 364 00:22:14,120 --> 00:22:17,270 active and infrared inactive. 365 00:22:17,270 --> 00:22:19,750 And it's related to this dipole moment. 366 00:22:19,750 --> 00:22:23,700 Now notice, I put a vector sign on it. 367 00:22:23,700 --> 00:22:29,610 It corresponds to motion, directions in the body frame 368 00:22:29,610 --> 00:22:31,140 where the dipole moment changes. 369 00:22:31,140 --> 00:22:33,570 When you do this, the dipole moment 370 00:22:33,570 --> 00:22:35,250 is perpendicular to the axis. 371 00:22:35,250 --> 00:22:37,740 When you do this, it's along the axis. 372 00:22:37,740 --> 00:22:39,690 And so there is stuff-- 373 00:22:39,690 --> 00:22:41,760 really a lot of stuff-- just looking 374 00:22:41,760 --> 00:22:44,560 at what vibrational modes are active. 375 00:22:44,560 --> 00:22:46,553 And then there's electronic. 376 00:22:50,810 --> 00:22:53,100 And that's mostly in the visible and UV. 377 00:22:53,100 --> 00:22:55,570 But the electronic spectrum could be in the X-ray region 378 00:22:55,570 --> 00:22:56,070 even. 379 00:22:56,070 --> 00:23:00,480 But mostly, molecules break when they get outside 380 00:23:00,480 --> 00:23:02,490 of the ordinary UV region. 381 00:23:02,490 --> 00:23:04,050 And so there's not much there. 382 00:23:06,751 --> 00:23:07,250 OK. 383 00:23:07,250 --> 00:23:08,720 What's needed. 384 00:23:08,720 --> 00:23:14,500 Well, what's needed is the electronic transition moment. 385 00:23:14,500 --> 00:23:17,450 Let's call this e1, e2. 386 00:23:17,450 --> 00:23:19,850 Going from two different electronic states, 387 00:23:19,850 --> 00:23:22,730 there is an electric dipole transition moment, 388 00:23:22,730 --> 00:23:25,070 which is not equal to zero. 389 00:23:25,070 --> 00:23:30,120 So H2, which has no vibrational spectrum 390 00:23:30,120 --> 00:23:34,540 and no rotational spectrum has an electronic spectrum. 391 00:23:34,540 --> 00:23:36,460 Everything has an electronic spectrum. 392 00:23:40,350 --> 00:23:40,870 OK. 393 00:23:40,870 --> 00:23:44,050 Now, a diatomic molecule. 394 00:23:48,970 --> 00:23:50,950 Here is sort of a template for everything 395 00:23:50,950 --> 00:23:53,890 a diatomic molecule can do. 396 00:23:53,890 --> 00:23:55,300 Now, they're cleverer than this. 397 00:23:55,300 --> 00:23:57,850 But this is sort of in preparation 398 00:23:57,850 --> 00:24:04,700 for dealing with greater complexity. 399 00:24:04,700 --> 00:24:06,830 And then we can have up here. 400 00:24:10,051 --> 00:24:10,550 OK. 401 00:24:10,550 --> 00:24:12,980 So this is the electronic ground state. 402 00:24:12,980 --> 00:24:17,300 And normally, if you have a diatomic molecule, 403 00:24:17,300 --> 00:24:21,080 you can predict what is the electronic ground state 404 00:24:21,080 --> 00:24:23,074 and how is it going to look-- 405 00:24:23,074 --> 00:24:24,740 how is its potential curve going to look 406 00:24:24,740 --> 00:24:28,560 relative to the excited states. 407 00:24:28,560 --> 00:24:30,300 That's something you should be able to do 408 00:24:30,300 --> 00:24:32,550 using LCAO-MO theory. 409 00:24:35,380 --> 00:24:37,580 OK. 410 00:24:37,580 --> 00:24:39,080 So this is the ground state. 411 00:24:43,310 --> 00:24:44,550 This is the repulsive state. 412 00:24:50,240 --> 00:24:52,970 And usually, the ground state correlates 413 00:24:52,970 --> 00:24:58,920 with ground state of the atoms. 414 00:24:58,920 --> 00:25:01,010 And so here we have a repulsive state, 415 00:25:01,010 --> 00:25:03,570 which also correlates with the ground state of the atom. 416 00:25:03,570 --> 00:25:06,920 So this is an excited state, and that correlates with an excited 417 00:25:06,920 --> 00:25:09,990 state of the atoms. 418 00:25:09,990 --> 00:25:14,915 And this is AB plus an electron. 419 00:25:18,070 --> 00:25:21,810 So that's one way the molecule breaks. 420 00:25:21,810 --> 00:25:25,140 And this dotted curve represents Rydberg 421 00:25:25,140 --> 00:25:30,220 states They're Rydberg states converging 422 00:25:30,220 --> 00:25:34,780 to every rotation vibration level of this excited state. 423 00:25:34,780 --> 00:25:36,110 And it can be complicated. 424 00:25:36,110 --> 00:25:37,810 But it's beautiful, because I know 425 00:25:37,810 --> 00:25:42,490 the magic decoder for how do you deal with Rydberg states. 426 00:25:42,490 --> 00:25:45,370 Because there's a lot of them, but they're 427 00:25:45,370 --> 00:25:47,090 closely related to each other. 428 00:25:47,090 --> 00:25:49,360 And we can exploit that relationship 429 00:25:49,360 --> 00:25:52,160 in guiding an experiment. 