1 00:00:00,120 --> 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 00:00:23,230 --> 00:00:25,930 PROFESSOR: Today, since the notes for this 9 00:00:25,930 --> 00:00:32,350 didn't exist until last night, I made copies for you. 10 00:00:32,350 --> 00:00:34,570 I will do a little more revisions 11 00:00:34,570 --> 00:00:37,370 on what appears on the web. 12 00:00:37,370 --> 00:00:40,970 But this is a really interesting and important topic. 13 00:00:40,970 --> 00:00:44,620 And so I think it's important that I give 14 00:00:44,620 --> 00:00:47,530 a little bit of extra guidance. 15 00:00:47,530 --> 00:00:53,090 So we're going to talk about what's in a spectrum. 16 00:00:53,090 --> 00:00:55,210 But in a very special sense, we're 17 00:00:55,210 --> 00:00:58,400 going to be talking about wave packet dynamics. 18 00:00:58,400 --> 00:01:05,810 And how do we understand what to expect in wave packet dynamics? 19 00:01:05,810 --> 00:01:09,240 And how do we approach it? 20 00:01:09,240 --> 00:01:13,490 And so let's begin talking about this. 21 00:01:13,490 --> 00:01:16,670 So you already know from my lectures 22 00:01:16,670 --> 00:01:22,220 earlier, if you have a time independent Hamiltonian, 23 00:01:22,220 --> 00:01:23,860 how do you create a wave packet? 24 00:01:27,680 --> 00:01:29,480 I mean, after this exam performance, 25 00:01:29,480 --> 00:01:31,886 I want some insights. 26 00:01:31,886 --> 00:01:36,464 AUDIENCE: [INAUDIBLE] the individual eigenstates. 27 00:01:36,464 --> 00:01:37,130 PROFESSOR: Yeah. 28 00:01:37,130 --> 00:01:39,830 So you use a short pulse. 29 00:01:39,830 --> 00:01:42,660 And you create some kind of linear combination. 30 00:01:42,660 --> 00:01:44,810 Usually, if you're not making eigenstates, 31 00:01:44,810 --> 00:01:46,340 that's a complication. 32 00:01:46,340 --> 00:01:48,140 And why would you want to do that? 33 00:01:48,140 --> 00:01:50,810 And the reason you do it is because these wave packets are 34 00:01:50,810 --> 00:01:52,190 going to move. 35 00:01:52,190 --> 00:01:54,470 And particles move. 36 00:01:54,470 --> 00:01:56,880 Your instincts are about particles moving. 37 00:01:56,880 --> 00:01:58,850 And there are rules for the particles moving, 38 00:01:58,850 --> 00:02:02,840 which are mostly what you have in your head. 39 00:02:02,840 --> 00:02:05,150 But there are some special stuff that has 40 00:02:05,150 --> 00:02:08,479 to do with quantum mechanics. 41 00:02:08,479 --> 00:02:13,260 And so I'd like to build from the bottom. 42 00:02:13,260 --> 00:02:18,690 And so I want you to be able to understand wave packets moving 43 00:02:18,690 --> 00:02:20,400 on a diatomic surface. 44 00:02:20,400 --> 00:02:23,490 And then we'll make a step into slightly more complicated 45 00:02:23,490 --> 00:02:24,400 situations. 46 00:02:30,536 --> 00:02:44,570 If we have a diatomic, we have a potential curve. 47 00:02:44,570 --> 00:02:48,500 And we have an initial state. 48 00:02:48,500 --> 00:02:52,070 And that initial state in V equals 49 00:02:52,070 --> 00:02:56,210 0 gives us access to a range of vibrational levels 50 00:02:56,210 --> 00:02:57,960 in the upper state. 51 00:02:57,960 --> 00:03:00,050 And if we have a short pulse, we can 52 00:03:00,050 --> 00:03:03,230 create some kind of a wave packet involving 53 00:03:03,230 --> 00:03:05,390 a linear combination of these vibrational levels. 54 00:03:10,410 --> 00:03:18,000 So suppose you had a polyatomic, 3N minus 6 vibrational modes. 55 00:03:21,650 --> 00:03:28,110 So what would you do to go from a diatonic molecule 56 00:03:28,110 --> 00:03:30,570 to a polyatomic molecule? 57 00:03:30,570 --> 00:03:34,110 Initially, you want to do something simple that 58 00:03:34,110 --> 00:03:40,830 doesn't involve anything that's especially new. 59 00:03:40,830 --> 00:03:45,420 So what would you say about dynamic diatomic polyatomic 60 00:03:45,420 --> 00:03:49,200 surface where there's 3N minus 6 as opposed to 1 61 00:03:49,200 --> 00:03:50,730 vibrational modes? 62 00:03:50,730 --> 00:03:58,389 What kind of a wave packet would you do, would you create? 63 00:03:58,389 --> 00:04:02,097 AUDIENCE: [INAUDIBLE] 64 00:04:02,097 --> 00:04:03,930 PROFESSOR: We had Franck-Condon bright modes 65 00:04:03,930 --> 00:04:06,210 and Franck-Condon dark modes. 66 00:04:06,210 --> 00:04:08,812 So it sounds like you're right on the verge 67 00:04:08,812 --> 00:04:10,020 of saying something profound. 68 00:04:10,020 --> 00:04:11,070 So say it. 69 00:04:11,070 --> 00:04:13,464 AUDIENCE: You're just going to have lot more coordinates. 70 00:04:13,464 --> 00:04:14,130 PROFESSOR: Yeah. 71 00:04:14,130 --> 00:04:15,880 You're going to have more coordinates. 72 00:04:15,880 --> 00:04:19,480 AUDIENCE: [INAUDIBLE] 73 00:04:19,480 --> 00:04:22,079 PROFESSOR: So you will create a wave packet 74 00:04:22,079 --> 00:04:25,320 on the Franck-Condon bright modes. 75 00:04:25,320 --> 00:04:29,280 And if there's more than one Franck-Condon bright mode, 76 00:04:29,280 --> 00:04:31,050 the wave packet will do just like what 77 00:04:31,050 --> 00:04:32,970 it would do on a diatomic molecule, 78 00:04:32,970 --> 00:04:35,130 except you might have two different frequencies. 79 00:04:38,010 --> 00:04:43,700 OK, well, there's additional complications. 80 00:04:43,700 --> 00:04:48,920 And that is, even a diatomic molecule, 81 00:04:48,920 --> 00:04:52,160 we don't have harmonic motion. 82 00:04:52,160 --> 00:04:56,560 So the wave packet, because of diagonal anharmonicity, 83 00:04:56,560 --> 00:04:58,810 will dephase. 84 00:04:58,810 --> 00:05:01,060 But the center of the wave packet 85 00:05:01,060 --> 00:05:04,030 will continue to move at the average 86 00:05:04,030 --> 00:05:07,780 of the vibrational frequencies for all these modes. 87 00:05:07,780 --> 00:05:10,540 You know that. 88 00:05:10,540 --> 00:05:13,480 And so in a polyatomic molecule, maybe there's two, 89 00:05:13,480 --> 00:05:16,550 maybe there's three Franck-Condon bright modes. 90 00:05:16,550 --> 00:05:20,900 And the wave packet will have several different components 91 00:05:20,900 --> 00:05:23,780 on the different Franck-Condon bright modes. 92 00:05:23,780 --> 00:05:25,920 And they'll be diagonal anharmonicities. 93 00:05:25,920 --> 00:05:30,780 And they will dephase and possibly do interesting things. 94 00:05:30,780 --> 00:05:36,680 So now, polyatomic molecules can do something else. 95 00:05:36,680 --> 00:05:40,800 Because there are, in fact, 3N minus 6 vibrational modes. 96 00:05:40,800 --> 00:05:44,930 And there's anharmonic couplings between the modes. 97 00:05:44,930 --> 00:05:51,970 So if you created some wave packet or set of wave packets 98 00:05:51,970 --> 00:05:56,260 in a polyatomic molecule, there will be anharmonic interactions 99 00:05:56,260 --> 00:05:57,010 between modes. 100 00:06:01,180 --> 00:06:04,240 And what you will have is the appearance 101 00:06:04,240 --> 00:06:08,770 of wave packets in the Franck-Condon dark modes 102 00:06:08,770 --> 00:06:10,060 with their own frequencies. 