1 00:00:00,090 --> 00:00:02,490 The following content is provided under a Creative 2 00:00:02,490 --> 00:00:04,030 Commons license. 3 00:00:04,030 --> 00:00:06,330 Your support will help MIT OpenCourseWare 4 00:00:06,330 --> 00:00:10,690 continue to offer high quality educational resources for free. 5 00:00:10,690 --> 00:00:13,320 To make a donation or view additional materials 6 00:00:13,320 --> 00:00:16,814 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:16,814 --> 00:00:17,850 at OCW.MIT.edu. 8 00:00:21,620 --> 00:00:27,750 PROFESSOR: Today's lecture is on intermolecular interactions. 9 00:00:27,750 --> 00:00:32,600 And it's really one of these favorite topics 10 00:00:32,600 --> 00:00:37,320 where I can say, well, this is the origin of life. 11 00:00:37,320 --> 00:00:41,150 How do gases actually condense? 12 00:00:41,150 --> 00:00:45,410 And we can build a really, really simple theory 13 00:00:45,410 --> 00:00:50,630 for how gases condense based on what we know, 14 00:00:50,630 --> 00:00:53,180 based on second order perturbation theory, 15 00:00:53,180 --> 00:00:55,580 based on some hand-waving stuff. 16 00:00:55,580 --> 00:00:58,340 And it really is something from nothing. 17 00:00:58,340 --> 00:01:02,070 But it's also an example of what we do in physical chemistry 18 00:01:02,070 --> 00:01:07,220 where we explain one phenomenon by taking something that 19 00:01:07,220 --> 00:01:11,210 seems to be unrelated and building a model around it, 20 00:01:11,210 --> 00:01:14,630 and explaining this surprising thing. 21 00:01:14,630 --> 00:01:19,110 And that's what's going to happen today. 22 00:01:19,110 --> 00:01:20,720 We'll talk more about that. 23 00:01:20,720 --> 00:01:27,760 OK, now last time we talked about wavepacket dynamics. 24 00:01:27,760 --> 00:01:34,060 And so with wavepacket dynamics, there is first the pluck. 25 00:01:34,060 --> 00:01:37,420 How do you start the system in some superposition 26 00:01:37,420 --> 00:01:40,300 of eigenstates? 27 00:01:40,300 --> 00:01:48,320 And then there's the evolution of the pluck, which is usually 28 00:01:48,320 --> 00:01:53,890 some kind of particle-like state that follows Newton's laws. 29 00:01:53,890 --> 00:01:59,150 The center of the wavepacket follows Newton's laws 30 00:01:59,150 --> 00:02:01,100 for the position and the momentum. 31 00:02:03,890 --> 00:02:10,470 But if you make a wavepacket in a real molecule, 32 00:02:10,470 --> 00:02:14,570 the molecule is going to be not a harmonic oscillator. 33 00:02:14,570 --> 00:02:18,590 There will be anharmonic effects which causes the wavepacket 34 00:02:18,590 --> 00:02:21,200 to de-phase, but the center of the wavepacket 35 00:02:21,200 --> 00:02:28,520 continues to move as if it were in a simple potential. 36 00:02:28,520 --> 00:02:31,130 And it can de-phase, and rephase, and all sorts 37 00:02:31,130 --> 00:02:32,180 of complicated stuff. 38 00:02:32,180 --> 00:02:34,850 And that can be very interesting. 39 00:02:34,850 --> 00:02:38,240 Now, when you take this step from a diatonic molecule 40 00:02:38,240 --> 00:02:41,210 to a polyatomic molecule, you have 41 00:02:41,210 --> 00:02:45,560 3n minus 6 oscillators cohabiting in the same house. 42 00:02:45,560 --> 00:02:47,180 And they're going to interact. 43 00:02:47,180 --> 00:02:51,770 And so if you start a wavepacket on one 44 00:02:51,770 --> 00:02:54,890 Franck-Condon bright mode because 45 00:02:54,890 --> 00:02:59,240 of anharmonic interactions handled by perturbation theory, 46 00:02:59,240 --> 00:03:04,430 between modes you start to get this wavepacket leaking out 47 00:03:04,430 --> 00:03:06,110 into other modes. 48 00:03:06,110 --> 00:03:09,800 And so the expectation value is going 49 00:03:09,800 --> 00:03:15,470 to exhibit an oscillating time dependence associated 50 00:03:15,470 --> 00:03:18,470 with the bright mode and some of the dark modes 51 00:03:18,470 --> 00:03:21,050 that are anharmonically coupled to the bright mode. 52 00:03:21,050 --> 00:03:23,870 And it can get pretty complicated. 53 00:03:23,870 --> 00:03:27,580 But you can understand all of it because you 54 00:03:27,580 --> 00:03:28,930 have perturbation theory. 55 00:03:28,930 --> 00:03:35,920 And so you can say, OK this time the wavepacket-- 56 00:03:35,920 --> 00:03:39,090 some amplitude of the initial wavepacket 57 00:03:39,090 --> 00:03:41,460 is transferred to a different mode. 58 00:03:41,460 --> 00:03:43,170 And when does that happen? 59 00:03:43,170 --> 00:03:45,350 Stationary phase. 60 00:03:45,350 --> 00:03:49,630 It happens at a particular geometry. 61 00:03:49,630 --> 00:03:52,590 And so you can be asking, OK, when is 62 00:03:52,590 --> 00:03:54,510 this excitation transferring? 63 00:03:54,510 --> 00:03:55,710 How strongly? 64 00:03:55,710 --> 00:03:59,040 This is a very, very complete picture. 65 00:03:59,040 --> 00:04:01,860 And it's very much like classical mechanics 66 00:04:01,860 --> 00:04:03,420 because it's localized. 67 00:04:05,940 --> 00:04:08,820 Then we talked about Landau-Zener. 68 00:04:08,820 --> 00:04:12,240 And the Landau-Zener is basically a simple idea 69 00:04:12,240 --> 00:04:16,760 that you also understand from real life. 70 00:04:16,760 --> 00:04:20,920 And it has to do with how does a wavepacket jump from one 71 00:04:20,920 --> 00:04:24,310 potential surface to another? 72 00:04:24,310 --> 00:04:27,453 And again, it's stationary phase. 73 00:04:27,453 --> 00:04:32,390 At the internuclear distance of the curve crossing, 74 00:04:32,390 --> 00:04:35,340 the wavepacket has to decide which 75 00:04:35,340 --> 00:04:37,140 curve it's going to follow. 76 00:04:37,140 --> 00:04:40,620 And it depends on the velocity of the packet 77 00:04:40,620 --> 00:04:45,500 at the curve crossing and the difference in slopes of the two 78 00:04:45,500 --> 00:04:47,535 paths it has to choose between. 79 00:04:47,535 --> 00:04:50,400 And all of this is very simple. 80 00:04:50,400 --> 00:04:55,290 Now, I am trying to create an exam problem which 81 00:04:55,290 --> 00:04:58,080 is related to Landau-Zener. 82 00:04:58,080 --> 00:05:00,540 And this is hard. 83 00:05:00,540 --> 00:05:05,490 But if I succeed, then you can expect to see it. 84 00:05:05,490 --> 00:05:08,550 The last thing I talked about the last lecture 85 00:05:08,550 --> 00:05:13,000 was the Zewail pump-probe experiments. 86 00:05:13,000 --> 00:05:18,060 And this is a beautifully graphical picture 87 00:05:18,060 --> 00:05:20,580 of how do you create something and then 88 00:05:20,580 --> 00:05:23,000 probe what's happening. 89 00:05:23,000 --> 00:05:27,800 And again, I hope that I can come up 90 00:05:27,800 --> 00:05:33,170 with something pictorial for the exam on the Zewail pump. 91 00:05:33,170 --> 00:05:39,480 OK, now before I talk about intermolecular interactions, 92 00:05:39,480 --> 00:05:44,120 I want to do a little bit of tying up loose ends. 93 00:05:44,120 --> 00:05:47,000 You know about non-degenerate perturbation theory. 