430 00:25:52,160 --> 00:25:55,510 And so here, now we have a curve crossing 431 00:25:55,510 --> 00:26:00,180 between a repulsive state and a bound state. 432 00:26:00,180 --> 00:26:03,088 And that leads to what we call predissociation. 433 00:26:07,870 --> 00:26:10,630 So the vibrational level of this state-- 434 00:26:10,630 --> 00:26:15,100 which would normally be bound above the curve crossing-- 435 00:26:15,100 --> 00:26:17,420 are not bound. 436 00:26:17,420 --> 00:26:20,540 And that's encoded in the spectrum too. 437 00:26:20,540 --> 00:26:24,930 And so for more complicated molecules, 438 00:26:24,930 --> 00:26:28,590 they're going to be these curve crossings or surface crossings. 439 00:26:28,590 --> 00:26:31,110 And we want to know how do we deal with them, 440 00:26:31,110 --> 00:26:36,190 and what is a diatomic-like way of dealing with them. 441 00:26:36,190 --> 00:26:43,240 And the important thing is at that internuclear distance 442 00:26:43,240 --> 00:26:50,030 where the curves cross, then you could be at a level-- 443 00:26:50,030 --> 00:26:52,730 starting at a level on this state-- 444 00:26:52,730 --> 00:26:54,890 the bound state. 445 00:26:54,890 --> 00:26:58,400 And it will have the same momentum at the crossing radius 446 00:26:58,400 --> 00:27:00,950 as the repulsive state. 447 00:27:00,950 --> 00:27:02,010 And that's where it goes. 448 00:27:02,010 --> 00:27:05,100 That's where it leaks out. 449 00:27:05,100 --> 00:27:10,350 I mean, we're normally used to thinking about processes-- 450 00:27:10,350 --> 00:27:13,620 non-radiative processes, all kinds of processes-- 451 00:27:13,620 --> 00:27:15,240 as an integral overall space. 452 00:27:18,192 --> 00:27:23,610 But because the momenta are the same 453 00:27:23,610 --> 00:27:26,990 on the two curves at this radius, 454 00:27:26,990 --> 00:27:29,090 they can go freely from one to the other. 455 00:27:29,090 --> 00:27:31,370 The molecules can go freely from one to the other. 456 00:27:31,370 --> 00:27:36,010 We get a tremendous simplification. 457 00:27:36,010 --> 00:27:39,880 And this is something that is always ignored in textbooks 458 00:27:39,880 --> 00:27:42,820 and is a profound insight. 459 00:27:42,820 --> 00:27:46,540 Because you know exactly where things happen and why 460 00:27:46,540 --> 00:27:48,480 they happen. 461 00:27:48,480 --> 00:27:51,990 And so you can arrange the information 462 00:27:51,990 --> 00:27:54,940 to describe what's going on at this point. 463 00:27:54,940 --> 00:27:58,110 And this is where semi-classical theory is really valuable. 464 00:27:58,110 --> 00:28:02,280 Because not only do you know that you have stationary phase 465 00:28:02,280 --> 00:28:02,910 at this point. 466 00:28:02,910 --> 00:28:07,800 But you know what the spatial oscillation frequency is. 467 00:28:07,800 --> 00:28:11,310 Because the spatial oscillation frequency is H-- 468 00:28:11,310 --> 00:28:14,270 or the wavelength-- is h over p. 469 00:28:17,830 --> 00:28:20,320 And you know what the momentum is at this point. 470 00:28:20,320 --> 00:28:23,140 And so you know where the nodes are. 471 00:28:23,140 --> 00:28:26,980 How far the nodes are apart and what the amplitude is here. 472 00:28:26,980 --> 00:28:28,630 And so it tells you exactly what you 473 00:28:28,630 --> 00:28:30,970 want to know in order to describe 474 00:28:30,970 --> 00:28:34,600 this non-radiative process. 475 00:28:34,600 --> 00:28:39,820 And my belief is that almost all of the complex things that 476 00:28:39,820 --> 00:28:43,570 molecules do happen at a predictable region 477 00:28:43,570 --> 00:28:45,490 in coordinated space. 478 00:28:45,490 --> 00:28:49,600 And you can get the information you need to understand them, 479 00:28:49,600 --> 00:28:51,700 because all of a sudden, the molecule 480 00:28:51,700 --> 00:28:55,750 is behaving in a kind of classical way. 481 00:28:55,750 --> 00:28:58,735 And we're entitled to think locally rather than globally. 482 00:29:02,160 --> 00:29:03,140 OK. 483 00:29:03,140 --> 00:29:05,210 I'm going very slowly. 484 00:29:05,210 --> 00:29:07,880 We may get through most of what I planned to talk about today. 485 00:29:07,880 --> 00:29:10,952 I have 11 pages of notes, and this is-- 486 00:29:10,952 --> 00:29:12,950 I'm on page three. 487 00:29:12,950 --> 00:29:16,160 So maybe we'll take off. 488 00:29:16,160 --> 00:29:17,450 OK. 489 00:29:17,450 --> 00:29:19,700 So how do you do spectra? 