103 00:06:12,665 --> 00:06:14,290 So this is starting to get interesting. 104 00:06:14,290 --> 00:06:18,550 Because you hit this molecule. 105 00:06:18,550 --> 00:06:21,520 And it starts revealing its secrets. 106 00:06:21,520 --> 00:06:23,295 Where you had Franck-Condon dark modes, 107 00:06:23,295 --> 00:06:24,670 they're not going to talk to you. 108 00:06:24,670 --> 00:06:27,970 But now, because of the anharmonic couplings, they do. 109 00:06:31,270 --> 00:06:42,670 And then with a diatomic molecule, 110 00:06:42,670 --> 00:06:45,780 we might have an excited state. 111 00:06:45,780 --> 00:06:48,210 And it might be crossed by a repulsive state 112 00:06:48,210 --> 00:06:50,610 or crossed by another bound state. 113 00:06:50,610 --> 00:06:57,070 But if you have a wave packet operating on this bound state, 114 00:06:57,070 --> 00:07:04,760 it's going to be crossing through this region. 115 00:07:04,760 --> 00:07:09,740 What's so special about where curves cross? 116 00:07:09,740 --> 00:07:10,680 You know this, too. 117 00:07:16,380 --> 00:07:19,819 If you're in this state and at this point, 118 00:07:19,819 --> 00:07:20,985 you have this much momentum. 119 00:07:24,940 --> 00:07:27,870 And if you're on this curve at that point, you have the same. 120 00:07:27,870 --> 00:07:32,250 So you have two rapidly oscillating things 121 00:07:32,250 --> 00:07:35,190 with the same spatial oscillation frequency. 122 00:07:35,190 --> 00:07:38,070 And so there's a possibility that the wave packet 123 00:07:38,070 --> 00:07:41,040 can leak from this curve, from the bound curve, 124 00:07:41,040 --> 00:07:42,120 to the repulsive curve. 125 00:07:44,890 --> 00:07:47,450 And there are all sorts of things of that nature. 126 00:07:47,450 --> 00:07:50,640 I'm going to talk about all of that today. 127 00:07:50,640 --> 00:07:52,230 And one of the things-- the reason I 128 00:07:52,230 --> 00:07:54,870 made these notes is because I've always 129 00:07:54,870 --> 00:08:06,760 wanted to say something about the Landau-Zener model 130 00:08:06,760 --> 00:08:11,740 for crossing between different potential curves. 131 00:08:11,740 --> 00:08:15,640 And we talked a little bit before about driving too fast 132 00:08:15,640 --> 00:08:16,560 on a curvy road. 133 00:08:16,560 --> 00:08:19,660 Do you stay on it or do you hit the tree? 134 00:08:19,660 --> 00:08:24,760 And that's basically this picture of-- 135 00:08:24,760 --> 00:08:31,915 you have two curves that can cross or they could do this. 136 00:08:35,250 --> 00:08:39,360 This is the adiabatic representation, 137 00:08:39,360 --> 00:08:44,250 which is amenable to quantum chemical calculations. 138 00:08:44,250 --> 00:08:48,660 And this crossing curve is the diabatic representation, 139 00:08:48,660 --> 00:08:50,880 which chemists like because the electronic wave 140 00:08:50,880 --> 00:08:54,130 functions don't change as a function of internuclear 141 00:08:54,130 --> 00:08:55,240 distance. 142 00:08:55,240 --> 00:08:59,700 And so the issue is, when you have crossing curves 143 00:08:59,700 --> 00:09:02,010 or avoided crossing curves, how do you 144 00:09:02,010 --> 00:09:03,480 understand what the wave packet is 145 00:09:03,480 --> 00:09:06,190 going to do with those points? 146 00:09:06,190 --> 00:09:09,480 And that's the point of this lecture. 147 00:09:09,480 --> 00:09:19,950 And there are two experiments done by Ahmed Zewail's research 148 00:09:19,950 --> 00:09:25,860 group which illustrate many of the effects of wave packets 149 00:09:25,860 --> 00:09:27,180 and avoided crossings. 150 00:09:27,180 --> 00:09:30,120 And so the question is, how do we 151 00:09:30,120 --> 00:09:34,290 observe this kind of evolution of the wave packet? 152 00:09:34,290 --> 00:09:35,970 What kind of experiments do we do? 153 00:09:39,200 --> 00:09:42,070 If we're talking about motion, we're 154 00:09:42,070 --> 00:09:44,830 probably not interested in the frequency domain. 155 00:09:44,830 --> 00:09:47,920 Although, you can get a lot of information 156 00:09:47,920 --> 00:09:50,230 from the frequency domain that enables 157 00:09:50,230 --> 00:09:54,510 you to understand what you might see in the time domain. 158 00:09:54,510 --> 00:10:01,090 And one thing about frequency domain experiments-- 159 00:10:01,090 --> 00:10:03,100 when you have a molecule, which has 160 00:10:03,100 --> 00:10:12,390 3N minus 6 vibrational modes, like benzene has 30, 161 00:10:12,390 --> 00:10:15,140 the vibrational density of states 162 00:10:15,140 --> 00:10:17,810 increases rapidly with excitation energy. 163 00:10:17,810 --> 00:10:21,320 So rapidly that, above the region 164 00:10:21,320 --> 00:10:25,430 of the fundamentals, the highest frequency fundamentals, 165 00:10:25,430 --> 00:10:30,220 they're no longer resolvable vibrational levels. 166 00:10:30,220 --> 00:10:32,950 The density of states is on the order of a billion 167 00:10:32,950 --> 00:10:34,480 per wave number. 168 00:10:34,480 --> 00:10:39,550 And you're going to be asking different kinds 169 00:10:39,550 --> 00:10:43,420 of experimental questions to say, do I understand what's 170 00:10:43,420 --> 00:10:48,610 going on in even a molecule as small, from your point of view, 171 00:10:48,610 --> 00:10:51,770 as benzene, big from my point of view. 172 00:10:51,770 --> 00:10:55,970 OK, so we're interested in mechanism. 173 00:11:01,100 --> 00:11:02,850 Why do things happen? 174 00:11:02,850 --> 00:11:04,430 What makes them happen? 175 00:11:04,430 --> 00:11:08,150 And how do we construct an experiment 176 00:11:08,150 --> 00:11:10,130 that reveals mechanism? 177 00:11:10,130 --> 00:11:13,130 And how do we interpret those sorts of experiments? 178 00:11:13,130 --> 00:11:16,220 And the Zewail experiments are beautiful examples 179 00:11:16,220 --> 00:11:20,580 of revealing mechanism, which is more complicated-- 180 00:11:20,580 --> 00:11:23,780 well, more revealing than what you might expect. 181 00:11:23,780 --> 00:11:31,260 For example, suppose you excite a molecule and it breaks. 182 00:11:31,260 --> 00:11:33,260 Well, how did it break? 183 00:11:33,260 --> 00:11:34,850 Did it just break? 184 00:11:34,850 --> 00:11:39,980 Or was there some motion preceding the breaking where 185 00:11:39,980 --> 00:11:46,850 the molecule arranged itself to receive this photon 186 00:11:46,850 --> 00:11:49,990 and do something with it? 187 00:11:49,990 --> 00:11:51,530 And it makes fragments. 188 00:11:51,530 --> 00:11:53,330 And what are the fragments? 189 00:11:53,330 --> 00:11:55,470 What state are the fragments in? 190 00:11:55,470 --> 00:11:57,620 There are all sorts of stuff there. 191 00:11:57,620 --> 00:11:59,870 And that's mechanism. 192 00:11:59,870 --> 00:12:02,510 And it's not just the breaking of a molecule 193 00:12:02,510 --> 00:12:06,680 and getting from the width of a spectral feature 194 00:12:06,680 --> 00:12:09,150 how fast that breaking occurred. 195 00:12:09,150 --> 00:12:13,101 There's things that happen in that time. 196 00:12:13,101 --> 00:12:13,600 OK. 197 00:12:13,600 --> 00:12:18,770 So we need to talk about diabatic versus adiabatic. 