94 00:05:49,850 --> 00:05:52,100 Now there's another kind of perturbation theory called 95 00:05:52,100 --> 00:05:53,510 degenerate perturbation theory. 96 00:05:57,900 --> 00:06:01,240 And this is when non-degenerate perturbation theory 97 00:06:01,240 --> 00:06:13,930 fails because H i j over E i 0 minus E j 0 98 00:06:13,930 --> 00:06:16,200 is greater than one. 99 00:06:16,200 --> 00:06:17,519 Well then you've got a problem. 100 00:06:17,519 --> 00:06:19,310 You can't use ordinary perturbation theory. 101 00:06:19,310 --> 00:06:21,940 You have to diagonalize. 102 00:06:21,940 --> 00:06:23,840 You diagonalize it two by two or you 103 00:06:23,840 --> 00:06:28,990 diagonalize all of the little cases where this is violated. 104 00:06:28,990 --> 00:06:31,350 That's called degenerative perturbation theory. 105 00:06:31,350 --> 00:06:33,850 And there is a little bit of extra recipe 106 00:06:33,850 --> 00:06:37,930 for how do you put information about remote states 107 00:06:37,930 --> 00:06:41,130 into the part of the Hamiltonian you diagonalize? 108 00:06:41,130 --> 00:06:43,120 That's called a Van Vleck transformation, 109 00:06:43,120 --> 00:06:45,830 and I'm not going to talk about it. 110 00:06:45,830 --> 00:06:49,930 But basically you fix things up by diagonalizing two 111 00:06:49,930 --> 00:06:52,090 by two or three by three. 112 00:06:52,090 --> 00:06:55,160 And then do ordinary perturbation theory. 113 00:06:55,160 --> 00:06:56,704 And then there's variational. 114 00:07:01,450 --> 00:07:06,220 The variational model is something 115 00:07:06,220 --> 00:07:12,130 we've used mostly in a minimum basis set for talking 116 00:07:12,130 --> 00:07:15,800 about the atomic molecules. 117 00:07:15,800 --> 00:07:21,290 And in the variational model, we minimize energies 118 00:07:21,290 --> 00:07:25,160 by optimizing mixing coefficients. 119 00:07:28,080 --> 00:07:34,220 Now, perturbation theory is implicitly an infinite basis. 120 00:07:34,220 --> 00:07:36,680 Usually the infinity doesn't matter 121 00:07:36,680 --> 00:07:39,470 because the interesting stuff comes in 122 00:07:39,470 --> 00:07:42,680 from relatively low-lying levels. 123 00:07:42,680 --> 00:07:45,860 And the infinite number of very far away levels 124 00:07:45,860 --> 00:07:49,520 just modifies the energy levels a little bit, 125 00:07:49,520 --> 00:07:52,050 and you don't have to worry about it. 126 00:07:52,050 --> 00:07:57,640 The variational method, you choose your own basis set. 127 00:07:57,640 --> 00:08:00,800 Now, the choice of basis set is frequently 128 00:08:00,800 --> 00:08:02,720 dictated by convenience. 129 00:08:02,720 --> 00:08:09,680 And the basis set is often not very large or not orthonormal. 130 00:08:09,680 --> 00:08:12,710 And so there are problems with how 131 00:08:12,710 --> 00:08:15,650 do we deal with basis functions which 132 00:08:15,650 --> 00:08:20,150 are not orthonormal, especially not orthogonal to each other? 133 00:08:20,150 --> 00:08:22,040 And there is a transformation that you 134 00:08:22,040 --> 00:08:24,860 can do to make that work out. 135 00:08:24,860 --> 00:08:27,890 I mean, the problem is basically-- 136 00:08:27,890 --> 00:08:35,990 the secular equation is this for the variational method. 137 00:08:35,990 --> 00:08:41,510 And this is the overlap matrix. 138 00:08:41,510 --> 00:08:46,550 And the overlap matrix is not the unit matrix. 139 00:08:46,550 --> 00:08:50,270 We have what's called the generalized variational 140 00:08:50,270 --> 00:08:51,410 calculation. 141 00:08:51,410 --> 00:08:53,270 And there are methods for handling that. 142 00:08:53,270 --> 00:08:57,520 And I have often introduced those methods on a final exam. 143 00:09:00,790 --> 00:09:03,380 Those are just statements. 144 00:09:03,380 --> 00:09:09,320 OK, so variational is not guaranteed 145 00:09:09,320 --> 00:09:12,900 to give the exact answer. 146 00:09:12,900 --> 00:09:17,310 And it can't if you don't use an infinite basis. 147 00:09:17,310 --> 00:09:20,909 Perturbation theory implicitly uses this infinite basis. 148 00:09:20,909 --> 00:09:22,950 And you should feel confident that you're getting 149 00:09:22,950 --> 00:09:25,070 close to the right answer. 150 00:09:25,070 --> 00:09:31,015 And you can know what sort of errors you're making. 151 00:09:31,015 --> 00:09:33,030 OK, yes? 152 00:09:33,030 --> 00:09:34,910 AUDIENCE: When you go to very high orders 153 00:09:34,910 --> 00:09:36,930 in perturbation theory, your convergence 154 00:09:36,930 --> 00:09:38,787 is not necessarily better-- 155 00:09:38,787 --> 00:09:39,870 PROFESSOR: That's correct. 156 00:09:39,870 --> 00:09:41,060 AUDIENCE: --on the real solutions. 157 00:09:41,060 --> 00:09:41,550 PROFESSOR: OK, but that's-- 158 00:09:41,550 --> 00:09:43,133 AUDIENCE: [INAUDIBLE] when to cool it? 159 00:09:45,690 --> 00:09:49,920 PROFESSOR: This is-- the problem there is that you cannot 160 00:09:49,920 --> 00:09:55,630 include correlation in perturbation theory. 161 00:09:55,630 --> 00:09:59,510 And so there are things that are outside 162 00:09:59,510 --> 00:10:07,420 of the network of things that I've described. 163 00:10:07,420 --> 00:10:10,650 And so correlation effects where the particles 164 00:10:10,650 --> 00:10:14,040 can move in some way that they know about each other. 165 00:10:14,040 --> 00:10:18,000 That's very difficult to build in via perturbation theory. 166 00:10:18,000 --> 00:10:20,400 But when we don't have correlation, 167 00:10:20,400 --> 00:10:23,130 length perturbation theory makes an error. 168 00:10:23,130 --> 00:10:25,770 And you know how big that error is. 169 00:10:25,770 --> 00:10:35,390 OK, and then we know that we can diagonalize the matrix 170 00:10:35,390 --> 00:10:39,590 by some kind of unitary transformation. 171 00:10:39,590 --> 00:10:48,930 And the columns of T dagger are the eigenvectors. 172 00:10:48,930 --> 00:10:55,150 And the rows of T are the inverse-- 173 00:10:55,150 --> 00:10:59,030 rows of T dagger or the columns of T are the inverse. 174 00:10:59,030 --> 00:11:02,200 And so if you want to make a wavepacket where 175 00:11:02,200 --> 00:11:09,280 you have some additional state to equal 0, 176 00:11:09,280 --> 00:11:12,430 you want to express that as a linear combination 177 00:11:12,430 --> 00:11:14,440 of eigenstates. 178 00:11:14,440 --> 00:11:17,937 And this sort of transformation provides that information 179 00:11:17,937 --> 00:11:19,020 if you know how to use it. 180 00:11:23,370 --> 00:11:25,910 And you can get approximations to all 181 00:11:25,910 --> 00:11:29,780 of these elements of T dagger from non-degenerate 182 00:11:29,780 --> 00:11:31,520 perturbation theory. 