490 00:29:19,700 --> 00:29:23,780 Well, in the old days, you had some light source 491 00:29:23,780 --> 00:29:28,020 like a candle, and you had a lens, 492 00:29:28,020 --> 00:29:31,410 and you had an absorption cell. 493 00:29:31,410 --> 00:29:33,870 And there would be some sort of a spectrometer here. 494 00:29:36,840 --> 00:29:37,660 Could be a grading. 495 00:29:37,660 --> 00:29:38,610 It could be a prism. 496 00:29:38,610 --> 00:29:39,930 It could be anything. 497 00:29:39,930 --> 00:29:42,600 But it's something that says, OK, I 498 00:29:42,600 --> 00:29:45,870 looked at the selected wavelengths 499 00:29:45,870 --> 00:29:49,620 at which the gas in this cell removed light 500 00:29:49,620 --> 00:29:51,840 from the continuum. 501 00:29:51,840 --> 00:29:54,780 And then you have a detector, which in the old days 502 00:29:54,780 --> 00:29:57,660 was a photographic plate. 503 00:29:57,660 --> 00:29:59,730 But it could be a photo multiplier, 504 00:29:59,730 --> 00:30:03,570 and you're looking at the spectrum 505 00:30:03,570 --> 00:30:08,720 by scanning the grading or something like that. 506 00:30:08,720 --> 00:30:11,950 And so what you would get is some kind 507 00:30:11,950 --> 00:30:21,030 of a record where you have dark regions corresponding to where 508 00:30:21,030 --> 00:30:26,710 there has been no absorption and, well, actually it 509 00:30:26,710 --> 00:30:27,960 would be the other way around. 510 00:30:27,960 --> 00:30:30,480 There'd be bright regions, because there'd 511 00:30:30,480 --> 00:30:34,740 be no exposure of the emulsion on the plate and dark regions 512 00:30:34,740 --> 00:30:37,320 where there-- 513 00:30:37,320 --> 00:30:39,221 yes-- where the light hits. 514 00:30:39,221 --> 00:30:39,720 OK. 515 00:30:39,720 --> 00:30:42,450 So but we're much cleverer than this. 516 00:30:42,450 --> 00:30:45,640 And we can do all sorts of wonderful things. 517 00:30:45,640 --> 00:30:49,620 And again, I've been around for a long time 518 00:30:49,620 --> 00:30:52,710 and lasers were just beginning to be used 519 00:30:52,710 --> 00:30:54,960 when I was a graduate student. 520 00:30:54,960 --> 00:30:58,440 And I was one of the first small molecules 521 00:30:58,440 --> 00:31:00,460 spectroscopists to use lasers. 522 00:31:00,460 --> 00:31:02,220 But not as a graduate student. 523 00:31:02,220 --> 00:31:05,400 I wanted them, but we had such-- 524 00:31:05,400 --> 00:31:14,610 lasers were so terrible in the region between 1965 and 1971 525 00:31:14,610 --> 00:31:16,290 when I was a graduate student. 526 00:31:16,290 --> 00:31:20,950 And so lasers were things to be admired, but hardly to be used. 527 00:31:20,950 --> 00:31:23,870 But one of the crucial things was dye lasers. 528 00:31:28,540 --> 00:31:31,610 Because these guys are monochromatic, 529 00:31:31,610 --> 00:31:36,740 and they can be tuned and tuned continuously over a wide region 530 00:31:36,740 --> 00:31:38,350 of the spectrum. 531 00:31:38,350 --> 00:31:43,100 And so that's way better than a candle or a light bulb. 532 00:31:43,100 --> 00:31:45,292 Because it's monochromatic, and you're asking one 533 00:31:45,292 --> 00:31:47,000 question at a time as you tune the laser. 534 00:31:52,050 --> 00:31:57,380 Now, lasers enable you to do many kinds of experiments. 535 00:31:57,380 --> 00:32:09,140 You can simply tune the laser through a series 536 00:32:09,140 --> 00:32:17,040 of transitions, and you get fluorescence 537 00:32:17,040 --> 00:32:19,870 every time the laser tunes through a transition. 538 00:32:22,500 --> 00:32:26,990 And if one laser is good, two lasers are better. 539 00:32:26,990 --> 00:32:29,180 And so you can do all sorts of things 540 00:32:29,180 --> 00:32:36,610 like suppose this spectrum is really complicated. 541 00:32:36,610 --> 00:32:39,420 And you want to be able to simplify it. 542 00:32:39,420 --> 00:32:41,760 And so you can do a double resonance experiment 543 00:32:41,760 --> 00:32:43,770 where you tune this laser to one line 544 00:32:43,770 --> 00:32:47,700 and then you tune this laser through a series of transition. 545 00:32:47,700 --> 00:32:49,594 That spectrum is going to be simple, 546 00:32:49,594 --> 00:32:51,510 and it's going to be telling you who this was. 547 00:32:54,070 --> 00:32:58,090 And there's just no end of tricks. 548 00:32:58,090 --> 00:33:05,860 And often, instead of detecting the fluorescence, 549 00:33:05,860 --> 00:33:08,080 you tune the laser. 550 00:33:08,080 --> 00:33:09,370 Let's do this. 551 00:33:09,370 --> 00:33:13,930 And so starting here is some kind of a continuum, ionization 552 00:33:13,930 --> 00:33:15,140 continuum. 