WEBVTT 00:00:00.090 --> 00:00:02.490 The following content is provided under a Creative 00:00:02.490 --> 00:00:04.030 Commons license. 00:00:04.030 --> 00:00:06.330 Your support will help MIT OpenCourseWare 00:00:06.330 --> 00:00:10.690 continue to offer high quality educational resources for free. 00:00:10.690 --> 00:00:13.320 To make a donation or view additional materials 00:00:13.320 --> 00:00:16.814 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:16.814 --> 00:00:17.850 at OCW.MIT.edu. 00:00:21.620 --> 00:00:27.750 PROFESSOR: Today's lecture is on intermolecular interactions. 00:00:27.750 --> 00:00:32.600 And it's really one of these favorite topics 00:00:32.600 --> 00:00:37.320 where I can say, well, this is the origin of life. 00:00:37.320 --> 00:00:41.150 How do gases actually condense? 00:00:41.150 --> 00:00:45.410 And we can build a really, really simple theory 00:00:45.410 --> 00:00:50.630 for how gases condense based on what we know, 00:00:50.630 --> 00:00:53.180 based on second order perturbation theory, 00:00:53.180 --> 00:00:55.580 based on some hand-waving stuff. 00:00:55.580 --> 00:00:58.340 And it really is something from nothing. 00:00:58.340 --> 00:01:02.070 But it's also an example of what we do in physical chemistry 00:01:02.070 --> 00:01:07.220 where we explain one phenomenon by taking something that 00:01:07.220 --> 00:01:11.210 seems to be unrelated and building a model around it, 00:01:11.210 --> 00:01:14.630 and explaining this surprising thing. 00:01:14.630 --> 00:01:19.110 And that's what's going to happen today. 00:01:19.110 --> 00:01:20.720 We'll talk more about that. 00:01:20.720 --> 00:01:27.760 OK, now last time we talked about wavepacket dynamics. 00:01:27.760 --> 00:01:34.060 And so with wavepacket dynamics, there is first the pluck. 00:01:34.060 --> 00:01:37.420 How do you start the system in some superposition 00:01:37.420 --> 00:01:40.300 of eigenstates? 00:01:40.300 --> 00:01:48.320 And then there's the evolution of the pluck, which is usually 00:01:48.320 --> 00:01:53.890 some kind of particle-like state that follows Newton's laws. 00:01:53.890 --> 00:01:59.150 The center of the wavepacket follows Newton's laws 00:01:59.150 --> 00:02:01.100 for the position and the momentum. 00:02:03.890 --> 00:02:10.470 But if you make a wavepacket in a real molecule, 00:02:10.470 --> 00:02:14.570 the molecule is going to be not a harmonic oscillator. 00:02:14.570 --> 00:02:18.590 There will be anharmonic effects which causes the wavepacket 00:02:18.590 --> 00:02:21.200 to de-phase, but the center of the wavepacket 00:02:21.200 --> 00:02:28.520 continues to move as if it were in a simple potential. 00:02:28.520 --> 00:02:31.130 And it can de-phase, and rephase, and all sorts 00:02:31.130 --> 00:02:32.180 of complicated stuff. 00:02:32.180 --> 00:02:34.850 And that can be very interesting. 00:02:34.850 --> 00:02:38.240 Now, when you take this step from a diatonic molecule 00:02:38.240 --> 00:02:41.210 to a polyatomic molecule, you have 00:02:41.210 --> 00:02:45.560 3n minus 6 oscillators cohabiting in the same house. 00:02:45.560 --> 00:02:47.180 And they're going to interact. 00:02:47.180 --> 00:02:51.770 And so if you start a wavepacket on one 00:02:51.770 --> 00:02:54.890 Franck-Condon bright mode because 00:02:54.890 --> 00:02:59.240 of anharmonic interactions handled by perturbation theory, 00:02:59.240 --> 00:03:04.430 between modes you start to get this wavepacket leaking out 00:03:04.430 --> 00:03:06.110 into other modes. 00:03:06.110 --> 00:03:09.800 And so the expectation value is going 00:03:09.800 --> 00:03:15.470 to exhibit an oscillating time dependence associated 00:03:15.470 --> 00:03:18.470 with the bright mode and some of the dark modes 00:03:18.470 --> 00:03:21.050 that are anharmonically coupled to the bright mode. 00:03:21.050 --> 00:03:23.870 And it can get pretty complicated. 