183 00:11:31,520 --> 00:11:34,100 And on the second exam that was one of the things 184 00:11:34,100 --> 00:11:35,830 you didn't do very well on. 185 00:11:35,830 --> 00:11:40,670 And so you can see that again. 186 00:11:40,670 --> 00:11:41,170 All right. 187 00:11:45,160 --> 00:11:47,710 OK, so now we're going to talk about intermolecular 188 00:11:47,710 --> 00:11:51,180 interactions. 189 00:11:51,180 --> 00:11:53,130 And up until now, we've been talking 190 00:11:53,130 --> 00:11:55,870 about isolated molecules. 191 00:11:55,870 --> 00:11:58,660 And you have a lot of insights about how isolated molecules 192 00:11:58,660 --> 00:12:01,600 work, or at least what kind of stuff 193 00:12:01,600 --> 00:12:04,510 you need to be able to make predictions about isolated 194 00:12:04,510 --> 00:12:05,260 molecules. 195 00:12:07,900 --> 00:12:12,100 And perturbation theory is a very important tool 196 00:12:12,100 --> 00:12:13,720 in building that insight. 197 00:12:13,720 --> 00:12:18,160 In almost every problem which is not an exactly soluble problem, 198 00:12:18,160 --> 00:12:21,220 perturbation theory gives you at least a hand-waving argument 199 00:12:21,220 --> 00:12:23,640 for what's going on. 200 00:12:23,640 --> 00:12:27,530 Everything we did with LCAO-MO is really 201 00:12:27,530 --> 00:12:32,180 based on perturbation theory, or at least the interpretations, 202 00:12:32,180 --> 00:12:35,810 especially when you go from homonuclear to heteronuclear. 203 00:12:35,810 --> 00:12:40,310 And when we talk about what various orbitals look like 204 00:12:40,310 --> 00:12:42,350 and why they look that way. 205 00:12:42,350 --> 00:12:47,380 OK, so perturbation theory can also be used to talk about-- 206 00:12:47,380 --> 00:12:52,660 we have two particles, A and B. And we 207 00:12:52,660 --> 00:12:54,560 have the origin of coordinates. 208 00:12:54,560 --> 00:12:56,245 And so this is the coordinate for A, 209 00:12:56,245 --> 00:13:00,940 and this is the coordinate for B. 210 00:13:00,940 --> 00:13:05,370 And this is the interatomic coordinate. 211 00:13:05,370 --> 00:13:09,390 And so we're interested in what happens 212 00:13:09,390 --> 00:13:14,370 as a function of this interatomic distance. 213 00:13:14,370 --> 00:13:16,890 And can we use perturbation theory 214 00:13:16,890 --> 00:13:20,252 to be able to say something about that? 215 00:13:20,252 --> 00:13:21,460 And the answer is you betcha. 216 00:13:24,060 --> 00:13:29,000 Because basically we understand how these two molecules work. 217 00:13:29,000 --> 00:13:34,550 And so we have to somehow build in some insight 218 00:13:34,550 --> 00:13:37,100 into what is the interaction between these, 219 00:13:37,100 --> 00:13:41,770 and then use perturbation theory to quantify it. 220 00:13:41,770 --> 00:13:44,400 OK, now this is a situation where 221 00:13:44,400 --> 00:13:52,860 we're going to say atoms A and B are our field shell, 222 00:13:52,860 --> 00:13:54,460 or whatever you want to call it. 223 00:13:54,460 --> 00:13:56,730 They don't need to make another bond. 224 00:13:56,730 --> 00:13:58,990 They're not going to bond to each other. 225 00:13:58,990 --> 00:14:01,480 There's no covalent bond between A and B. 226 00:14:01,480 --> 00:14:07,980 There is no charge on A and B. There is no donor/acceptor. 227 00:14:07,980 --> 00:14:10,560 All of the things that we can understand 228 00:14:10,560 --> 00:14:12,830 by looking at individual systems, 229 00:14:12,830 --> 00:14:14,760 they are all turned off. 230 00:14:14,760 --> 00:14:19,670 These molecules do have energy levels. 231 00:14:19,670 --> 00:14:21,940 And so if we could find some way to talk 232 00:14:21,940 --> 00:14:27,330 about the weak interaction between them, 233 00:14:27,330 --> 00:14:29,690 well then we have the basis for perturbation theory. 234 00:14:29,690 --> 00:14:32,360 Because we have weak and we have energy levels 235 00:14:32,360 --> 00:14:35,750 that we know for the two molecules. 236 00:14:35,750 --> 00:14:40,410 And so if we're going to do perturbation theory 237 00:14:40,410 --> 00:14:42,360 it'll be second order perturbation theory 238 00:14:42,360 --> 00:14:48,990 because we are going to be interested in how interaction 239 00:14:48,990 --> 00:14:52,550 of A with B causes excited states of B to mix 240 00:14:52,550 --> 00:14:57,150 into B, and vice versa. 241 00:14:57,150 --> 00:15:00,440 And so one of the things you know about perturbation theory, 242 00:15:00,440 --> 00:15:05,170 if you're talking about the ground state of a system 243 00:15:05,170 --> 00:15:13,034 and you do perturbation theory, what happens? 244 00:15:13,034 --> 00:15:14,450 What happens to the energy levels? 245 00:15:17,660 --> 00:15:19,450 I mean, you have-- 246 00:15:19,450 --> 00:15:21,260 for second order perturbation theory-- 247 00:15:24,921 --> 00:15:25,420 I'm sorry? 248 00:15:25,420 --> 00:15:27,490 AUDIENCE: Everything gets-- things 249 00:15:27,490 --> 00:15:31,290 can increase or decrease. 250 00:15:31,290 --> 00:15:35,290 The H term gets squared. 251 00:15:35,290 --> 00:15:35,938 PROFESSOR: Yes. 252 00:15:35,938 --> 00:15:37,562 AUDIENCE: Some [INAUDIBLE] information. 253 00:15:37,562 --> 00:15:41,217 But depending on whether a level that's lower-- 254 00:15:41,217 --> 00:15:43,550 PROFESSOR: But if we're talking about the ground state-- 255 00:15:43,550 --> 00:15:46,092 AUDIENCE: If it's that ground state it has to go down. 256 00:15:46,092 --> 00:15:46,800 PROFESSOR: Right. 257 00:15:46,800 --> 00:15:48,620 And so that's the key. 258 00:15:48,620 --> 00:15:52,850 We know that if we're going to do second order perturbation 259 00:15:52,850 --> 00:15:55,370 theory and we're going to be asking questions 260 00:15:55,370 --> 00:15:58,170 about molecules in their ground state, 261 00:15:58,170 --> 00:16:00,120 they're going to be stabilized by interaction 262 00:16:00,120 --> 00:16:01,710 with other molecules. 263 00:16:01,710 --> 00:16:04,830 It doesn't matter, as long as there's 264 00:16:04,830 --> 00:16:11,340 no bonding interactions, or very strong short range 265 00:16:11,340 --> 00:16:14,550 interactions, this sort of a problem 266 00:16:14,550 --> 00:16:19,310 is guaranteed to give stabilization of the ground 267 00:16:19,310 --> 00:16:21,095 state of the system. 268 00:16:23,700 --> 00:16:26,330 And that's really a surprise, because-- 269 00:16:26,330 --> 00:16:26,860 Yes? 270 00:16:26,860 --> 00:16:28,360 AUDIENCE: You're treating the system 271 00:16:28,360 --> 00:16:30,860 as though it has like a ladder of energies and everything 272 00:16:30,860 --> 00:16:32,960 is already at its lowest? 273 00:16:32,960 --> 00:16:33,920 PROFESSOR: Yes. 274 00:16:33,920 --> 00:16:38,390 Well, OK-- you know, I'm not allowed 275 00:16:38,390 --> 00:16:40,070 to talk about statistics. 