553 00:33:15,140 --> 00:33:21,670 And so you have this photon being used twice. 554 00:33:21,670 --> 00:33:24,730 One here and one to take it above the ionization limit. 555 00:33:24,730 --> 00:33:29,080 And so you have an excitation, which you do 556 00:33:29,080 --> 00:33:30,970 want to know how strong it is. 557 00:33:30,970 --> 00:33:33,070 And so you might monitor the fluorescence. 558 00:33:33,070 --> 00:33:35,380 But you don't know who this is, and you find out 559 00:33:35,380 --> 00:33:36,640 by tuning this. 560 00:33:36,640 --> 00:33:41,290 But you would detect the excitation here 561 00:33:41,290 --> 00:33:43,630 by subsequent ionization. 562 00:33:43,630 --> 00:33:46,860 It's easy to collect ions. 563 00:33:46,860 --> 00:33:49,890 Every ion you produce, you can detect. 564 00:33:49,890 --> 00:33:51,900 Every photon you produce, you can't detect. 565 00:33:51,900 --> 00:33:55,230 Because you have a solid angle consideration and photo 566 00:33:55,230 --> 00:33:57,120 multipliers are not perfect. 567 00:33:57,120 --> 00:34:01,066 And so ionization detection is way more sensitive. 568 00:34:01,066 --> 00:34:02,440 So you can do that kind of thing. 569 00:34:08,280 --> 00:34:12,989 You can also do-- 570 00:34:12,989 --> 00:34:16,230 this is a kind of sequential excitation. 571 00:34:16,230 --> 00:34:18,989 You could imagine doing an experiment where 572 00:34:18,989 --> 00:34:24,460 you have an energy level here, and you have a laser, 573 00:34:24,460 --> 00:34:26,920 which is not on resonance. 574 00:34:26,920 --> 00:34:28,520 That's a coherent process. 575 00:34:28,520 --> 00:34:34,449 It uses the oscilltor strength at this level to get to here. 576 00:34:34,449 --> 00:34:39,230 And that's related to many other kinds of current experiments. 577 00:34:39,230 --> 00:34:39,940 And that's neat. 578 00:34:50,659 --> 00:34:53,810 Now, we're recording spectra, and we need 579 00:34:53,810 --> 00:34:57,460 to know what the rules are. 580 00:34:57,460 --> 00:35:02,060 And so there are certain transitions that are allowed 581 00:35:02,060 --> 00:35:05,050 and certain transitions that are forbidden. 582 00:35:05,050 --> 00:35:11,500 Now, I talked about the transition requirements 583 00:35:11,500 --> 00:35:15,930 for rotation, vibration, and electronic. 584 00:35:15,930 --> 00:35:18,190 But let's just talk about the electronic spectrum, 585 00:35:18,190 --> 00:35:21,060 because the other two are simple. 586 00:35:21,060 --> 00:35:29,570 The transition operator is equal to the sum over electrons 587 00:35:29,570 --> 00:35:35,550 of e times r sub i. 588 00:35:35,550 --> 00:35:38,840 It's a one electron operator. 589 00:35:38,840 --> 00:35:40,910 And that means if we have wave functions which 590 00:35:40,910 --> 00:35:53,000 are Slater determinates of spin orbitals like 2s alpha. 591 00:35:57,390 --> 00:36:01,560 This one electron operator can only 592 00:36:01,560 --> 00:36:06,030 have a non-zero matrix element if the two states 593 00:36:06,030 --> 00:36:09,760 differ by one spin orbital. 594 00:36:09,760 --> 00:36:12,780 That's a big simplification. 595 00:36:12,780 --> 00:36:17,960 And so one can actually use this to selectively access 596 00:36:17,960 --> 00:36:21,140 different kinds of states by designing an experiment. 597 00:36:21,140 --> 00:36:24,800 But the important thing is that for electronic transitions, 598 00:36:24,800 --> 00:36:30,180 we have the selection rule delta so is equal to 1. 599 00:36:30,180 --> 00:36:31,680 Not 2. 600 00:36:31,680 --> 00:36:34,540 Not 0. 601 00:36:34,540 --> 00:36:42,150 And now, the operator doesn't have any spin involved with it. 602 00:36:42,150 --> 00:36:47,060 And so that means delta s equals 0. 603 00:36:47,060 --> 00:36:50,350 You did not change, you did not go from a singlet state 604 00:36:50,350 --> 00:36:52,780 to a triplet state. 605 00:36:52,780 --> 00:36:56,830 And the only way you can get from a singlet state 606 00:36:56,830 --> 00:36:59,620 to a triplet state is if the triplet state is 607 00:36:59,620 --> 00:37:01,270 perturbed by a singlet state. 608 00:37:03,930 --> 00:37:08,330 So this picture I drew where I had a repulsive state crossing 609 00:37:08,330 --> 00:37:11,450 through a bound state, it might have been that one of those 610 00:37:11,450 --> 00:37:13,010 was a triplet. 611 00:37:13,010 --> 00:37:16,940 And as a result, and the other is a singlet, 612 00:37:16,940 --> 00:37:19,610 and you have spin orbit interactions, 613 00:37:19,610 --> 00:37:23,900 and you get extra states, extra lines appearing. 