198 00:12:18,770 --> 00:12:25,150 And so if you did a quantum chemical calculation-- 199 00:12:30,370 --> 00:12:30,960 and you can. 200 00:12:30,960 --> 00:12:36,300 Because you still have access to the Athena cluster. 201 00:12:36,300 --> 00:12:41,020 Or is your membership expired? 202 00:12:41,020 --> 00:12:43,150 AUDIENCE: We should always have access to it. 203 00:12:43,150 --> 00:12:43,950 PROFESSOR: OK. 204 00:12:43,950 --> 00:12:47,640 Well, you do a calculation using the computer programs. 205 00:12:47,640 --> 00:12:53,100 And they're basically clamped nuclei. 206 00:12:57,650 --> 00:13:01,700 And that's just because you want to reduce the complexity 207 00:13:01,700 --> 00:13:03,350 of the calculation. 208 00:13:03,350 --> 00:13:06,650 It's an impossible calculation anyway to do exactly. 209 00:13:06,650 --> 00:13:09,530 And if we say, let's keep the heavy particles 210 00:13:09,530 --> 00:13:11,090 from moving around, we just solve 211 00:13:11,090 --> 00:13:13,410 the electronic Schrodinger equation, 212 00:13:13,410 --> 00:13:20,540 well, then we get what we call an adiabatic potential energy 213 00:13:20,540 --> 00:13:24,000 surface, or one for each electronic state. 214 00:13:30,340 --> 00:13:34,190 And well, if we clamp the nuclei, 215 00:13:34,190 --> 00:13:36,020 molecules don't hit the nuclei clamp. 216 00:13:36,020 --> 00:13:37,820 We have to unclamp them. 217 00:13:37,820 --> 00:13:51,470 And the effect unclamping leads to some perturbation term, H1. 218 00:13:51,470 --> 00:13:56,480 And so when we solve the clamped nuclei Schrodinger equation, 219 00:13:56,480 --> 00:13:57,890 we don't have vibrations. 220 00:13:57,890 --> 00:14:01,180 We don't have rotations. 221 00:14:01,180 --> 00:14:04,670 And so one of the things we don't know about 222 00:14:04,670 --> 00:14:10,090 is partial derivatives with respect to nuclear coordinates, 223 00:14:10,090 --> 00:14:13,620 which we have the nuclear kinetic energy, which 224 00:14:13,620 --> 00:14:18,077 is the second derivative. 225 00:14:18,077 --> 00:14:19,660 And when you have a second derivative, 226 00:14:19,660 --> 00:14:23,466 you can apply one derivative to the nuclear part 227 00:14:23,466 --> 00:14:25,090 of the wave function and one derivative 228 00:14:25,090 --> 00:14:26,740 to the electronic part. 229 00:14:26,740 --> 00:14:28,600 Or you can apply both. 230 00:14:28,600 --> 00:14:36,490 And so the electronic wave function you get is-- 231 00:14:36,490 --> 00:14:39,520 you get a wave function, which is an explicit function 232 00:14:39,520 --> 00:14:42,810 of the electron positions and parametrically dependent 233 00:14:42,810 --> 00:14:46,990 on internuclear distances or nuclear geometry. 234 00:14:46,990 --> 00:14:49,330 Well, that's there. 235 00:14:49,330 --> 00:14:56,360 And this nuclear kinetic energy term, which is present, 236 00:14:56,360 --> 00:15:02,100 is going to operate on these guys and lead to trouble. 237 00:15:02,100 --> 00:15:07,950 Or maybe not, depending on what the potential curves look like 238 00:15:07,950 --> 00:15:10,120 and what the their problem is. 239 00:15:10,120 --> 00:15:16,980 But certainly, this nuclear kinetic energy 240 00:15:16,980 --> 00:15:19,674 can operate on these functions and give something 241 00:15:19,674 --> 00:15:21,090 that we have to at least consider. 242 00:15:28,260 --> 00:15:33,060 So this guy has secrets embedded in it that we're 243 00:15:33,060 --> 00:15:34,950 going to have to look at. 244 00:15:34,950 --> 00:15:37,590 And they're going to be surprises. 245 00:15:47,940 --> 00:15:49,650 So let's think about diatomics. 246 00:15:49,650 --> 00:15:52,980 Because with diatomics, most of the things 247 00:15:52,980 --> 00:15:55,680 you're going to encounter in polyatomic molecules 248 00:15:55,680 --> 00:15:59,810 have examples that you can understand really clearly. 249 00:15:59,810 --> 00:16:04,330 So suppose there are two electronic states that 250 00:16:04,330 --> 00:16:06,370 cross or do something. 251 00:16:06,370 --> 00:16:09,835 And we can have a picture like this. 252 00:16:15,670 --> 00:16:19,050 So the calculation you do will give you 253 00:16:19,050 --> 00:16:22,470 these avoided crossing potential curves. 254 00:16:26,810 --> 00:16:29,270 Well, why do they avoid each other? 255 00:16:29,270 --> 00:16:35,910 If you have two states of the same symmetry, 256 00:16:35,910 --> 00:16:37,470 they can perturb each other. 257 00:16:37,470 --> 00:16:41,750 The Hamiltonian is totally symmetric. 258 00:16:44,300 --> 00:16:46,610 Any kind of term in the Hamiltonian 259 00:16:46,610 --> 00:16:48,770 that is totally symmetric can cause 260 00:16:48,770 --> 00:16:53,580 interactions between states of the same symmetry. 261 00:16:53,580 --> 00:16:58,400 Now, many of these terms of the same symmetry-- 262 00:16:58,400 --> 00:17:00,830 totally symmetric terms in the Hamiltonian-- 263 00:17:00,830 --> 00:17:03,920 are excluded when we do a calculation. 264 00:17:03,920 --> 00:17:06,319 Because we can. 265 00:17:06,319 --> 00:17:07,190 One is spin R.? 266 00:17:07,190 --> 00:17:17,000 But there are things that could affect the states belonging 267 00:17:17,000 --> 00:17:18,930 to these two potential curves. 268 00:17:18,930 --> 00:17:24,500 And so there are going to be interactions between states 269 00:17:24,500 --> 00:17:26,300 of the same symmetry. 270 00:17:26,300 --> 00:17:31,460 And this is the place. 271 00:17:31,460 --> 00:17:35,210 So you have the ab initio calculation. 272 00:17:35,210 --> 00:17:36,930 You get these curves. 273 00:17:36,930 --> 00:17:41,960 And you can see that there is some coordinate 274 00:17:41,960 --> 00:17:48,050 at which the difference between potential curves is a minimum. 275 00:17:48,050 --> 00:17:51,750 Nature tips its hand. 276 00:17:51,750 --> 00:17:55,660 These avoided crossings that are really important. 277 00:17:55,660 --> 00:17:58,540 And it tells you-- the quantum chemical calculations tell you 278 00:17:58,540 --> 00:17:59,500 two things. 279 00:17:59,500 --> 00:18:02,320 They tell you the internuclear distance 280 00:18:02,320 --> 00:18:04,030 at which the crossing occurs. 281 00:18:13,990 --> 00:18:18,460 And let's call this one-- 282 00:18:18,460 --> 00:18:21,260 let's have several names for these curves. 283 00:18:21,260 --> 00:18:22,340 OK, I have to-- 284 00:18:22,340 --> 00:18:32,380 so we can call this 2, 1, 1, 2 plus plus minus minus. 285 00:18:32,380 --> 00:18:35,530 I have two sets of labels. 286 00:18:35,530 --> 00:18:40,830 So let's say that the 2 and 1 describe 287 00:18:40,830 --> 00:18:45,180 the electronic character of this state, the kind of thing 288 00:18:45,180 --> 00:18:48,590 that the chemists care about. 289 00:18:48,590 --> 00:18:52,730 And plus and minus as upper and lower. 290 00:18:52,730 --> 00:18:54,964 And that's what you get from quantum chemistry. 291 00:18:57,690 --> 00:19:03,970 And so the quantum chemistry gives you 292 00:19:03,970 --> 00:19:06,730 the place at which these two curves 293 00:19:06,730 --> 00:19:08,920 have this minimum separation. 