276 00:16:40,070 --> 00:16:40,880 I can't, actually. 277 00:16:40,880 --> 00:16:41,870 Because we do it-- 278 00:16:41,870 --> 00:16:45,150 we now do some statistical mechanics in 560. 279 00:16:45,150 --> 00:16:46,550 So you know a little bit. 280 00:16:46,550 --> 00:16:53,300 And you know that the molecules like to be in the lowest energy 281 00:16:53,300 --> 00:16:53,940 levels. 282 00:16:53,940 --> 00:16:58,250 And so we have these ground state molecules in a gas. 283 00:16:58,250 --> 00:17:04,250 And we understand they're isolated properties, 284 00:17:04,250 --> 00:17:05,720 or we could understand. 285 00:17:05,720 --> 00:17:07,670 And what happens here? 286 00:17:07,670 --> 00:17:10,770 Somehow there's interactions between them. 287 00:17:10,770 --> 00:17:18,104 And now first order perturbation theory, 288 00:17:18,104 --> 00:17:19,354 you're allowed to have a sign. 289 00:17:23,119 --> 00:17:27,200 But you're allowed to have a sign for diagonal elements 290 00:17:27,200 --> 00:17:30,700 of the perturbation operator. 291 00:17:30,700 --> 00:17:32,600 OK, so we'll just continue with this. 292 00:17:38,600 --> 00:17:42,965 So we want to talk about Hamiltonian 293 00:17:42,965 --> 00:17:50,110 for the system, which is the Hamiltonian for particle 294 00:17:50,110 --> 00:17:52,900 A, the Hamiltonian for particle B, 295 00:17:52,900 --> 00:18:00,770 plus some interaction term between A and B. 296 00:18:00,770 --> 00:18:08,550 OK, now the typical interaction between two particles 297 00:18:08,550 --> 00:18:10,770 is a dipole-dipole interaction. 298 00:18:10,770 --> 00:18:13,560 If molecules have dipole moments, 299 00:18:13,560 --> 00:18:16,216 then we already know something about what they 300 00:18:16,216 --> 00:18:17,340 want to do with each other. 301 00:18:19,870 --> 00:18:22,960 And from electricity and magnetism-- 302 00:18:22,960 --> 00:18:25,100 we know this isn't quantum mechanics-- 303 00:18:25,100 --> 00:18:29,020 we know that we can write a general formula 304 00:18:29,020 --> 00:18:31,840 for the dipole-dipole interaction. 305 00:18:49,430 --> 00:18:50,890 So that's the general formula for 306 00:18:50,890 --> 00:18:53,904 the dipole-dipole interaction. 307 00:18:53,904 --> 00:18:55,320 And it's a little more complicated 308 00:18:55,320 --> 00:18:56,850 than you would like because there's two terms. 309 00:18:56,850 --> 00:18:58,530 And this one is kind of a puzzling term 310 00:18:58,530 --> 00:19:03,900 because you're projecting a dipole on the interatomic axis. 311 00:19:03,900 --> 00:19:08,190 And so we'd like to be able to simplify this. 312 00:19:08,190 --> 00:19:14,160 And we can simplify it by simply drawing 313 00:19:14,160 --> 00:19:17,500 some vectors corresponding to the dipole moment. 314 00:19:17,500 --> 00:19:23,970 And so we have the dipoles oriented like that. 315 00:19:23,970 --> 00:19:28,010 Or we have the dipoles oriented like this, 316 00:19:28,010 --> 00:19:35,430 or like this and like that. 317 00:19:35,430 --> 00:19:37,780 Now, this is a reduction. 318 00:19:37,780 --> 00:19:40,200 You know, these are the simplified terms 319 00:19:40,200 --> 00:19:45,450 where these guys are irrelevant or at least you 320 00:19:45,450 --> 00:19:48,540 don't have to worry about the projections. 321 00:19:48,540 --> 00:19:54,280 And we know that this is an attractive interaction, 322 00:19:54,280 --> 00:19:56,560 this is an attractive interaction. 323 00:19:56,560 --> 00:20:00,120 This is repulsive and this is repulsive. 324 00:20:00,120 --> 00:20:04,120 And we know also by plugging into the formulas, 325 00:20:04,120 --> 00:20:13,050 we know that the energy for this one is minus mu-A mu-B over two 326 00:20:13,050 --> 00:20:17,085 pi epsilon 0 R cubed. 327 00:20:19,700 --> 00:20:23,020 And this is minus the same stuff, 328 00:20:23,020 --> 00:20:26,870 except a four in the denominator. 329 00:20:26,870 --> 00:20:30,580 So that means it's less attractive than this one. 330 00:20:30,580 --> 00:20:31,990 Minus means attractive. 331 00:20:31,990 --> 00:20:35,110 And then we have two terms that resemble these 332 00:20:35,110 --> 00:20:37,440 except they're repulsive. 333 00:20:37,440 --> 00:20:38,670 So there's four terms. 334 00:20:42,900 --> 00:20:48,395 So there's two terms that are attractive 335 00:20:48,395 --> 00:20:50,960 and two terms that are repulsive. 336 00:20:50,960 --> 00:20:56,120 And they come in pairs, one positive and one negative. 337 00:20:56,120 --> 00:21:00,020 And so you might naively say well, you can't win. 338 00:21:00,020 --> 00:21:04,550 If the particles are randomly distributed and randomly 339 00:21:04,550 --> 00:21:09,570 oriented, there will be no stabilization, even 340 00:21:09,570 --> 00:21:12,200 with dipoles. 341 00:21:12,200 --> 00:21:16,100 But that's not true because the molecules 342 00:21:16,100 --> 00:21:23,840 try to find an orientation which is 343 00:21:23,840 --> 00:21:29,870 more stable than the sum of the energies without the dipoles. 344 00:21:29,870 --> 00:21:31,810 And so maybe we could draw a picture 345 00:21:31,810 --> 00:21:34,230 and try to figure out, well, how that would be. 346 00:21:39,730 --> 00:21:45,040 So does anybody want to give me guidance to a picture 347 00:21:45,040 --> 00:21:49,920 showing, let's say, four dipoles oriented 348 00:21:49,920 --> 00:21:53,800 in the corner of a square? 349 00:21:53,800 --> 00:21:55,630 And what would be the stable arrangement 350 00:21:55,630 --> 00:21:59,980 for four dipoles oriented on the corners of a square? 351 00:22:05,350 --> 00:22:11,099 OK, right? 352 00:22:16,480 --> 00:22:22,290 So we have two attractive interactions, 353 00:22:22,290 --> 00:22:24,630 two more attractive interactions, 354 00:22:24,630 --> 00:22:31,630 and then there are repulsive interactions this way. 355 00:22:31,630 --> 00:22:34,240 But we have nearest neighbors. 356 00:22:34,240 --> 00:22:37,660 And they're stronger than the repulsive interactions. 357 00:22:37,660 --> 00:22:40,880 And so for one layer, that can win. 358 00:22:40,880 --> 00:22:46,610 Now, we have another layer of four dipoles. 359 00:22:46,610 --> 00:22:48,770 And if that layer is-- 360 00:22:54,690 --> 00:22:58,854 so we have another layer, and there will be four of them. 361 00:22:58,854 --> 00:22:59,770 And what would happen? 362 00:22:59,770 --> 00:23:01,270 What would be the stable arrangement 363 00:23:01,270 --> 00:23:04,440 if I have a dipole above this one? 364 00:23:04,440 --> 00:23:06,630 Is it going to be this way? 365 00:23:06,630 --> 00:23:07,380 Right. 366 00:23:07,380 --> 00:23:11,370 And so the next layer, all the dipoles are reversed. 367 00:23:11,370 --> 00:23:15,300 And so you can conceive of an arrangement 368 00:23:15,300 --> 00:23:23,480 where the dipoles would adopt a more stable arrangement. 