614 00:37:23,900 --> 00:37:29,750 But you get this wonderful selection rule. 615 00:37:29,750 --> 00:37:36,075 There is also for the electric dipole 616 00:37:36,075 --> 00:37:42,980 that we have plus to minus parity. 617 00:37:42,980 --> 00:37:44,970 Now what's parity? 618 00:37:44,970 --> 00:37:46,710 I don't like talking about parity, 619 00:37:46,710 --> 00:37:51,970 because the useful definition leads to complexity. 620 00:37:51,970 --> 00:37:54,450 But basically, parity corresponds 621 00:37:54,450 --> 00:38:03,060 to the symmetry, the inversion symmetry in the laboratory 622 00:38:03,060 --> 00:38:04,130 frame. 623 00:38:04,130 --> 00:38:06,960 Now, you say, well, a molecule doesn't have any inversion 624 00:38:06,960 --> 00:38:07,470 symmetry. 625 00:38:07,470 --> 00:38:10,180 But space is isotopic. 626 00:38:10,180 --> 00:38:12,510 And so you can go from a left handed to a right handed 627 00:38:12,510 --> 00:38:13,760 coordinate system. 628 00:38:13,760 --> 00:38:16,720 And that's what happens when you invert space. 629 00:38:16,720 --> 00:38:19,740 And so you can classify levels according 630 00:38:19,740 --> 00:38:22,200 to whether they're odd or even with respect 631 00:38:22,200 --> 00:38:23,520 to space inversion. 632 00:38:23,520 --> 00:38:25,010 This is close to the truth. 633 00:38:25,010 --> 00:38:28,500 This is close to all you need to know unless you're actually 634 00:38:28,500 --> 00:38:32,190 going to do stuff with the parity operator. 635 00:38:32,190 --> 00:38:34,360 But it's a useful way of saying, OK, 636 00:38:34,360 --> 00:38:35,760 I put parity labels on things. 637 00:38:35,760 --> 00:38:38,280 I learned how to do that, and that's enough. 638 00:38:42,880 --> 00:38:50,290 Now, you follow selection rules where good quantum numbers 639 00:38:50,290 --> 00:38:51,940 are conserved. 640 00:38:51,940 --> 00:38:54,940 Or they change in a way that you predict 641 00:38:54,940 --> 00:38:57,960 based on the way you did the experiment. 642 00:38:57,960 --> 00:39:01,300 A good quantum number, I remind you, 643 00:39:01,300 --> 00:39:03,940 is the eigenvalue of an operator that commutes 644 00:39:03,940 --> 00:39:05,494 with the exact Hamiltonian. 645 00:39:13,780 --> 00:39:16,910 There are very few rigorously good quantum numbers. 646 00:39:16,910 --> 00:39:19,160 But if a molecule has any symmetry, 647 00:39:19,160 --> 00:39:23,120 group theory tells you a bunch of things 648 00:39:23,120 --> 00:39:25,580 that commute with the Hamiltonian, 649 00:39:25,580 --> 00:39:28,160 and it gives you symmetry labels. 650 00:39:28,160 --> 00:39:30,470 And that's very important in inorganic chemistry 651 00:39:30,470 --> 00:39:34,940 where you have either molecules with symmetry or molecules 652 00:39:34,940 --> 00:39:40,460 with atoms with ligands in a symmetric arrangement. 653 00:39:40,460 --> 00:39:43,820 And since the transition is on the center atom 654 00:39:43,820 --> 00:39:48,770 usually that you can classify them using group theory as 655 00:39:48,770 --> 00:39:50,200 allowed or forbidden. 656 00:39:56,900 --> 00:40:05,550 So if we have an electronic transition, 657 00:40:05,550 --> 00:40:11,690 the easiest thing to observe is vibrational bands. 658 00:40:11,690 --> 00:40:14,870 If you have a relatively low resolution spectrum, 659 00:40:14,870 --> 00:40:17,150 you're going to see vibrational bands. 660 00:40:17,150 --> 00:40:20,210 You have to work harder to see the rotational transitions 661 00:40:20,210 --> 00:40:25,190 in each vibrational band, but you get an enormous amount 662 00:40:25,190 --> 00:40:28,040 of qualitative information just looking 663 00:40:28,040 --> 00:40:29,810 at the vibrational bands. 664 00:40:29,810 --> 00:40:43,060 Because the vibrational bands encode the difference 665 00:40:43,060 --> 00:40:49,070 between the ground state potential and the excited state 666 00:40:49,070 --> 00:40:50,060 potential. 667 00:40:50,060 --> 00:40:52,370 Now, this is a universal notation. 668 00:40:52,370 --> 00:40:54,560 Ground state is always double prime. 669 00:40:54,560 --> 00:40:56,570 Upper state is always single prime. 670 00:40:56,570 --> 00:40:59,480 Very strange, but that's the way it is. 671 00:40:59,480 --> 00:41:02,980 And so if these potentials are different, 672 00:41:02,980 --> 00:41:05,941 the vibrational bands encode the difference. 