00:03:23.870 --> 00:03:27.580 But you can understand all of it because you 00:03:27.580 --> 00:03:28.930 have perturbation theory. 00:03:28.930 --> 00:03:35.920 And so you can say, OK this time the wavepacket-- 00:03:35.920 --> 00:03:39.090 some amplitude of the initial wavepacket 00:03:39.090 --> 00:03:41.460 is transferred to a different mode. 00:03:41.460 --> 00:03:43.170 And when does that happen? 00:03:43.170 --> 00:03:45.350 Stationary phase. 00:03:45.350 --> 00:03:49.630 It happens at a particular geometry. 00:03:49.630 --> 00:03:52.590 And so you can be asking, OK, when is 00:03:52.590 --> 00:03:54.510 this excitation transferring? 00:03:54.510 --> 00:03:55.710 How strongly? 00:03:55.710 --> 00:03:59.040 This is a very, very complete picture. 00:03:59.040 --> 00:04:01.860 And it's very much like classical mechanics 00:04:01.860 --> 00:04:03.420 because it's localized. 00:04:05.940 --> 00:04:08.820 Then we talked about Landau-Zener. 00:04:08.820 --> 00:04:12.240 And the Landau-Zener is basically a simple idea 00:04:12.240 --> 00:04:16.760 that you also understand from real life. 00:04:16.760 --> 00:04:20.920 And it has to do with how does a wavepacket jump from one 00:04:20.920 --> 00:04:24.310 potential surface to another? 00:04:24.310 --> 00:04:27.453 And again, it's stationary phase. 00:04:27.453 --> 00:04:32.390 At the internuclear distance of the curve crossing, 00:04:32.390 --> 00:04:35.340 the wavepacket has to decide which 00:04:35.340 --> 00:04:37.140 curve it's going to follow. 00:04:37.140 --> 00:04:40.620 And it depends on the velocity of the packet 00:04:40.620 --> 00:04:45.500 at the curve crossing and the difference in slopes of the two 00:04:45.500 --> 00:04:47.535 paths it has to choose between. 00:04:47.535 --> 00:04:50.400 And all of this is very simple. 00:04:50.400 --> 00:04:55.290 Now, I am trying to create an exam problem which 00:04:55.290 --> 00:04:58.080 is related to Landau-Zener. 00:04:58.080 --> 00:05:00.540 And this is hard. 00:05:00.540 --> 00:05:05.490 But if I succeed, then you can expect to see it. 00:05:05.490 --> 00:05:08.550 The last thing I talked about the last lecture 00:05:08.550 --> 00:05:13.000 was the Zewail pump-probe experiments. 00:05:13.000 --> 00:05:18.060 And this is a beautifully graphical picture 00:05:18.060 --> 00:05:20.580 of how do you create something and then 00:05:20.580 --> 00:05:23.000 probe what's happening. 00:05:23.000 --> 00:05:27.800 And again, I hope that I can come up 00:05:27.800 --> 00:05:33.170 with something pictorial for the exam on the Zewail pump. 00:05:33.170 --> 00:05:39.480 OK, now before I talk about intermolecular interactions, 00:05:39.480 --> 00:05:44.120 I want to do a little bit of tying up loose ends. 00:05:44.120 --> 00:05:47.000 You know about non-degenerate perturbation theory. 00:05:49.850 --> 00:05:52.100 Now there's another kind of perturbation theory called 00:05:52.100 --> 00:05:53.510 degenerate perturbation theory. 00:05:57.900 --> 00:06:01.240 And this is when non-degenerate perturbation theory 00:06:01.240 --> 00:06:13.930 fails because H i j over E i 0 minus E j 0 00:06:13.930 --> 00:06:16.200 is greater than one. 00:06:16.200 --> 00:06:17.519 Well then you've got a problem. 00:06:17.519 --> 00:06:19.310 You can't use ordinary perturbation theory. 00:06:19.310 --> 00:06:21.940 You have to diagonalize. 00:06:21.940 --> 00:06:23.840 You diagonalize it two by two or you 00:06:23.840 --> 00:06:28.990 diagonalize all of the little cases where this is violated. 00:06:28.990 --> 00:06:31.350 That's called degenerative perturbation theory. 