369 00:23:23,480 --> 00:23:26,990 And statistical mechanics, or whatever you want to call it, 370 00:23:26,990 --> 00:23:28,700 guides that. 371 00:23:28,700 --> 00:23:35,700 And so we can say, all right, the ensemble 372 00:23:35,700 --> 00:23:38,250 of dipoles interacting with each other 373 00:23:38,250 --> 00:23:44,070 can arrange to be energetically stabilizing. 374 00:23:44,070 --> 00:23:50,125 And so we can reduce the problem to just one picture, 375 00:23:50,125 --> 00:23:53,820 say this type or that dipole-dipole arrangement, 376 00:23:53,820 --> 00:23:57,090 and do second order perturbation theory. 377 00:23:57,090 --> 00:24:00,830 And that will be an overestimate of the stabilization. 378 00:24:00,830 --> 00:24:04,460 But it will be proportional to the stabilization. 379 00:24:04,460 --> 00:24:08,780 And so we can get what we need for a very 380 00:24:08,780 --> 00:24:13,480 complicated infinite number of particle problem. 381 00:24:13,480 --> 00:24:16,300 OK, another thing that I want to ask you, 382 00:24:16,300 --> 00:24:23,735 suppose you have two particles and they come together. 383 00:24:26,720 --> 00:24:31,140 And so they come together with some kinetic energy. 384 00:24:31,140 --> 00:24:33,980 And they can't get rid of that kinetic energy, 385 00:24:33,980 --> 00:24:36,080 so they come apart. 386 00:24:36,080 --> 00:24:38,130 Now, if you have three particles, 387 00:24:38,130 --> 00:24:44,160 if something drives them together by mutual attraction, 388 00:24:44,160 --> 00:24:47,070 then one particle can take away the excess energy. 389 00:24:47,070 --> 00:24:51,130 You can end up leaving the two particles together. 390 00:24:51,130 --> 00:24:53,690 That's how you make liquids. 391 00:24:53,690 --> 00:24:56,330 So the driving force for the attraction 392 00:24:56,330 --> 00:25:00,080 between atoms or molecules is somehow 393 00:25:00,080 --> 00:25:03,740 related to what we're going to drive here. 394 00:25:03,740 --> 00:25:07,580 And that is the only way you make liquids from gases. 395 00:25:13,100 --> 00:25:15,300 And it's really strange because we're 396 00:25:15,300 --> 00:25:19,140 going to be talking about energy levels and interactions 397 00:25:19,140 --> 00:25:23,520 between molecules, and somehow coming up 398 00:25:23,520 --> 00:25:31,830 with a picture that leads to universal attraction 399 00:25:31,830 --> 00:25:36,000 between molecules that are not charged, 400 00:25:36,000 --> 00:25:39,510 not reactive, not anything. 401 00:25:39,510 --> 00:25:43,110 And they will always attract each other. 402 00:25:43,110 --> 00:25:45,390 And that's the first step towards making liquids. 403 00:25:48,720 --> 00:25:52,070 OK, now the important thing in all of this 404 00:25:52,070 --> 00:25:54,830 is that the interactions are small 405 00:25:54,830 --> 00:25:59,070 compared to the energy levels of the individual molecules. 406 00:25:59,070 --> 00:26:01,370 And so that's why we're doing perturbation theory. 407 00:26:14,890 --> 00:26:21,920 OK, so in perturbation theory we have an H 0. 408 00:26:21,920 --> 00:26:23,090 And the H 0 is-- 409 00:26:27,300 --> 00:26:30,010 I've got to make sure I use consistent notation. 410 00:26:30,010 --> 00:26:33,110 Well, I already see I have inconsistent 411 00:26:33,110 --> 00:26:34,200 notation in my notes. 412 00:26:34,200 --> 00:26:41,810 But anyway, OK, so we have the Hamiltonian 413 00:26:41,810 --> 00:26:45,470 is the individual system Hamiltonians, 414 00:26:45,470 --> 00:26:47,000 the sum of the two. 415 00:26:47,000 --> 00:26:50,990 And the first order term is VAB. 416 00:26:55,820 --> 00:27:00,590 And so whenever we can separate a system, 417 00:27:00,590 --> 00:27:06,250 we know that the zero order wave functions 418 00:27:06,250 --> 00:27:09,980 are products of the eigenfunctions of HA and HB. 419 00:27:09,980 --> 00:27:12,500 And the energy levels of 0 [INAUDIBLE] energy levels 420 00:27:12,500 --> 00:27:16,340 are sum of the energy levels for HA and HB. 421 00:27:16,340 --> 00:27:27,660 So we know that we can write H 0 psi A alpha psi B alpha. 422 00:27:27,660 --> 00:27:29,880 So this guy-- 423 00:27:29,880 --> 00:27:35,520 HA operates on psi A and treats this as a constant. 424 00:27:35,520 --> 00:27:39,400 HB operates on psi B and treats this as a constant. 425 00:27:39,400 --> 00:27:43,170 And when you do that, what you discover is that we get-- 426 00:28:03,340 --> 00:28:07,160 OK, so these are our energy levels. 427 00:28:07,160 --> 00:28:09,240 And these are our basis functions. 428 00:28:09,240 --> 00:28:10,725 And we are off to the races. 429 00:28:16,470 --> 00:28:21,140 So this is E 0. 430 00:28:21,140 --> 00:28:25,860 And this is psi 0 for the system. 431 00:28:25,860 --> 00:28:36,490 OK, now the first thing we do in perturbation theory is we say, 432 00:28:36,490 --> 00:28:39,840 let's look for the first order correction of the energy. 433 00:28:39,840 --> 00:28:45,640 That's the diagonal matrix element of the H1 term, this. 434 00:28:52,530 --> 00:28:56,660 So suppose we do have dipole moments on both particle A 435 00:28:56,660 --> 00:28:59,640 and particle B 436 00:29:21,730 --> 00:29:23,260 So that's the first order energy. 437 00:29:30,120 --> 00:29:34,180 Well, this is just the dipole-dipole interaction. 438 00:29:34,180 --> 00:29:37,890 And so we can take the reduced formula. 439 00:29:37,890 --> 00:29:46,040 We're just going to regard VAB as mu A dot mu B over R cubed. 440 00:29:46,040 --> 00:30:01,110 And so the energy is minus one over two pi epsilon 0 R cubed. 441 00:30:01,110 --> 00:30:06,002 I left out the two pi epsilon 0 on this formula. 442 00:30:13,260 --> 00:30:15,490 And so skipping a couple of steps, 443 00:30:15,490 --> 00:30:26,650 we have mu A mu B, the expectation value of these. 444 00:30:26,650 --> 00:30:29,560 So if there's a dipole-dipole interaction, 445 00:30:29,560 --> 00:30:36,710 there will be a first order correction term. 446 00:30:36,710 --> 00:30:42,580 And keep in mind that these guys can have signs. 447 00:30:42,580 --> 00:30:45,950 And that sign does appear in the first order correction 448 00:30:45,950 --> 00:30:46,670 to the energy. 449 00:30:49,450 --> 00:30:52,900 But that's basically these pictures we drew here. 450 00:30:52,900 --> 00:30:54,530 We have a dipole pointing this way, 451 00:30:54,530 --> 00:30:56,230 we have a dipole pointing that way. 452 00:30:56,230 --> 00:30:58,120 That's repulsive. 453 00:30:58,120 --> 00:31:01,270 And that's all-- so there's no surprises here. 454 00:31:06,150 --> 00:31:09,680 OK, but the important thing is that the energy-- 455 00:31:09,680 --> 00:31:12,220 If we have two molecules with dipoles, 456 00:31:12,220 --> 00:31:17,080 we get an interaction energy that we can calculate. 