673 00:41:10,860 --> 00:41:13,740 And this comes from the Franck-Condon principle, which 674 00:41:13,740 --> 00:41:19,120 says nuclei move slowly, electrons move fast. 675 00:41:19,120 --> 00:41:21,780 The transition is an instantaneous process, 676 00:41:21,780 --> 00:41:23,670 as far as the nuclei are concerned. 677 00:41:23,670 --> 00:41:28,500 And so there's no change in nuclear coordinates, 678 00:41:28,500 --> 00:41:32,280 and there is no change in nuclear momentum. 679 00:41:32,280 --> 00:41:34,146 This is what's in all the textbooks, 680 00:41:34,146 --> 00:41:35,520 and nobody ever talks about this, 681 00:41:35,520 --> 00:41:43,420 because we don't really normally know or think about momentum. 682 00:41:43,420 --> 00:41:45,370 But we do know what it is. 683 00:41:45,370 --> 00:41:47,170 We do know what the operator is. 684 00:41:47,170 --> 00:41:51,040 And we know that kinetic energy is 685 00:41:51,040 --> 00:41:55,240 related to the momentum squared over 2 times the mass. 686 00:41:58,660 --> 00:41:59,361 So what is this? 687 00:41:59,361 --> 00:42:00,860 This means transitions are vertical. 688 00:42:03,525 --> 00:42:09,970 In other words, if we have a pair of electronic states, 689 00:42:09,970 --> 00:42:12,880 we draw these vertical lines. 690 00:42:12,880 --> 00:42:15,760 Not slanting lines. 691 00:42:15,760 --> 00:42:19,340 This means momentum is conserved. 692 00:42:19,340 --> 00:42:24,110 And this, here at this vertical point, 693 00:42:24,110 --> 00:42:25,980 we have this much momentum. 694 00:42:25,980 --> 00:42:30,080 And, well, there's the same amount here. 695 00:42:30,080 --> 00:42:32,660 Now usually, this just means-- 696 00:42:32,660 --> 00:42:35,000 the delta p is equal to 0-- 697 00:42:35,000 --> 00:42:38,300 means that of all the strong transitions, 698 00:42:38,300 --> 00:42:43,570 turning point to turning point transitions are the strongest. 699 00:42:43,570 --> 00:42:49,450 Because at a turning point, the vibrational amplitude is large. 700 00:42:49,450 --> 00:42:51,100 But there's more to it than that, 701 00:42:51,100 --> 00:42:53,730 because there are secondary maxima 702 00:42:53,730 --> 00:42:55,780 in the vibrational transition intensities. 703 00:42:55,780 --> 00:43:00,760 These correspond to stationary phase between the initial state 704 00:43:00,760 --> 00:43:02,230 and the final state. 705 00:43:02,230 --> 00:43:05,640 And so in addition to the strongest transitions, 706 00:43:05,640 --> 00:43:10,980 you get other transitions that you can explain by this delta p 707 00:43:10,980 --> 00:43:11,610 equals 0. 708 00:43:14,810 --> 00:43:19,255 Now, you don't know anything when you start. 709 00:43:19,255 --> 00:43:22,130 You know something maybe about the ground state. 710 00:43:22,130 --> 00:43:24,955 And this could be a polyatomic molecule 711 00:43:24,955 --> 00:43:27,490 if we're just looking at one mode. 712 00:43:27,490 --> 00:43:31,820 And suppose we have an excited state where 713 00:43:31,820 --> 00:43:36,160 the vibrational frequency is the same. 714 00:43:36,160 --> 00:43:40,850 In other words, there is no change in bonding character. 715 00:43:40,850 --> 00:43:45,950 And so what you end up getting is just 716 00:43:45,950 --> 00:43:49,740 the zero to zero transition in absorption. 717 00:43:49,740 --> 00:43:54,080 Or if you have many vibrational levels up here, you see v to v, 718 00:43:54,080 --> 00:43:55,280 delta v equals 0. 719 00:44:01,490 --> 00:44:05,170 Now, you could have a bound state, 720 00:44:05,170 --> 00:44:10,470 and it's usually true that the excited state is less bound. 721 00:44:10,470 --> 00:44:13,780 And so you would have-- 722 00:44:17,250 --> 00:44:20,850 the Franck-Condon active region corresponds 723 00:44:20,850 --> 00:44:23,820 to turning point to turning point in the lower state. 724 00:44:23,820 --> 00:44:25,390 So I shouldn't draw this. 725 00:44:25,390 --> 00:44:31,580 Let's draw a vertical transition. 726 00:44:31,580 --> 00:44:33,180 Well, I missed. 727 00:44:33,180 --> 00:44:34,950 Let me just start again. 728 00:44:34,950 --> 00:44:39,150 So we have an excited state, and we have a ground state. 729 00:44:39,150 --> 00:44:40,920 Ground state is bound. 730 00:44:40,920 --> 00:44:42,640 And this is v equal 0. 731 00:44:42,640 --> 00:44:45,890 And so we go from here to there. 732 00:44:50,270 --> 00:44:54,540 So if the excited state is less bound, 733 00:44:54,540 --> 00:44:58,460 it's a larger inner nucleus and smaller vibrational frequency. 734 00:44:58,460 --> 00:45:02,480 We have many vibrational levels accessed 735 00:45:02,480 --> 00:45:04,552 by the Franck-Condon principle. 