00:06:31.350 --> 00:06:33.850 And there is a little bit of extra recipe 00:06:33.850 --> 00:06:37.930 for how do you put information about remote states 00:06:37.930 --> 00:06:41.130 into the part of the Hamiltonian you diagonalize? 00:06:41.130 --> 00:06:43.120 That's called a Van Vleck transformation, 00:06:43.120 --> 00:06:45.830 and I'm not going to talk about it. 00:06:45.830 --> 00:06:49.930 But basically you fix things up by diagonalizing two 00:06:49.930 --> 00:06:52.090 by two or three by three. 00:06:52.090 --> 00:06:55.160 And then do ordinary perturbation theory. 00:06:55.160 --> 00:06:56.704 And then there's variational. 00:07:01.450 --> 00:07:06.220 The variational model is something 00:07:06.220 --> 00:07:12.130 we've used mostly in a minimum basis set for talking 00:07:12.130 --> 00:07:15.800 about the atomic molecules. 00:07:15.800 --> 00:07:21.290 And in the variational model, we minimize energies 00:07:21.290 --> 00:07:25.160 by optimizing mixing coefficients. 00:07:28.080 --> 00:07:34.220 Now, perturbation theory is implicitly an infinite basis. 00:07:34.220 --> 00:07:36.680 Usually the infinity doesn't matter 00:07:36.680 --> 00:07:39.470 because the interesting stuff comes in 00:07:39.470 --> 00:07:42.680 from relatively low-lying levels. 00:07:42.680 --> 00:07:45.860 And the infinite number of very far away levels 00:07:45.860 --> 00:07:49.520 just modifies the energy levels a little bit, 00:07:49.520 --> 00:07:52.050 and you don't have to worry about it. 00:07:52.050 --> 00:07:57.640 The variational method, you choose your own basis set. 00:07:57.640 --> 00:08:00.800 Now, the choice of basis set is frequently 00:08:00.800 --> 00:08:02.720 dictated by convenience. 00:08:02.720 --> 00:08:09.680 And the basis set is often not very large or not orthonormal. 00:08:09.680 --> 00:08:12.710 And so there are problems with how 00:08:12.710 --> 00:08:15.650 do we deal with basis functions which 00:08:15.650 --> 00:08:20.150 are not orthonormal, especially not orthogonal to each other? 00:08:20.150 --> 00:08:22.040 And there is a transformation that you 00:08:22.040 --> 00:08:24.860 can do to make that work out. 00:08:24.860 --> 00:08:27.890 I mean, the problem is basically-- 00:08:27.890 --> 00:08:35.990 the secular equation is this for the variational method. 00:08:35.990 --> 00:08:41.510 And this is the overlap matrix. 00:08:41.510 --> 00:08:46.550 And the overlap matrix is not the unit matrix. 00:08:46.550 --> 00:08:50.270 We have what's called the generalized variational 00:08:50.270 --> 00:08:51.410 calculation. 00:08:51.410 --> 00:08:53.270 And there are methods for handling that. 00:08:53.270 --> 00:08:57.520 And I have often introduced those methods on a final exam. 00:09:00.790 --> 00:09:03.380 Those are just statements. 00:09:03.380 --> 00:09:09.320 OK, so variational is not guaranteed 00:09:09.320 --> 00:09:12.900 to give the exact answer. 00:09:12.900 --> 00:09:17.310 And it can't if you don't use an infinite basis. 00:09:17.310 --> 00:09:20.909 Perturbation theory implicitly uses this infinite basis. 00:09:20.909 --> 00:09:22.950 And you should feel confident that you're getting 00:09:22.950 --> 00:09:25.070 close to the right answer. 00:09:25.070 --> 00:09:31.015 And you can know what sort of errors you're making. 00:09:31.015 --> 00:09:33.030 OK, yes? 00:09:33.030 --> 00:09:34.910 AUDIENCE: When you go to very high orders 00:09:34.910 --> 00:09:36.930 in perturbation theory, your convergence 00:09:36.930 --> 00:09:38.787 is not necessarily better-- 00:09:38.787 --> 00:09:39.870 PROFESSOR: That's correct. 00:09:39.870 --> 00:09:41.060 AUDIENCE: --on the real solutions. 