294 00:19:08,920 --> 00:19:19,670 And they have the plus at RC minus V minus at RC 295 00:19:19,670 --> 00:19:21,610 is equal to 2 H 12. 296 00:19:24,320 --> 00:19:27,710 You know from perturbation theory 297 00:19:27,710 --> 00:19:31,940 that, if you have two levels which are degenerate 298 00:19:31,940 --> 00:19:35,950 and they're interacting by some coupling term, 299 00:19:35,950 --> 00:19:40,800 the separation will end up being twice the matrix element. 300 00:19:40,800 --> 00:19:45,500 So nature gives you this and this 301 00:19:45,500 --> 00:19:51,080 from which you can construct the crossing curves, 302 00:19:51,080 --> 00:19:53,270 the curves that chemists like. 303 00:19:53,270 --> 00:19:56,900 So even though quantum chemistry doesn't 304 00:19:56,900 --> 00:20:01,880 know about diabatic curves, it tells you 305 00:20:01,880 --> 00:20:03,260 how you could construct them. 306 00:20:06,930 --> 00:20:09,780 And now, there's two limiting cases. 307 00:20:09,780 --> 00:20:12,660 We can have a very weak interaction. 308 00:20:12,660 --> 00:20:15,050 And so the curves get really close together. 309 00:20:15,050 --> 00:20:18,540 Or we can have a very strong interaction 310 00:20:18,540 --> 00:20:21,060 and get something like this. 311 00:20:24,790 --> 00:20:30,820 So these are the two limits. 312 00:20:30,820 --> 00:20:37,140 And it turns out that the way you would handle these two 313 00:20:37,140 --> 00:20:41,820 limits is experimentally and theoretically profoundly 314 00:20:41,820 --> 00:20:44,215 different. 315 00:20:44,215 --> 00:20:44,715 Why? 316 00:20:54,615 --> 00:20:58,380 This term, the nuclear kinetic energy term, 317 00:20:58,380 --> 00:21:02,350 does terrible things here. 318 00:21:02,350 --> 00:21:05,365 Because when it operates on the electronic wave function, 319 00:21:05,365 --> 00:21:08,110 it says, the electronic wave function 320 00:21:08,110 --> 00:21:12,810 is changing rapidly in this region. 321 00:21:12,810 --> 00:21:21,850 And so if we're going to try to set up a Hamiltonian that 322 00:21:21,850 --> 00:21:26,150 describes the energy levels of these two states, 323 00:21:26,150 --> 00:21:29,390 because of this term, there will be 324 00:21:29,390 --> 00:21:33,950 enormous couplings between the vibrational levels of the two 325 00:21:33,950 --> 00:21:35,790 states. 326 00:21:35,790 --> 00:21:42,340 And as a result, H0, the thing that ignores those coupling 327 00:21:42,340 --> 00:21:47,030 effects, will bear almost no resemblance to the observed 328 00:21:47,030 --> 00:21:50,770 energy level structure. 329 00:21:50,770 --> 00:21:53,890 To get from these kinds of curves 330 00:21:53,890 --> 00:21:57,730 to the observed energy levels is a lot of work. 331 00:21:57,730 --> 00:22:03,070 And it's work that people who use pre-written computer 332 00:22:03,070 --> 00:22:06,640 programs are ill prepared to do. 333 00:22:06,640 --> 00:22:09,850 The spectrum will not be predicted in any way 334 00:22:09,850 --> 00:22:15,550 by this, unless you do a huge matrix of interactions 335 00:22:15,550 --> 00:22:20,530 between vibrational levels of the different electronic 336 00:22:20,530 --> 00:22:21,550 states. 337 00:22:21,550 --> 00:22:25,270 And when you have a matrix element, which is large 338 00:22:25,270 --> 00:22:28,900 compared to the differences between levels, 339 00:22:28,900 --> 00:22:31,150 you can't use perturbation theory. 340 00:22:31,150 --> 00:22:32,605 You have to diagonalize a matrix. 341 00:22:35,200 --> 00:22:37,660 And usually, when you diagonalize a matrix, 342 00:22:37,660 --> 00:22:42,890 the eigenvalues are a bit of a surprise. 343 00:22:42,890 --> 00:22:47,740 Over here, the electronic wave function 344 00:22:47,740 --> 00:22:50,890 doesn't change very rapidly. 345 00:22:50,890 --> 00:22:53,430 And so we have no problem of the interactions 346 00:22:53,430 --> 00:22:57,430 among the vibrational levels of these two states. 347 00:22:57,430 --> 00:23:01,140 And so the adiabatic representation 348 00:23:01,140 --> 00:23:05,370 provides a good zero order picture of the energy levels. 349 00:23:05,370 --> 00:23:06,780 And there's a little bit of stuff 350 00:23:06,780 --> 00:23:12,120 that you can add to improve the fit. 351 00:23:12,120 --> 00:23:18,960 So when you have what's called a weakly avoided crossing, 352 00:23:18,960 --> 00:23:22,650 it tells you that the adiabatic representation 353 00:23:22,650 --> 00:23:25,011 is the wrong one. 354 00:23:25,011 --> 00:23:27,890 But when you have a strongly avoided crossing, 355 00:23:27,890 --> 00:23:28,990 well, it's the right one. 356 00:23:28,990 --> 00:23:32,770 Because all of the effects of this repulsion between the two 357 00:23:32,770 --> 00:23:36,130 states are included in your zero order Hamiltonian. 358 00:23:36,130 --> 00:23:38,620 Whereas here, none are. 359 00:23:38,620 --> 00:23:40,270 And you say, well, I could just go 360 00:23:40,270 --> 00:23:42,740 cruising through this region. 361 00:23:42,740 --> 00:23:44,270 And I go fast through it. 362 00:23:44,270 --> 00:23:49,250 And I don't notice the bad effects. 363 00:23:49,250 --> 00:23:51,100 Well, we'll see. 364 00:23:51,100 --> 00:23:51,600 OK. 365 00:24:10,640 --> 00:24:21,720 So now, if we had a diabatic representation like this, 366 00:24:21,720 --> 00:24:24,045 well, diabatic curves can cross. 367 00:24:29,640 --> 00:24:32,460 In fact, that's the whole point. 368 00:24:32,460 --> 00:24:33,900 We want them to cross. 369 00:24:33,900 --> 00:24:38,300 Because we'd like to keep track of electronic character 1 370 00:24:38,300 --> 00:24:40,440 and electronic character 2. 371 00:24:40,440 --> 00:24:43,140 And so I'm just going to use this here. 372 00:24:52,450 --> 00:24:53,210 OK. 373 00:24:53,210 --> 00:25:00,770 But the really annoying thing is, 374 00:25:00,770 --> 00:25:03,950 when we did an initial calculation, 375 00:25:03,950 --> 00:25:05,610 we clamped the nuclei. 376 00:25:05,610 --> 00:25:09,560 We said there are certain terms in the exact Hamiltonian we're 377 00:25:09,560 --> 00:25:14,030 going to push aside and consider later. 378 00:25:14,030 --> 00:25:16,760 But for the diabatic picture, well, it's 379 00:25:16,760 --> 00:25:19,970 clearly there's some term in the electronic Hamiltonian 380 00:25:19,970 --> 00:25:22,330 that we'd like to turn off. 381 00:25:22,330 --> 00:25:24,700 But there isn't one. 382 00:25:24,700 --> 00:25:26,950 It's all or nothing. 383 00:25:26,950 --> 00:25:29,290 There is nothing you can identify 384 00:25:29,290 --> 00:25:32,320 in the electronic Hamiltonian that enables 385 00:25:32,320 --> 00:25:36,270 the two curves to cross. 386 00:25:36,270 --> 00:25:39,240 So it's not that it's a bad idea. 387 00:25:39,240 --> 00:25:43,410 It's just there is no simple way of dealing with this. 388 00:25:43,410 --> 00:25:48,750 Now sometimes, we have the Hamiltonian. 389 00:25:48,750 --> 00:25:52,590 And we have, say, the spin orbit Hamiltonian. 