736 00:45:07,930 --> 00:45:10,810 And if we have, in the other sense, 737 00:45:10,810 --> 00:45:14,080 we have an excited state, which is more bound than the ground 738 00:45:14,080 --> 00:45:23,165 state then the Franck-Condon region is narrower. 739 00:45:27,620 --> 00:45:30,290 Because this wall is nearly vertical. 740 00:45:30,290 --> 00:45:35,000 You have more transitions when it's displaced this way. 741 00:45:35,000 --> 00:45:39,230 And this branch of the potential is nearly much flatter, 742 00:45:39,230 --> 00:45:42,560 and you have fewer transitions here. 743 00:45:42,560 --> 00:45:47,420 So the vibrational pattern tells you qualitatively from the 744 00:45:47,420 --> 00:45:51,031 get go whether you're going to a more bound or a less bound 745 00:45:51,031 --> 00:45:51,530 state. 746 00:45:54,050 --> 00:45:56,017 That's very useful information. 747 00:46:06,090 --> 00:46:09,590 Now, you want to go further. 748 00:46:09,590 --> 00:46:11,860 You want to say, oh, well, I'm observing 749 00:46:11,860 --> 00:46:15,840 a bunch of vibrational levels in the excited state. 750 00:46:15,840 --> 00:46:18,900 What are their quantum numbers? 751 00:46:18,900 --> 00:46:21,810 Are we seeing the v equals zero level? 752 00:46:21,810 --> 00:46:27,130 How do we know what vibrational level we're observing? 753 00:46:27,130 --> 00:46:29,380 Or levels? 754 00:46:29,380 --> 00:46:31,260 And we can calculate Franck-Condon factors. 755 00:46:35,160 --> 00:46:38,400 But you can do many things. 756 00:46:38,400 --> 00:46:42,550 So I mean, if you have a situation like this, 757 00:46:42,550 --> 00:46:45,630 you might not get to the lowest vibrational level. 758 00:46:45,630 --> 00:46:47,260 But if you have a situation like this, 759 00:46:47,260 --> 00:46:49,740 you probably will get to the lowest vibrational level. 760 00:46:49,740 --> 00:46:52,050 So one way to get vibrational numbering 761 00:46:52,050 --> 00:46:56,100 is seeing the vibrational pattern terminate. 762 00:46:56,100 --> 00:46:58,500 That's a good sign it's v equals 0. 763 00:46:58,500 --> 00:47:00,370 If it terminates abruptly, it's v equal 0. 764 00:47:00,370 --> 00:47:04,010 But if it terminates slowly, you're not sure. 765 00:47:04,010 --> 00:47:08,150 But there are other really wonderful things about this. 766 00:47:08,150 --> 00:47:12,240 And you can do isotope separation. 767 00:47:12,240 --> 00:47:14,940 Isotope shifts. 768 00:47:18,220 --> 00:47:20,740 Since the vibrational frequency is 769 00:47:20,740 --> 00:47:27,490 the square root of k over mu, we can change and reduce mass. 770 00:47:27,490 --> 00:47:31,180 That changes which vibrational bands you observe, 771 00:47:31,180 --> 00:47:33,250 and it changes it in a quantitative way, 772 00:47:33,250 --> 00:47:37,510 and so you can often use that to tell. 773 00:47:37,510 --> 00:47:43,330 Another thing you can do is to now. 774 00:47:43,330 --> 00:47:47,480 If you have an excited state, the vibrational wave function 775 00:47:47,480 --> 00:47:51,030 is going to look like this. 776 00:47:51,030 --> 00:47:53,020 And there's a node here, and a node here, 777 00:47:53,020 --> 00:47:55,230 and node here, node here, node here. 778 00:47:55,230 --> 00:47:59,010 And so if we're looking at the vibrational progression 779 00:47:59,010 --> 00:48:02,760 observed from such a level, the intensities 780 00:48:02,760 --> 00:48:07,920 will have minima corresponding to how many nodes there 781 00:48:07,920 --> 00:48:09,150 are in the excited state. 782 00:48:09,150 --> 00:48:12,270 And the number of nodes is the vibrational quantum number. 783 00:48:15,570 --> 00:48:18,450 So there are lots of ways of doing it. 784 00:48:18,450 --> 00:48:21,200 And then there is something else. 785 00:48:21,200 --> 00:48:24,320 If you observe at moderately low resolution 786 00:48:24,320 --> 00:48:28,700 a vibrational band in an electronic spectrum, 787 00:48:28,700 --> 00:48:40,250 it will have the peculiar shape like that or like that. 788 00:48:40,250 --> 00:48:42,950 This is called a band head. 789 00:48:42,950 --> 00:48:45,290 And it corresponds to the fact that because 790 00:48:45,290 --> 00:48:48,560 the rotational constants in the upper and lower state 791 00:48:48,560 --> 00:48:51,980 are different, one of the branches-- 792 00:48:51,980 --> 00:48:58,460 you have delta J equals plus or minus 1, 0. 793 00:48:58,460 --> 00:49:04,340 And the delta J plus 1 is called the R branch and minus 1 794 00:49:04,340 --> 00:49:06,550 is called the P branch. 