00:09:41.060 --> 00:09:41.550 PROFESSOR: OK, but that's-- 00:09:41.550 --> 00:09:43.133 AUDIENCE: [INAUDIBLE] when to cool it? 00:09:45.690 --> 00:09:49.920 PROFESSOR: This is-- the problem there is that you cannot 00:09:49.920 --> 00:09:55.630 include correlation in perturbation theory. 00:09:55.630 --> 00:09:59.510 And so there are things that are outside 00:09:59.510 --> 00:10:07.420 of the network of things that I've described. 00:10:07.420 --> 00:10:10.650 And so correlation effects where the particles 00:10:10.650 --> 00:10:14.040 can move in some way that they know about each other. 00:10:14.040 --> 00:10:18.000 That's very difficult to build in via perturbation theory. 00:10:18.000 --> 00:10:20.400 But when we don't have correlation, 00:10:20.400 --> 00:10:23.130 length perturbation theory makes an error. 00:10:23.130 --> 00:10:25.770 And you know how big that error is. 00:10:25.770 --> 00:10:35.390 OK, and then we know that we can diagonalize the matrix 00:10:35.390 --> 00:10:39.590 by some kind of unitary transformation. 00:10:39.590 --> 00:10:48.930 And the columns of T dagger are the eigenvectors. 00:10:48.930 --> 00:10:55.150 And the rows of T are the inverse-- 00:10:55.150 --> 00:10:59.030 rows of T dagger or the columns of T are the inverse. 00:10:59.030 --> 00:11:02.200 And so if you want to make a wavepacket where 00:11:02.200 --> 00:11:09.280 you have some additional state to equal 0, 00:11:09.280 --> 00:11:12.430 you want to express that as a linear combination 00:11:12.430 --> 00:11:14.440 of eigenstates. 00:11:14.440 --> 00:11:17.937 And this sort of transformation provides that information 00:11:17.937 --> 00:11:19.020 if you know how to use it. 00:11:23.370 --> 00:11:25.910 And you can get approximations to all 00:11:25.910 --> 00:11:29.780 of these elements of T dagger from non-degenerate 00:11:29.780 --> 00:11:31.520 perturbation theory. 00:11:31.520 --> 00:11:34.100 And on the second exam that was one of the things 00:11:34.100 --> 00:11:35.830 you didn't do very well on. 00:11:35.830 --> 00:11:40.670 And so you can see that again. 00:11:40.670 --> 00:11:41.170 All right. 00:11:45.160 --> 00:11:47.710 OK, so now we're going to talk about intermolecular 00:11:47.710 --> 00:11:51.180 interactions. 00:11:51.180 --> 00:11:53.130 And up until now, we've been talking 00:11:53.130 --> 00:11:55.870 about isolated molecules. 00:11:55.870 --> 00:11:58.660 And you have a lot of insights about how isolated molecules 00:11:58.660 --> 00:12:01.600 work, or at least what kind of stuff 00:12:01.600 --> 00:12:04.510 you need to be able to make predictions about isolated 00:12:04.510 --> 00:12:05.260 molecules. 00:12:07.900 --> 00:12:12.100 And perturbation theory is a very important tool 00:12:12.100 --> 00:12:13.720 in building that insight. 00:12:13.720 --> 00:12:18.160 In almost every problem which is not an exactly soluble problem, 00:12:18.160 --> 00:12:21.220 perturbation theory gives you at least a hand-waving argument 00:12:21.220 --> 00:12:23.640 for what's going on. 00:12:23.640 --> 00:12:27.530 Everything we did with LCAO-MO is really 00:12:27.530 --> 00:12:32.180 based on perturbation theory, or at least the interpretations, 00:12:32.180 --> 00:12:35.810 especially when you go from homonuclear to heteronuclear. 00:12:35.810 --> 00:12:40.310 And when we talk about what various orbitals look like 00:12:40.310 --> 00:12:42.350 and why they look that way. 00:12:42.350 --> 00:12:47.380 OK, so perturbation theory can also be used to talk about-- 00:12:47.380 --> 00:12:52.660 we have two particles, A and B. And we 00:12:52.660 --> 00:12:54.560 have the origin of coordinates. 00:12:54.560 --> 00:12:56.245 And so this is the coordinate for A, 00:12:56.245 --> 00:13:00.940 and this is the coordinate for B. 00:13:00.940 --> 00:13:05.370 And this is the interatomic coordinate. 