390 00:25:52,590 --> 00:25:54,330 And this is something we could turn off. 391 00:25:57,200 --> 00:26:00,720 But that's gilding the lily. 392 00:26:00,720 --> 00:26:04,340 I am a spin orbit aficionado. 393 00:26:04,340 --> 00:26:06,470 So I could deal with this. 394 00:26:06,470 --> 00:26:10,880 But if we don't say there are specific, named, 395 00:26:10,880 --> 00:26:14,240 small terms in the electronic Hamiltonian, 396 00:26:14,240 --> 00:26:19,550 there is no way we can get the diabatic limit directly. 397 00:26:19,550 --> 00:26:21,260 You have to use a trick. 398 00:26:26,270 --> 00:26:30,096 So let's draw a picture now, which illustrates the trick. 399 00:26:30,096 --> 00:26:31,720 And we're almost ready to start talking 400 00:26:31,720 --> 00:26:33,930 about Zewail's experiment. 401 00:26:33,930 --> 00:26:36,560 So here I'm going to draw. 402 00:26:44,960 --> 00:26:45,460 OK. 403 00:26:45,460 --> 00:26:49,730 So the adiabatic curves are the ones that don't cross. 404 00:26:49,730 --> 00:26:50,230 And The. 405 00:26:50,230 --> 00:26:52,015 Dash lines are the diabatic curves. 406 00:26:56,450 --> 00:27:02,900 And we know RC and H12. 407 00:27:02,900 --> 00:27:07,300 Or at least we know H12 at that internuclear distance. 408 00:27:07,300 --> 00:27:09,480 And it's a reasonable thing to say, well, 409 00:27:09,480 --> 00:27:12,320 let's let it not change with internuclear distance. 410 00:27:15,150 --> 00:27:17,140 And let's make some other approximation. 411 00:27:17,140 --> 00:27:20,100 We can say the diabatic curves are 412 00:27:20,100 --> 00:27:24,650 linear in the region of the curve crossing. 413 00:27:24,650 --> 00:27:29,290 We can always-- especially when the adiabatic picture is bad, 414 00:27:29,290 --> 00:27:34,400 that curve crossing is very, very compressed. 415 00:27:34,400 --> 00:27:36,320 And it's not a big step to say, OK, 416 00:27:36,320 --> 00:27:39,320 over this relatively small range of internuclear distance 417 00:27:39,320 --> 00:27:42,300 and energy, we're going to approximate the diabatic curves 418 00:27:42,300 --> 00:27:42,800 as linear. 419 00:27:45,770 --> 00:27:51,560 And so we know the exact Hamiltonian, whatever it is, 420 00:27:51,560 --> 00:27:52,640 be gotten from. 421 00:27:57,940 --> 00:27:58,570 OK. 422 00:27:58,570 --> 00:28:08,230 So here, we have what's available from quantum 423 00:28:08,230 --> 00:28:08,951 chemistry. 424 00:28:08,951 --> 00:28:11,200 These are the potential curves for the upper and lower 425 00:28:11,200 --> 00:28:12,180 adiabatic states. 426 00:28:14,760 --> 00:28:19,050 And here we have a potential curve for state 1 427 00:28:19,050 --> 00:28:23,010 and a potential curve for state 2 at H12. 428 00:28:28,150 --> 00:28:33,900 And a unitary transformation of this matrix 429 00:28:33,900 --> 00:28:35,085 has to be equal that matrix. 430 00:28:39,546 --> 00:28:42,020 Well, we don't know these. 431 00:28:42,020 --> 00:28:44,930 But we're going to reduce them to one number, 432 00:28:44,930 --> 00:28:47,580 the slope at the crossing. 433 00:28:47,580 --> 00:28:50,670 And so we have one number here, one number here. 434 00:28:50,670 --> 00:28:53,820 And all of a sudden, we have enough 435 00:28:53,820 --> 00:28:58,080 to iteratively determine the difference between the slopes 436 00:28:58,080 --> 00:29:03,810 by fitting to the observed adiabatic potentials. 437 00:29:03,810 --> 00:29:05,940 And this is basically how it's done. 438 00:29:05,940 --> 00:29:10,560 Now, people who do these calculations for a living 439 00:29:10,560 --> 00:29:14,370 have a much encrusted picture of how they do this. 440 00:29:14,370 --> 00:29:17,190 But this is basically what's going on. 441 00:29:17,190 --> 00:29:20,670 Because people do want the diabatic picture. 442 00:29:20,670 --> 00:29:25,140 And a lot of insight is gained from the diabatic picture. 443 00:29:32,111 --> 00:29:32,610 OK. 444 00:29:32,610 --> 00:29:37,330 So if we're going to be approaching a spectrum where 445 00:29:37,330 --> 00:29:40,810 we have energy levels or an experiment where we learn 446 00:29:40,810 --> 00:29:46,680 something about the dynamics, we're 447 00:29:46,680 --> 00:29:51,950 going to want to think about this problem with two limits-- 448 00:29:51,950 --> 00:29:54,890 weakly avoided and strongly avoided. 449 00:29:54,890 --> 00:29:57,620 And completely different methods for dealing 450 00:29:57,620 --> 00:30:02,900 with both spectroscopic and dynamical information 451 00:30:02,900 --> 00:30:05,540 are appropriate for the two limits. 452 00:30:12,496 --> 00:30:13,390 OK. 453 00:30:13,390 --> 00:30:22,060 Landau-Zener-- this is a model for saying, OK, 454 00:30:22,060 --> 00:30:27,780 what is the probability of going for one curve to the other? 455 00:30:27,780 --> 00:30:32,610 And so we have a formula, which you can derive. 456 00:30:32,610 --> 00:30:33,600 I don't recommend it. 457 00:30:33,600 --> 00:30:38,010 Because derivations are things you do when you need it. 458 00:30:38,010 --> 00:30:40,500 And then you forget them. 459 00:30:40,500 --> 00:30:43,780 But I do want to give you a sense of what's in it. 460 00:30:43,780 --> 00:30:49,450 So we're going from one adiabatic state to another. 461 00:30:49,450 --> 00:30:57,200 And the probability is 1 minus e to the minus pi gamma. 462 00:30:57,200 --> 00:30:59,150 The important parameter is gamma, right? 463 00:30:59,150 --> 00:31:02,250 Everything in this. 464 00:31:02,250 --> 00:31:02,750 OK. 465 00:31:02,750 --> 00:31:11,530 And so P12 is small when gamma is-- 466 00:31:11,530 --> 00:31:13,660 well, we want this to be 1-- 467 00:31:13,660 --> 00:31:15,790 so when gamma is small. 468 00:31:18,630 --> 00:31:21,140 And P12 is large-- 469 00:31:21,140 --> 00:31:23,310 now, I just want to make sure that I have it-- 470 00:31:23,310 --> 00:31:27,940 yeah, when gamma is large. 471 00:31:27,940 --> 00:31:29,460 Now, what's gamma? 472 00:31:29,460 --> 00:31:32,310 Gamma is expressed-- and this is what 473 00:31:32,310 --> 00:31:36,630 you would derive if you were going to do this for real life. 474 00:31:36,630 --> 00:31:42,210 Because you would never accept somebody else's speculation, 475 00:31:42,210 --> 00:31:43,950 you derive it yourself. 476 00:31:43,950 --> 00:31:48,000 So V12 squared-- that's the matrix element. 477 00:31:48,000 --> 00:31:50,460 That's half the gap between the diabatic curves. 478 00:31:54,620 --> 00:32:02,600 And it's over H bar velocity S1 minus S2. 479 00:32:05,420 --> 00:32:10,950 This is the difference in slopes of the diabatic curves 480 00:32:10,950 --> 00:32:14,400 at the crossing point. 481 00:32:14,400 --> 00:32:16,590 This is the velocity of the particle going 482 00:32:16,590 --> 00:32:19,950 through the crossing point. 483 00:32:19,950 --> 00:32:22,050 And you can, of course, get the velocity 484 00:32:22,050 --> 00:32:25,590 from the momentum divided by the mass. 485 00:32:25,590 --> 00:32:28,560 And so all of this stuff is there. 486 00:32:28,560 --> 00:32:42,700 And so gamma is small when V is large 487 00:32:42,700 --> 00:32:46,840 and when S1 is approximately equal to S2. 