457 00:31:17,080 --> 00:31:18,670 But the interesting thing is going 458 00:31:18,670 --> 00:31:22,210 to be when we have molecules where at least one 459 00:31:22,210 --> 00:31:24,070 of the molecules doesn't have a dipole. 460 00:31:27,140 --> 00:31:29,330 Then we're getting into territory 461 00:31:29,330 --> 00:31:31,657 that we didn't think we understood. 462 00:31:41,610 --> 00:31:50,000 But we're continuing with the problem 463 00:31:50,000 --> 00:31:51,100 where we do have dipoles. 464 00:31:56,950 --> 00:31:58,390 I just want to make sure that-- 465 00:32:04,210 --> 00:32:05,760 yeah, OK. 466 00:32:05,760 --> 00:32:14,280 So for the second order correction to the lowest energy 467 00:32:14,280 --> 00:32:21,470 level, we have the sum over m, which is two quantum numbers. 468 00:32:21,470 --> 00:32:24,740 And m is not equal to the 0-0 state. 469 00:32:24,740 --> 00:32:28,550 Alpha is not 0, beta is not 0 sum over everything 470 00:32:28,550 --> 00:32:31,140 except the ground state. 471 00:32:31,140 --> 00:32:47,660 And what we have is an integral of psi m 0 star VAB psi 0 0 472 00:32:47,660 --> 00:33:01,940 d tau squared over then E 0 0 minus E 0 m. 473 00:33:01,940 --> 00:33:05,870 Standard formula for second order variation theory. 474 00:33:05,870 --> 00:33:07,320 This is the ground state. 475 00:33:07,320 --> 00:33:08,850 This is some excited state. 476 00:33:08,850 --> 00:33:10,079 This is negative. 477 00:33:15,460 --> 00:33:18,002 This is squared. 478 00:33:18,002 --> 00:33:18,835 And this is squared. 479 00:33:21,480 --> 00:33:24,604 So this overall correction is negative. 480 00:33:29,450 --> 00:33:31,930 Now we can unpack all this and we 481 00:33:31,930 --> 00:33:38,450 can write the various terms in more instructive form. 482 00:33:38,450 --> 00:33:44,990 And what we'll have in the denominator will be E 0-- 483 00:33:44,990 --> 00:34:11,760 well, E A 0 minus E A alpha plus E B 0 E 0 E minus E B beta. 484 00:34:11,760 --> 00:34:24,719 Now we can rearrange that so we have E 0 A plus E 0 B minus E A 485 00:34:24,719 --> 00:34:30,400 alpha plus E B beta. 486 00:34:30,400 --> 00:34:33,909 So we have the ground state energy and the known excited 487 00:34:33,909 --> 00:34:34,780 state energies. 488 00:34:34,780 --> 00:34:37,179 And there is the difference between these. 489 00:34:37,179 --> 00:34:40,330 So that part is simple. 490 00:34:40,330 --> 00:34:48,870 And the matrix element also simplifies to the product 491 00:34:48,870 --> 00:34:50,075 of the dipole moment. 492 00:34:54,170 --> 00:34:58,210 OK, now suppose-- 493 00:35:06,200 --> 00:35:07,640 I just want to make sure that I'm 494 00:35:07,640 --> 00:35:10,520 doing this in the order it appears in the notes. 495 00:35:19,860 --> 00:35:28,400 Suppose mu A is not 0 and mu B is equal to 0. 496 00:35:31,580 --> 00:35:38,510 OK, well then all these formulas go out the window 497 00:35:38,510 --> 00:35:41,860 because nominally we think well, this B 498 00:35:41,860 --> 00:35:43,070 doesn't have a dipole moment. 499 00:35:45,690 --> 00:35:52,250 But suppose we have a particle B in an electric field. 500 00:35:52,250 --> 00:35:56,465 The electric field will polarize B. 501 00:35:56,465 --> 00:36:03,810 And if we have molecule A, which has a dipole moment, near B, 502 00:36:03,810 --> 00:36:07,320 it will also polarize B. And second order perturbation 503 00:36:07,320 --> 00:36:09,510 theory tells us how to handle that. 504 00:36:13,550 --> 00:36:22,850 And so we get a formula, a second order 505 00:36:22,850 --> 00:36:28,220 correction to the energy B. And we call this B comma induction. 506 00:36:32,020 --> 00:36:45,860 And that's going to be mu A squared over four pi 507 00:36:45,860 --> 00:36:52,340 squared epsilon 0 squared R to the six times 508 00:36:52,340 --> 00:36:59,220 sum beta not equal to 0 of the squared matrix element 509 00:36:59,220 --> 00:37:20,480 integral psi B star beta mu B psi B 0 d tau B squared 510 00:37:20,480 --> 00:37:27,290 over E B 0 minus E B beta. 511 00:37:27,290 --> 00:37:30,860 Now, I just told you that the dipole moment-- 512 00:37:30,860 --> 00:37:34,400 the permanent dipole moment for particle B is 0. 513 00:37:34,400 --> 00:37:39,260 But this is not the expectation value of the dipole moment. 514 00:37:39,260 --> 00:37:44,610 This is an off-diagonal matrix element of the dipole moment. 515 00:37:44,610 --> 00:37:46,860 This is the transition dipole. 516 00:37:46,860 --> 00:37:49,230 Transition between v equals 0-- 517 00:37:49,230 --> 00:37:53,100 well, between the ground state of B 518 00:37:53,100 --> 00:37:57,800 and some excited state, the beta, excited state of B. Well, 519 00:37:57,800 --> 00:38:02,730 we can write something like that because we 520 00:38:02,730 --> 00:38:08,220 know for all molecules there are transitions, which 521 00:38:08,220 --> 00:38:11,240 are electric dipole allowed. 522 00:38:11,240 --> 00:38:16,290 Now they could be between the ground state 523 00:38:16,290 --> 00:38:19,020 and an electronically excited state or the ground 524 00:38:19,020 --> 00:38:22,770 state and a vibrationally excited state. 525 00:38:22,770 --> 00:38:24,800 So we know something about these: 526 00:38:24,800 --> 00:38:26,930 we know the rules for when they're not 0. 527 00:38:26,930 --> 00:38:29,030 And this is a perfectly reasonable thing 528 00:38:29,030 --> 00:38:32,970 because it says this is related to the transition of-- 529 00:38:32,970 --> 00:38:36,640 the intensity of the observable transition. 530 00:38:36,640 --> 00:38:38,620 And this should really start to make 531 00:38:38,620 --> 00:38:43,450 you a little bit appreciative of perturbation theory. 532 00:38:43,450 --> 00:38:48,020 Because we're taking something which you can observe, 533 00:38:48,020 --> 00:38:51,220 a transition strength, and using it 534 00:38:51,220 --> 00:38:57,850 to describe an interaction between two molecules. 535 00:38:57,850 --> 00:39:03,400 So it's a relationship between two apparently unrelated 536 00:39:03,400 --> 00:39:05,260 phenomena. 537 00:39:05,260 --> 00:39:07,810 Physical chemistry is full of that. 538 00:39:07,810 --> 00:39:08,930 You build a model. 539 00:39:08,930 --> 00:39:11,950 And the model is telling you that certain things 540 00:39:11,950 --> 00:39:13,150 are related. 541 00:39:13,150 --> 00:39:14,380 And how are they related? 542 00:39:14,380 --> 00:39:17,560 Well, they're related by one over R to the sixth. 543 00:39:17,560 --> 00:39:20,830 They're related by the strength of the dipole on one, 544 00:39:20,830 --> 00:39:23,860 the permanent dipole on one, and some transition 545 00:39:23,860 --> 00:39:26,500 intensity-derived quantity on the other. 546 00:39:26,500 --> 00:39:29,290 Isn't that neat? 547 00:39:29,290 --> 00:39:30,520 And since the dipole-- 548 00:39:33,520 --> 00:39:37,070 we say well, this is a dipole-dipole interaction. 