795 00:49:06,550 --> 00:49:09,160 And delta J of 0 is called the Q branch. 796 00:49:09,160 --> 00:49:12,830 Now, these two guys are such that depending 797 00:49:12,830 --> 00:49:15,780 on the sign of the difference in B values, 798 00:49:15,780 --> 00:49:21,290 you get ahead on the high frequency side 799 00:49:21,290 --> 00:49:25,380 or ahead on the low frequency side. 800 00:49:25,380 --> 00:49:27,350 Well, actually, it's the other way around. 801 00:49:27,350 --> 00:49:31,800 But it tells you then the rotational constant, 802 00:49:31,800 --> 00:49:35,360 which is also a signal of how bound the state is. 803 00:49:35,360 --> 00:49:40,130 The rotational constant-- the shading of these band heads-- 804 00:49:40,130 --> 00:49:43,430 confirms your vibrational assignment. 805 00:49:43,430 --> 00:49:48,000 And typically, if you have a smaller vibrational frequency, 806 00:49:48,000 --> 00:49:52,120 you'll also have a smaller rotational constant. 807 00:49:52,120 --> 00:49:53,240 It's not always true. 808 00:49:53,240 --> 00:49:55,990 And it's always interesting when it doesn't happen. 809 00:49:55,990 --> 00:49:59,850 And so the vibrational bands have this asymmetric shape 810 00:49:59,850 --> 00:50:07,480 unless the rotational constant upstairs and downstairs 811 00:50:07,480 --> 00:50:08,380 is the same. 812 00:50:08,380 --> 00:50:11,170 And then you have a branch going this way and a branch 813 00:50:11,170 --> 00:50:11,800 going this way. 814 00:50:11,800 --> 00:50:13,633 And a gap in the middle that might be filled 815 00:50:13,633 --> 00:50:15,790 with some Q branch lines. 816 00:50:15,790 --> 00:50:18,460 And the Q branch lines tell you, oh, yeah, that 817 00:50:18,460 --> 00:50:26,470 was a transition where the lambda, the projection of L 818 00:50:26,470 --> 00:50:29,970 on the internuclear axis changed. 819 00:50:29,970 --> 00:50:32,600 And if it doesn't have this Q branch, 820 00:50:32,600 --> 00:50:35,270 it'll tell you this delta lambda equals 0. 821 00:50:35,270 --> 00:50:37,620 All sorts of stuff. 822 00:50:37,620 --> 00:50:42,790 And if you have no sign of band heads, 823 00:50:42,790 --> 00:50:45,040 but just this double hump structure, 824 00:50:45,040 --> 00:50:48,500 it tells you that the rotational constant is about the same as 825 00:50:48,500 --> 00:50:49,520 in the ground state. 826 00:50:49,520 --> 00:50:52,160 And it tells you also that the vibrational frequency is 827 00:50:52,160 --> 00:50:53,570 expected to be about the same. 828 00:50:58,750 --> 00:51:00,940 I better stop. 829 00:51:00,940 --> 00:51:05,120 So as soon as you go to polyatomic molecules. 830 00:51:05,120 --> 00:51:10,430 Instead of having just one vibration, you have 3n minus 6. 831 00:51:10,430 --> 00:51:14,090 And so 3n minus 6 downstairs, 3n minus 6 upstairs, 832 00:51:14,090 --> 00:51:16,590 that looks bad. 833 00:51:16,590 --> 00:51:21,550 But only some of the vibrational modes 834 00:51:21,550 --> 00:51:24,070 correspond to a distortion from-- 835 00:51:24,070 --> 00:51:26,010 a difference in geometry between the ground 836 00:51:26,010 --> 00:51:27,860 state and the excited state. 837 00:51:27,860 --> 00:51:33,160 And so most of the vibrational modes 838 00:51:33,160 --> 00:51:35,650 correspond to identical frequencies. 839 00:51:35,650 --> 00:51:38,830 And what we call that is Franck-Condon dark. 840 00:51:38,830 --> 00:51:42,520 Because the only transitions that are allowed are delta v 841 00:51:42,520 --> 00:51:45,070 equals 0 for that mode. 842 00:51:45,070 --> 00:51:47,710 And so only the modes that correspond 843 00:51:47,710 --> 00:51:53,180 to a change in structure appear as long progressions. 844 00:51:53,180 --> 00:51:55,370 And so there are only a few, and so the spectrum 845 00:51:55,370 --> 00:51:59,240 of a polyatomic molecule is not too much different 846 00:51:59,240 --> 00:52:01,220 from a diatomic molecule, because 847 00:52:01,220 --> 00:52:05,550 of the small number of Franck-Condon active modes. 848 00:52:05,550 --> 00:52:09,960 But you look closer and you see big differences. 849 00:52:09,960 --> 00:52:10,460 OK. 850 00:52:10,460 --> 00:52:16,400 So well, we'll talk more about this on Friday. 851 00:52:16,400 --> 00:52:19,340 And we might go to three lectures 852 00:52:19,340 --> 00:52:20,910 on this, because this is really-- 853 00:52:20,910 --> 00:52:22,700 you know the whole point of this course 854 00:52:22,700 --> 00:52:27,760 is to be able to understand how molecules talk to us.