488 00:32:46,840 --> 00:32:48,730 Well, if the slopes are nearly equal, 489 00:32:48,730 --> 00:32:53,500 that's a strongly avoided crossing, right? 490 00:32:53,500 --> 00:32:57,640 And V is large-- well, if you're going through a bend 491 00:32:57,640 --> 00:32:59,560 in the road and it's too fast, well, 492 00:32:59,560 --> 00:33:01,330 then you're not going to-- 493 00:33:01,330 --> 00:33:03,910 yes, you're not going to be able to make the curve. 494 00:33:03,910 --> 00:33:11,080 So anyway, this is the physical basis behind Landau-Zener. 495 00:33:11,080 --> 00:33:14,560 And it contains the connections to the stuff 496 00:33:14,560 --> 00:33:19,120 you have from your picture of the potentials. 497 00:33:19,120 --> 00:33:23,036 And this is basically how you organize a Zewail type 498 00:33:23,036 --> 00:33:24,910 experiment, which I've got to talk about now. 499 00:33:28,140 --> 00:33:33,750 You have those notes already on Zewail electronically. 500 00:33:33,750 --> 00:33:37,110 So I didn't make copies of them for you. 501 00:33:37,110 --> 00:33:42,980 So you're going to have to accept 502 00:33:42,980 --> 00:33:46,120 what I'm doing without looking at notes for this part, 503 00:33:46,120 --> 00:33:49,620 unless you have them already printed out. 504 00:33:49,620 --> 00:34:02,710 So Zewail had the 1999 Nobel Prize in chemistry. 505 00:34:02,710 --> 00:34:03,880 And it was really special. 506 00:34:03,880 --> 00:34:09,064 Because usually, these things are divided three ways. 507 00:34:09,064 --> 00:34:10,230 He got the whole damn thing. 508 00:34:13,469 --> 00:34:19,300 And this was because he offered something that we really want. 509 00:34:19,300 --> 00:34:24,219 We want to have an idea of, what is the mechanism by which 510 00:34:24,219 --> 00:34:27,810 dynamical processes occur? 511 00:34:27,810 --> 00:34:31,760 In other words, we're not just getting rate. 512 00:34:31,760 --> 00:34:34,080 But we're getting, what are the nuclei doing? 513 00:34:34,080 --> 00:34:36,870 And how does the motion of the nuclei 514 00:34:36,870 --> 00:34:38,930 affect the rate of the process? 515 00:34:42,917 --> 00:34:45,250 AUDIENCE: I think you want S1 is very different than S2. 516 00:34:47,840 --> 00:34:49,159 PROFESSOR: That's quite likely. 517 00:34:49,159 --> 00:34:51,750 And I'm very poor with logic. 518 00:34:51,750 --> 00:34:53,630 Let's see. 519 00:34:53,630 --> 00:35:06,320 So I want you to decide for yourself, OK? 520 00:35:06,320 --> 00:35:08,660 And let me just-- 521 00:35:08,660 --> 00:35:09,160 OK. 522 00:35:13,620 --> 00:35:19,700 So this is an example of a pump/probe experiment. 523 00:35:23,930 --> 00:35:31,480 So the pump pulse at t equals 0 starts things. 524 00:35:31,480 --> 00:35:37,840 And the probe pulse at t equals tau asks the question, 525 00:35:37,840 --> 00:35:40,840 has the wave packet gotten to what's called 526 00:35:40,840 --> 00:35:55,050 the OCR, the optically coupled region? 527 00:35:55,050 --> 00:35:58,170 In other words, you create a wave packet 528 00:35:58,170 --> 00:36:01,380 and you have a probe which can tell you 529 00:36:01,380 --> 00:36:05,850 the time at which the wave packet passes through where 530 00:36:05,850 --> 00:36:09,110 you're probing it. 531 00:36:09,110 --> 00:36:11,380 And this is really neat. 532 00:36:11,380 --> 00:36:17,740 And it is, I think, the essence of how Zewail created 533 00:36:17,740 --> 00:36:21,520 a really simple experiment using the crude technology that 534 00:36:21,520 --> 00:36:25,710 was available at the time to ask the kind of question he needed 535 00:36:25,710 --> 00:36:28,250 to ask. 536 00:36:28,250 --> 00:36:30,590 Now, I didn't like this whale experiment 537 00:36:30,590 --> 00:36:33,980 when it first came out. 538 00:36:33,980 --> 00:36:37,140 Well, because I'm a frequency domain spectroscopist. 539 00:36:37,140 --> 00:36:39,470 And I don't really care about dynamics, 540 00:36:39,470 --> 00:36:42,900 except how it's encoded in the frequency domain spectrum. 541 00:36:42,900 --> 00:36:45,110 So this is completely different. 542 00:36:45,110 --> 00:36:49,250 But anyway, so now, I want to describe the essence 543 00:36:49,250 --> 00:36:53,330 of what he did, and why it worked, and what it reveals. 544 00:36:56,960 --> 00:37:01,060 All right, and so I want to pick a blackboard. 545 00:37:09,010 --> 00:37:13,390 So there are two experiments that he did. 546 00:37:13,390 --> 00:37:18,370 One was dissociation of I-CN. 547 00:37:18,370 --> 00:37:22,675 And the other was dissociation of sodium iodide. 548 00:37:25,440 --> 00:37:28,690 Now, one thing to notice is we've 549 00:37:28,690 --> 00:37:32,040 got a big heavy atom in here. 550 00:37:32,040 --> 00:37:36,140 So that means the vibrational frequencies are low. 551 00:37:36,140 --> 00:37:39,250 That means that, with a not too short pulse, 552 00:37:39,250 --> 00:37:43,790 you could create a coherent superposition. 553 00:37:43,790 --> 00:37:45,470 When you think about what you would 554 00:37:45,470 --> 00:37:48,210 need to create a coherent superposition 555 00:37:48,210 --> 00:37:53,090 of vibrational levels, different by the canonical 1,000 wave 556 00:37:53,090 --> 00:37:59,660 numbers of a vibration, you realize that that experiment 557 00:37:59,660 --> 00:38:01,670 was only doable fairly recently when 558 00:38:01,670 --> 00:38:08,560 you have on the order of a few femtosecond time resolution. 559 00:38:08,560 --> 00:38:12,700 What Zewail did-- he had maybe a 100 femtosecond or maybe only 560 00:38:12,700 --> 00:38:13,960 a picosecond. 561 00:38:13,960 --> 00:38:15,580 And so he had to have a heavy atom. 562 00:38:19,510 --> 00:38:26,860 For I-CN, the beautiful thing about I-CN is there's CN here. 563 00:38:26,860 --> 00:38:32,860 CN is a diatomic molecule that has a very convenient spectrum. 564 00:38:32,860 --> 00:38:36,260 And so we're asking the question, 565 00:38:36,260 --> 00:38:44,850 how does the presence of the iodine affect the CN spectrum? 566 00:38:44,850 --> 00:38:46,860 Because the CN is going to give us 567 00:38:46,860 --> 00:38:50,305 the signal in this experiment. 568 00:38:55,890 --> 00:38:59,880 So here is-- it's really neat. 569 00:38:59,880 --> 00:39:02,360 This is a triatomic molecule. 570 00:39:02,360 --> 00:39:05,600 But we always use one dimensional pictures 571 00:39:05,600 --> 00:39:08,720 to describe everything that's going on. 572 00:39:08,720 --> 00:39:12,590 And you can do that when the one dimensional pictures are 573 00:39:12,590 --> 00:39:15,050 each applicable in separate times 574 00:39:15,050 --> 00:39:18,090 or at separate aspects of the experiment. 575 00:39:18,090 --> 00:39:20,490 But it's another beautiful example. 576 00:39:20,490 --> 00:39:23,990 If Zewail had been attempting to describe the full three 577 00:39:23,990 --> 00:39:26,690 dimensional potential , the pictures wouldn't have done 578 00:39:26,690 --> 00:39:28,220 anything for anybody. 