549 00:39:37,070 --> 00:39:43,990 And the mu A, the dipole on A, is inducing a dipole on B. 550 00:39:43,990 --> 00:39:45,670 It's not inducing anything. 551 00:39:45,670 --> 00:39:48,460 It's using transition dipoles. 552 00:39:48,460 --> 00:39:52,960 But it's creating something that looks like a permanent dipole. 553 00:39:52,960 --> 00:39:55,270 And you have dipole-dipole interaction. 554 00:39:55,270 --> 00:39:59,080 But the only difference is that it's one over R to the sixth 555 00:39:59,080 --> 00:40:00,445 because it's an induced dipole. 556 00:40:05,200 --> 00:40:09,500 OK, so we're using transition-- 557 00:40:09,500 --> 00:40:14,050 well, electric dipole transition moments. 558 00:40:14,050 --> 00:40:18,786 (There used to be two erasers!) 559 00:40:33,930 --> 00:40:38,290 So these electric dipole transition moments 560 00:40:38,290 --> 00:40:42,194 play the role of producing a dipole that can 561 00:40:42,194 --> 00:40:43,485 interact with the other dipole. 562 00:40:49,530 --> 00:40:53,390 And so-- well, do we need this? 563 00:40:53,390 --> 00:40:55,330 Well, we don't really. 564 00:40:55,330 --> 00:40:58,030 If you had dipoles, then the interaction 565 00:40:58,030 --> 00:41:00,550 would be one over R cubed. 566 00:41:00,550 --> 00:41:04,000 But if one of the dipoles is 0, then the interaction 567 00:41:04,000 --> 00:41:07,360 is one over R to the sixth. 568 00:41:07,360 --> 00:41:09,170 And it's one over R to the sixth times 569 00:41:09,170 --> 00:41:13,670 the permanent dipole of one and this thing from perturbation 570 00:41:13,670 --> 00:41:16,310 theory, which is the induced dipole of the other. 571 00:41:25,840 --> 00:41:31,270 Suppose mu A equals 0 and mu B equals 0. 572 00:41:34,740 --> 00:41:36,610 Will the particles interact with each other? 573 00:41:36,610 --> 00:41:38,170 Will they attract each other? 574 00:41:38,170 --> 00:41:40,360 And the answer is yes. 575 00:41:40,360 --> 00:41:42,960 And it's going to be an induced dipole-induced dipole 576 00:41:42,960 --> 00:41:44,810 interaction. 577 00:41:44,810 --> 00:41:51,840 Which, now since most people who write for general science have 578 00:41:51,840 --> 00:41:54,390 no clue about what quantum mechanics is, 579 00:41:54,390 --> 00:41:57,870 they will talk about all sorts of fanciful things-- 580 00:41:57,870 --> 00:42:01,230 you get an institute-- 581 00:42:01,230 --> 00:42:05,910 an instantaneous fluctuation on one particle which 582 00:42:05,910 --> 00:42:08,350 creates an instantaneous fluctuation 583 00:42:08,350 --> 00:42:11,220 on the other particle, and bam, you've got stabilization. 584 00:42:11,220 --> 00:42:11,970 Well, yeah, maybe. 585 00:42:11,970 --> 00:42:16,070 But you know, this is a simple perturbation theory. 586 00:42:16,070 --> 00:42:20,770 And so I don't really need to actually write out everything. 587 00:42:20,770 --> 00:42:27,730 Because one, you do that and you include the second order 588 00:42:27,730 --> 00:42:33,010 perturbation expression for particle A and particle B, 589 00:42:33,010 --> 00:42:35,740 you get a term that looks just like this. 590 00:42:35,740 --> 00:42:38,900 It has one over R to the sixth. 591 00:42:38,900 --> 00:42:43,250 But it now involves transition moments, particle A 592 00:42:43,250 --> 00:42:44,990 and particle B-- 593 00:42:44,990 --> 00:42:45,710 nothing new. 594 00:42:45,710 --> 00:42:48,650 You don't have to induce anything more. 595 00:42:48,650 --> 00:42:52,702 You don't get another factor of one over R cubed. 596 00:42:52,702 --> 00:42:54,660 You've already got the one over R to the sixth. 597 00:42:57,576 --> 00:43:07,090 So I'm going to skip a lot of steps 598 00:43:07,090 --> 00:43:09,250 because the notes are really clear in there. 599 00:43:09,250 --> 00:43:11,630 And we don't have a whole lot of time. 600 00:43:11,630 --> 00:43:15,430 And so I'm just going to write down the result. 601 00:43:15,430 --> 00:43:27,900 And now the name of this term is called dispersion, or London 602 00:43:27,900 --> 00:43:32,220 because a fellow by the name of London invented this. 603 00:43:32,220 --> 00:43:38,400 And the energy level formula for this correction term 604 00:43:38,400 --> 00:43:43,410 is one over four pi squared epsilon 0 squared 605 00:43:43,410 --> 00:43:45,990 R to the sixth. 606 00:43:45,990 --> 00:43:54,990 And then this sum of alpha equals one beta equals one 607 00:43:54,990 --> 00:43:56,350 to infinity. 608 00:43:56,350 --> 00:44:02,080 In other words, not alpha equals 0, not beta equals 0. 609 00:44:02,080 --> 00:44:02,610 And then-- 610 00:44:29,720 --> 00:44:30,430 Sorry about that. 611 00:44:33,870 --> 00:44:38,520 So we have a numerator, which is the squared transition 612 00:44:38,520 --> 00:44:44,320 moment between the 0 level of particle A and the alpha level 613 00:44:44,320 --> 00:44:48,040 and the 0 level of B and the beta level. 614 00:44:48,040 --> 00:44:50,090 We have the energy of the ground state 615 00:44:50,090 --> 00:44:53,170 and we have the energy of these excited states. 616 00:44:53,170 --> 00:44:56,340 All of this stuff, you know or at least you 617 00:44:56,340 --> 00:44:59,520 can parametrize it in terms of this strength 618 00:44:59,520 --> 00:45:01,040 of our transition. 619 00:45:01,040 --> 00:45:04,590 And there is an explicit formula that relates the dipole moment 620 00:45:04,590 --> 00:45:09,180 to the intensity depending on what kind of experiment you do. 621 00:45:09,180 --> 00:45:11,610 So I'm not going to take you through that. 622 00:45:11,610 --> 00:45:15,870 But you know you can measure the intensity, the molar 623 00:45:15,870 --> 00:45:22,260 absorptivity, or whatever you want that is related to this. 624 00:45:22,260 --> 00:45:24,510 So these are experimentally observable. 625 00:45:24,510 --> 00:45:26,490 These are experimentally observable. 626 00:45:26,490 --> 00:45:32,970 And usually this infinite sum is dominated 627 00:45:32,970 --> 00:45:37,630 by the lowest allowed transition from the ground state. 628 00:45:37,630 --> 00:45:40,260 And that can be an electronic transition 629 00:45:40,260 --> 00:45:42,660 or a vibrational transition. 630 00:45:42,660 --> 00:45:45,270 So for our homonuclear molecule, there 631 00:45:45,270 --> 00:45:48,000 isn't going to be an allowed vibrational transition. 632 00:45:48,000 --> 00:45:52,620 So nitrogen and oxygen cannot talk to each other through 633 00:45:52,620 --> 00:45:53,820 transition dipole-- 634 00:45:53,820 --> 00:45:57,870 vibrational transition dipoles because the dipole moment 635 00:45:57,870 --> 00:46:00,390 is not dependent on [INAUDIBLE] but they 636 00:46:00,390 --> 00:46:02,620 have electronic transitions. 637 00:46:02,620 --> 00:46:04,980 And so there is this electronic-- as 638 00:46:04,980 --> 00:46:08,200 opposed to a vibrational polarizability of the two. 639 00:46:08,200 --> 00:46:15,540 And that's what gives rise to attraction between N2 and O2, 640 00:46:15,540 --> 00:46:16,290 or N2 and N2. 