579 00:39:28,220 --> 00:39:30,770 Because you would have to work too hard to understand what 580 00:39:30,770 --> 00:39:33,421 the pictures are telling you. 581 00:39:33,421 --> 00:39:33,920 OK. 582 00:39:33,920 --> 00:39:38,780 So we now have a repulsive curve. 583 00:39:38,780 --> 00:39:40,520 And we have another repulsive curve. 584 00:39:44,910 --> 00:39:46,890 OK. 585 00:39:46,890 --> 00:39:49,240 So we have V equals 0. 586 00:39:49,240 --> 00:39:52,005 And so we're exciting a wave packet. 587 00:39:55,070 --> 00:39:59,510 And because we have a few vibrational levels that 588 00:39:59,510 --> 00:40:03,985 are accessible with the Franck-Condon principle from V 589 00:40:03,985 --> 00:40:08,360 equals 0 and within the necessary Fourier 590 00:40:08,360 --> 00:40:10,670 transform of the post duration, you 591 00:40:10,670 --> 00:40:12,680 get a wave packet starting here. 592 00:40:17,630 --> 00:40:19,520 OK. 593 00:40:19,520 --> 00:40:23,940 So this wave packet is going to be toodling along 594 00:40:23,940 --> 00:40:24,890 at this energy. 595 00:40:27,840 --> 00:40:33,710 I'm sorry it's going to be moving on this potential. 596 00:40:33,710 --> 00:40:35,280 But it has a definite energy. 597 00:40:35,280 --> 00:40:36,160 OK. 598 00:40:36,160 --> 00:40:45,590 And then you can probe it this way, or this way, or this way. 599 00:40:45,590 --> 00:40:50,390 And the important thing is that these three places 600 00:40:50,390 --> 00:40:57,350 at which you probe correspond to different excitation energies. 601 00:40:57,350 --> 00:41:04,880 It's saying that, as the iodine atom is leaving-- 602 00:41:04,880 --> 00:41:07,370 these are transitions between CN. 603 00:41:10,250 --> 00:41:14,030 And in here, the iodine atom is close to the CN. 604 00:41:14,030 --> 00:41:17,390 And it is affecting the bonding in the CN. 605 00:41:17,390 --> 00:41:19,070 Over here, it's gone. 606 00:41:19,070 --> 00:41:21,790 And the CN is free. 607 00:41:21,790 --> 00:41:25,750 And what's happening is the transition frequency changes 608 00:41:25,750 --> 00:41:27,880 with time. 609 00:41:27,880 --> 00:41:35,810 And so if he has the probe pulse here, 610 00:41:35,810 --> 00:41:41,450 well, then when the wave packet reaches this point, 611 00:41:41,450 --> 00:41:43,850 you get a response. 612 00:41:46,590 --> 00:41:50,480 Before the wave packet gets there, there is no exaltation. 613 00:41:50,480 --> 00:41:54,580 So the OCR is defined-- 614 00:41:54,580 --> 00:41:56,180 the optically coupled region-- it's 615 00:41:56,180 --> 00:41:58,700 defined by the frequency, the center frequency, 616 00:41:58,700 --> 00:41:59,570 of the probe pulse. 617 00:42:11,360 --> 00:42:15,110 So if we then look at-- 618 00:42:18,450 --> 00:42:20,100 OK, now, what's happening here? 619 00:42:20,100 --> 00:42:23,430 When you excite to this state, you 620 00:42:23,430 --> 00:42:26,340 have CN electronically excited. 621 00:42:26,340 --> 00:42:29,470 When you look at this limit, you have CN 622 00:42:29,470 --> 00:42:31,500 not electronically excited. 623 00:42:31,500 --> 00:42:39,720 The CN A double pi x double sigma plus transition 624 00:42:39,720 --> 00:42:44,430 is one of the most studied things in the atomic molecules. 625 00:42:46,960 --> 00:42:51,090 And so it's known. 626 00:42:51,090 --> 00:43:01,800 And like all transitions, the intrinsic radiative lifetime 627 00:43:01,800 --> 00:43:07,360 is on the order of 10 nanoseconds. 628 00:43:10,690 --> 00:43:16,600 So if we excite to this state, we're 629 00:43:16,600 --> 00:43:19,310 going to get a photon out. 630 00:43:19,310 --> 00:43:23,590 But the lifetime of that photon is 10 nanoseconds. 631 00:43:23,590 --> 00:43:24,965 Are we going to measure dynamics? 632 00:43:29,280 --> 00:43:31,380 This is the beauty of the experiment right here. 633 00:43:33,890 --> 00:43:38,700 He catches the wave packet at a particular position. 634 00:43:38,700 --> 00:43:40,020 And he puts it in the bank. 635 00:43:43,160 --> 00:43:47,180 And then eventually, the CN flouresces. 636 00:43:47,180 --> 00:43:49,220 We don't care anything about when it flouresces 637 00:43:49,220 --> 00:43:51,380 or what frequency it flouresces at. 638 00:43:51,380 --> 00:43:54,260 All we cared about is that-- 639 00:43:54,260 --> 00:43:57,500 if you're looking at if the wave packet is 640 00:43:57,500 --> 00:44:00,140 in the optically coupled region, it 641 00:44:00,140 --> 00:44:04,820 gets excited to this surface. 642 00:44:04,820 --> 00:44:07,910 And then when it chooses to say, I 643 00:44:07,910 --> 00:44:17,000 got excited, here is my photon, then you can create a plot of-- 644 00:44:20,530 --> 00:44:22,445 so versus time. 645 00:44:35,010 --> 00:44:36,290 So you see nothing. 646 00:44:36,290 --> 00:44:41,740 And then you get something at the optically coupled region. 647 00:44:41,740 --> 00:44:43,870 And you do a sequence of experiments 648 00:44:43,870 --> 00:44:51,700 where you vary where the optically coupled region is 649 00:44:51,700 --> 00:44:56,530 centered by adjusting the center frequency of the probe laser. 650 00:44:56,530 --> 00:45:00,430 So sorry. 651 00:45:00,430 --> 00:45:03,650 I should really say, OK, here is the intensity 652 00:45:03,650 --> 00:45:04,820 of the fluorescence. 653 00:45:04,820 --> 00:45:11,580 And here is the 1 over the wavelength of the probe. 654 00:45:11,580 --> 00:45:15,000 So this is related to the energy of the probe. 655 00:45:15,000 --> 00:45:17,180 And so you do a series of experiments. 656 00:45:17,180 --> 00:45:25,250 And you discover that the wave packet reaches this. 657 00:45:25,250 --> 00:45:26,840 And then you move that. 658 00:45:26,840 --> 00:45:29,510 And you get another. 659 00:45:29,510 --> 00:45:31,670 And so you get a picture. 660 00:45:46,580 --> 00:45:48,490 So what do I want to say about this? 661 00:45:52,440 --> 00:46:01,320 So it basically is telling you that, when the molecule breaks, 662 00:46:01,320 --> 00:46:06,270 the frequency of the CN excitation 663 00:46:06,270 --> 00:46:10,170 is changing as a function of the distance of the iodine atom 664 00:46:10,170 --> 00:46:12,560 from the CN. 665 00:46:12,560 --> 00:46:14,810 And so that's mechanism. 666 00:46:14,810 --> 00:46:17,370 Now, it's really a very crude mechanism. 667 00:46:17,370 --> 00:46:21,360 But because it's reduced to one question, 668 00:46:21,360 --> 00:46:24,730 it's a function of one geometry. 669 00:46:24,730 --> 00:46:28,450 But nobody's ever be able to say they 670 00:46:28,450 --> 00:46:34,400 observed this motion of the wave packet in real time. 671 00:46:34,400 --> 00:46:38,460 And here is where people really get annoyed with Zewail. 672 00:46:38,460 --> 00:46:43,360 Because he's looking at real motion in real time. 673 00:46:43,360 --> 00:46:44,380 And it ain't real. 674 00:46:48,710 --> 00:46:52,840 He's probing one time at a time. 675 00:46:52,840 --> 00:46:55,400 But it's still mechanism. 676 00:46:55,400 --> 00:46:59,150 It's more than just looking at when does something break. 677 00:46:59,150 --> 00:47:03,275 What's the lifetime of something? 678 00:47:03,275 --> 00:47:05,990 There's more than one thing going on. 679 00:47:05,990 --> 00:47:10,406 Now, I really like the next experiment. 680 00:47:13,880 --> 00:47:19,400 And the next experiment involves sodium iodide. 681 00:47:19,400 --> 00:47:22,370 Now, if I were to ask you, what is 682 00:47:22,370 --> 00:47:30,450 the nature of sodium iodide at equilibrium? 683 00:47:30,450 --> 00:47:33,210 You'd say, it's probably ionic. 684 00:47:33,210 --> 00:47:37,770 But because iodine is at the bottom 685 00:47:37,770 --> 00:47:42,810 as opposed to the top of the periodic potential, 686 00:47:42,810 --> 00:47:44,655 it's maybe a little bit covalent. 687 00:47:52,180 --> 00:47:56,720 So here is the ground state potential. 688 00:47:56,720 --> 00:48:03,220 And I just have to make sure I understand 689 00:48:03,220 --> 00:48:04,900 what I'm trying to do. 690 00:48:04,900 --> 00:48:09,890 And here is an excited state potential. 691 00:48:19,961 --> 00:48:20,460 OK. 692 00:48:20,460 --> 00:48:24,270 So what I've tried is crossing curves. 693 00:48:24,270 --> 00:48:26,940 Those are the diabatic curves. 694 00:48:26,940 --> 00:48:31,560 Now, I've connected-- I should do it with dotted lines. 695 00:48:31,560 --> 00:48:33,010 Those are the adiabatic curves. 696 00:48:35,970 --> 00:48:41,050 And so the question is, suppose we create a wave packet 697 00:48:41,050 --> 00:48:45,280 on this potential. 698 00:48:51,390 --> 00:48:54,560 So it's created here. 699 00:48:54,560 --> 00:48:58,979 And this wave packet feels a force that way. 700 00:48:58,979 --> 00:49:00,770 Because it's the gradient of the potential. 701 00:49:00,770 --> 00:49:01,557 It's a particle. 702 00:49:01,557 --> 00:49:02,390 It's in a potential. 703 00:49:02,390 --> 00:49:04,670 And it's going this way. 704 00:49:04,670 --> 00:49:08,930 And then something happens when it 705 00:49:08,930 --> 00:49:12,325 goes through this curve crossing region. 706 00:49:12,325 --> 00:49:13,450 But what's going to happen? 707 00:49:17,810 --> 00:49:21,886 I should just say this is the ionic. 708 00:49:21,886 --> 00:49:22,760 This is the covalent. 709 00:49:25,590 --> 00:49:29,980 So this is NA plus plus I minus. 710 00:49:29,980 --> 00:49:37,810 And this is NA plus I. When you dissociate a neutral molecule, 711 00:49:37,810 --> 00:49:41,110 you always get neutral atoms. 712 00:49:41,110 --> 00:49:44,580 So there is a higher energy limit where you can get ions. 713 00:49:49,330 --> 00:49:52,940 And so even if you create this wave packet here, 714 00:49:52,940 --> 00:49:57,600 which is below the threshold for making ions, 715 00:49:57,600 --> 00:50:01,544 it only can leave by this path. 716 00:50:01,544 --> 00:50:02,960 So what happens is the wave packet 717 00:50:02,960 --> 00:50:07,130 is going back and forth, back and forth, back and forth. 718 00:50:07,130 --> 00:50:15,175 And each time it crosses through this region, it has to decide, 719 00:50:15,175 --> 00:50:16,860 am I ionic or covalent? 720 00:50:21,390 --> 00:50:27,360 And if I'm covalent, I can leave. 721 00:50:27,360 --> 00:50:29,240 And if I'm ionic, I'm stuck. 722 00:50:29,240 --> 00:50:34,680 I've got to go back and try again and again and again. 723 00:50:38,460 --> 00:50:42,020 So this is where Landau-Zener comes in. 724 00:50:42,020 --> 00:50:43,960 Because what's happening is the molecule 725 00:50:43,960 --> 00:50:47,840 is addressing this ionic covalent curve crossing. 726 00:50:47,840 --> 00:50:52,130 And it's deciding each time it goes through, 727 00:50:52,130 --> 00:50:54,560 what is the probability of being able to get out? 728 00:50:58,600 --> 00:51:00,750 OK. 729 00:51:00,750 --> 00:51:02,960 But then, there's something else. 730 00:51:02,960 --> 00:51:08,960 And that is, well, what are we probing? 731 00:51:08,960 --> 00:51:12,530 So we probe to an excited state. 732 00:51:17,720 --> 00:51:27,530 And so this is where the figures, even for this reduced 733 00:51:27,530 --> 00:51:29,900 representation, are confusing. 734 00:51:29,900 --> 00:51:33,410 But basically, you're probing by exciting this sodium 735 00:51:33,410 --> 00:51:39,670 atom to an excited state, which puts it in the bank. 736 00:51:39,670 --> 00:51:43,200 And it fluoresces when it chooses to, 737 00:51:43,200 --> 00:51:44,970 which is in 10 nanoseconds. 738 00:51:44,970 --> 00:51:47,280 Because sodium's radiative lifetime really 739 00:51:47,280 --> 00:51:49,300 is about 10 nanoseconds. 740 00:51:49,300 --> 00:51:52,590 And so how to describe this? 741 00:52:03,690 --> 00:52:08,760 So you have a choice of being able to excite 742 00:52:08,760 --> 00:52:13,650 at the free sodium transition frequency. 743 00:52:13,650 --> 00:52:16,680 And then what you're only seeing is these little packets 744 00:52:16,680 --> 00:52:19,020 of sodium that make it out. 745 00:52:21,630 --> 00:52:27,540 Or you can excite at a-- 746 00:52:27,540 --> 00:52:28,880 well, this is coming out wrong. 747 00:52:28,880 --> 00:52:31,320 But if you excite to the red of that, 748 00:52:31,320 --> 00:52:41,130 you will only see the sodium atoms 749 00:52:41,130 --> 00:52:45,890 while they are still close to the iodine. 750 00:52:45,890 --> 00:52:49,340 And so there are two sorts of signals you get. 751 00:52:49,340 --> 00:52:57,650 One is you get a series of pulses separated 752 00:52:57,650 --> 00:53:01,320 by the round trip time-- 753 00:53:01,320 --> 00:53:03,570 the round trip time is the period. 754 00:53:03,570 --> 00:53:07,480 And so the period is related the vibrational frequency. 755 00:53:07,480 --> 00:53:10,650 And so each time this packet goes 756 00:53:10,650 --> 00:53:12,990 through the optically coupled region-- 757 00:53:12,990 --> 00:53:16,800 I'm sorry-- goes through the curve crossing region, 758 00:53:16,800 --> 00:53:20,180 you get a little puff of sodium. 759 00:53:20,180 --> 00:53:21,770 If you're exciting so that you can 760 00:53:21,770 --> 00:53:29,200 see only the free sodium, well, then instead of this, 761 00:53:29,200 --> 00:53:35,820 you see the signal, another signal. 762 00:53:35,820 --> 00:53:37,920 So you get little steps. 763 00:53:37,920 --> 00:53:45,170 Each wave packet that makes it out dies at the turning point. 764 00:53:45,170 --> 00:53:46,640 And I'm way over time. 765 00:53:46,640 --> 00:53:48,300 But this is really the-- 766 00:53:48,300 --> 00:53:49,300 so there's two pictures. 767 00:53:49,300 --> 00:53:53,120 There is this sampling of the wave packet each time 768 00:53:53,120 --> 00:53:54,320 it goes through. 769 00:53:54,320 --> 00:53:59,870 And there is the delivery of the goods. 770 00:53:59,870 --> 00:54:03,280 And again, the signal is a slow signal. 771 00:54:03,280 --> 00:54:08,360 But the pump/probe delay is femtosecond time resolution. 772 00:54:08,360 --> 00:54:13,150 And so it can reveal this wave packet propagation. 773 00:54:13,150 --> 00:54:15,140 So that's all I have to say on the subject. 774 00:54:15,140 --> 00:54:16,990 I think it's really beautiful. 775 00:54:16,990 --> 00:54:20,690 I like it more and more the more I understand it. 776 00:54:20,690 --> 00:54:27,600 And the next lecture will be on why gases condense.