641 00:46:20,320 --> 00:46:23,500 So everything is knowable and predictable. 642 00:46:23,500 --> 00:46:26,910 And even though the perturbation's sum is infinite, 643 00:46:26,910 --> 00:46:30,500 and even though we've made a gross approximation and say, 644 00:46:30,500 --> 00:46:34,270 well, we're just interested in this sort of an interaction 645 00:46:34,270 --> 00:46:39,730 or this sort of an interaction, and we 646 00:46:39,730 --> 00:46:42,850 know that the true interaction will 647 00:46:42,850 --> 00:46:47,290 be less because there will be an average over all 648 00:46:47,290 --> 00:46:48,670 of the random orientations. 649 00:46:48,670 --> 00:46:51,610 But there's going to be a lowest energy picture. 650 00:46:51,610 --> 00:46:54,310 And you can get that from statistical mechanics. 651 00:46:54,310 --> 00:46:56,740 And so you can do the configurational average 652 00:46:56,740 --> 00:47:00,220 over all possible orientations of all of the interacting 653 00:47:00,220 --> 00:47:01,120 particles. 654 00:47:01,120 --> 00:47:04,100 And you can calculate this. 655 00:47:04,100 --> 00:47:07,490 So that's beyond the level of this course. 656 00:47:07,490 --> 00:47:10,480 But the engine of the calculation is this formula. 657 00:47:26,900 --> 00:47:31,040 So molecules are polarizable. 658 00:47:31,040 --> 00:47:35,020 They react to an external electric field. 659 00:47:35,020 --> 00:47:37,660 And there are several ways you can describe 660 00:47:37,660 --> 00:47:39,600 the defined polarizability. 661 00:47:39,600 --> 00:47:42,600 One is to say well the polarizability-- 662 00:47:42,600 --> 00:47:44,430 let's call it alpha-- 663 00:47:44,430 --> 00:47:47,700 is equal to the second derivative 664 00:47:47,700 --> 00:47:53,610 of the energy with respect to the electric field. 665 00:47:57,350 --> 00:47:59,860 But the polarizability, you know, 666 00:47:59,860 --> 00:48:04,180 you also get from perturbation theory. 667 00:48:04,180 --> 00:48:08,310 And so each of these terms or the energy 668 00:48:08,310 --> 00:48:11,110 is associated with the interaction 669 00:48:11,110 --> 00:48:13,300 of a dipole with an electric field 670 00:48:13,300 --> 00:48:15,770 is the electric field times the dipole. 671 00:48:15,770 --> 00:48:17,470 And so we have electric field squared. 672 00:48:17,470 --> 00:48:19,700 We take the second derivative of that. 673 00:48:19,700 --> 00:48:21,070 And we've got back to this. 674 00:48:25,420 --> 00:48:28,037 So now here we come with intuition. 675 00:48:31,620 --> 00:48:36,830 If I asked you to tell me the van der Waals 676 00:48:36,830 --> 00:48:40,910 or the dispersion interaction between helium and helium, 677 00:48:40,910 --> 00:48:43,872 and compare that to xenon to xenon, 678 00:48:43,872 --> 00:48:45,080 would you be able to tell me? 679 00:48:48,550 --> 00:48:49,970 So which one is stronger? 680 00:48:49,970 --> 00:48:57,064 Which one-- which pair of inert gases is stickier? 681 00:48:57,064 --> 00:48:57,880 AUDIENCE: Xenon. 682 00:48:57,880 --> 00:48:58,880 PROFESSOR: Xenon, right. 683 00:48:58,880 --> 00:48:59,650 Why? 684 00:48:59,650 --> 00:49:04,690 Because the energy denominator for the first excited state 685 00:49:04,690 --> 00:49:06,880 is much lower for xenon than for helium. 686 00:49:09,410 --> 00:49:17,444 OK, what about large molecule versus small molecule? 687 00:49:25,360 --> 00:49:29,890 Do I need to provide a hint? 688 00:49:29,890 --> 00:49:31,060 3n minus six. 689 00:49:33,970 --> 00:49:37,170 The more vibrational modes, the more transitions 690 00:49:37,170 --> 00:49:40,230 are available for polarization. 691 00:49:40,230 --> 00:49:43,897 And so again, now we're talking about molecules 692 00:49:43,897 --> 00:49:44,730 which can be floppy. 693 00:49:44,730 --> 00:49:48,620 And the floppier the molecule-- 694 00:49:48,620 --> 00:49:52,190 that means they have low frequency vibrations-- 695 00:49:52,190 --> 00:49:55,930 you can polarize the daylights out of them. 696 00:49:55,930 --> 00:50:00,510 And so large molecules are really sticky, stickier 697 00:50:00,510 --> 00:50:02,580 than inert gases. 698 00:50:02,580 --> 00:50:05,640 Because you have this dispersion interaction 699 00:50:05,640 --> 00:50:07,120 that brings them together. 700 00:50:07,120 --> 00:50:10,050 And once they're together, then they can do magic. 701 00:50:10,050 --> 00:50:13,230 But the thing that drives the condensation, or the formation 702 00:50:13,230 --> 00:50:17,300 of droplets, or whatever, is this dispersion interaction. 703 00:50:17,300 --> 00:50:18,190 And it's universal. 704 00:50:18,190 --> 00:50:23,790 Every pair of molecules or atoms is going to attract each other. 705 00:50:23,790 --> 00:50:26,940 And the strength is something that you 706 00:50:26,940 --> 00:50:29,430 can rate in terms of what you know 707 00:50:29,430 --> 00:50:32,670 about atomic and molecular properties. 708 00:50:32,670 --> 00:50:34,560 And isn't that neat? 709 00:50:34,560 --> 00:50:38,190 Because you're saying the viscosity, 710 00:50:38,190 --> 00:50:44,120 or whatever it is, the property of a gas, all of those things 711 00:50:44,120 --> 00:50:49,250 are related to the properties of the individual atoms 712 00:50:49,250 --> 00:50:53,161 or molecules embedded in perturbation theory. 713 00:50:53,161 --> 00:50:54,910 This is why I think perturbation theory is 714 00:50:54,910 --> 00:51:00,550 so wonderful, because it enables you to relate different things. 715 00:51:00,550 --> 00:51:02,530 And even if you're not going to evaluate 716 00:51:02,530 --> 00:51:04,780 the magnitudes of the terms, you can 717 00:51:04,780 --> 00:51:07,530 make comparisons which are always going to be right. 718 00:51:10,680 --> 00:51:11,950 And that's kind of wonderful. 719 00:51:11,950 --> 00:51:15,970 I mean, you don't have to use the equations. 720 00:51:15,970 --> 00:51:19,660 Or you just know that these particles will attract 721 00:51:19,660 --> 00:51:22,960 each other and the driving force is either 722 00:51:22,960 --> 00:51:25,330 a permanent or an induced dipole. 723 00:51:25,330 --> 00:51:29,380 And the induced dipole will be strong if you 724 00:51:29,380 --> 00:51:34,060 have a strong transition to a low-lying state, 725 00:51:34,060 --> 00:51:36,700 or a lot of strong transitions to a lot of low-lying states 726 00:51:36,700 --> 00:51:38,020 as you have in big molecules. 727 00:51:41,170 --> 00:51:45,650 OK, so I'm not positive what I'm going to talk about on Friday. 728 00:51:45,650 --> 00:51:48,030 It's probably going to be photochemistry. 729 00:51:48,030 --> 00:51:53,510 But I may panic and do something else. 730 00:51:53,510 --> 00:51:56,500 OK, I'll see you Friday.