00:13:05.370 --> 00:13:09.390 And so we're interested in what happens 00:13:09.390 --> 00:13:14.370 as a function of this interatomic distance. 00:13:14.370 --> 00:13:16.890 And can we use perturbation theory 00:13:16.890 --> 00:13:20.252 to be able to say something about that? 00:13:20.252 --> 00:13:21.460 And the answer is you betcha. 00:13:24.060 --> 00:13:29.000 Because basically we understand how these two molecules work. 00:13:29.000 --> 00:13:34.550 And so we have to somehow build in some insight 00:13:34.550 --> 00:13:37.100 into what is the interaction between these, 00:13:37.100 --> 00:13:41.770 and then use perturbation theory to quantify it. 00:13:41.770 --> 00:13:44.400 OK, now this is a situation where 00:13:44.400 --> 00:13:52.860 we're going to say atoms A and B are our field shell, 00:13:52.860 --> 00:13:54.460 or whatever you want to call it. 00:13:54.460 --> 00:13:56.730 They don't need to make another bond. 00:13:56.730 --> 00:13:58.990 They're not going to bond to each other. 00:13:58.990 --> 00:14:01.480 There's no covalent bond between A and B. 00:14:01.480 --> 00:14:07.980 There is no charge on A and B. There is no donor/acceptor. 00:14:07.980 --> 00:14:10.560 All of the things that we can understand 00:14:10.560 --> 00:14:12.830 by looking at individual systems, 00:14:12.830 --> 00:14:14.760 they are all turned off. 00:14:14.760 --> 00:14:19.670 These molecules do have energy levels. 00:14:19.670 --> 00:14:21.940 And so if we could find some way to talk 00:14:21.940 --> 00:14:27.330 about the weak interaction between them, 00:14:27.330 --> 00:14:29.690 well then we have the basis for perturbation theory. 00:14:29.690 --> 00:14:32.360 Because we have weak and we have energy levels 00:14:32.360 --> 00:14:35.750 that we know for the two molecules. 00:14:35.750 --> 00:14:40.410 And so if we're going to do perturbation theory 00:14:40.410 --> 00:14:42.360 it'll be second order perturbation theory 00:14:42.360 --> 00:14:48.990 because we are going to be interested in how interaction 00:14:48.990 --> 00:14:52.550 of A with B causes excited states of B to mix 00:14:52.550 --> 00:14:57.150 into B, and vice versa. 00:14:57.150 --> 00:15:00.440 And so one of the things you know about perturbation theory, 00:15:00.440 --> 00:15:05.170 if you're talking about the ground state of a system 00:15:05.170 --> 00:15:13.034 and you do perturbation theory, what happens? 00:15:13.034 --> 00:15:14.450 What happens to the energy levels? 00:15:17.660 --> 00:15:19.450 I mean, you have-- 00:15:19.450 --> 00:15:21.260 for second order perturbation theory-- 00:15:24.921 --> 00:15:25.420 I'm sorry? 00:15:25.420 --> 00:15:27.490 AUDIENCE: Everything gets-- things 00:15:27.490 --> 00:15:31.290 can increase or decrease. 00:15:31.290 --> 00:15:35.290 The H term gets squared. 00:15:35.290 --> 00:15:35.938 PROFESSOR: Yes. 00:15:35.938 --> 00:15:37.562 AUDIENCE: Some [INAUDIBLE] information. 00:15:37.562 --> 00:15:41.217 But depending on whether a level that's lower-- 00:15:41.217 --> 00:15:43.550 PROFESSOR: But if we're talking about the ground state-- 00:15:43.550 --> 00:15:46.092 AUDIENCE: If it's that ground state it has to go down. 00:15:46.092 --> 00:15:46.800 PROFESSOR: Right. 00:15:46.800 --> 00:15:48.620 And so that's the key. 00:15:48.620 --> 00:15:52.850 We know that if we're going to do second order perturbation 00:15:52.850 --> 00:15:55.370 theory and we're going to be asking questions 00:15:55.370 --> 00:15:58.170 about molecules in their ground state, 00:15:58.170 --> 00:16:00.120 they're going to be stabilized by interaction 00:16:00.120 --> 00:16:01.710 with other molecules. 00:16:01.710 --> 00:16:04.830 It doesn't matter, as long as there's 00:16:04.830 --> 00:16:11.340 no bonding interactions, or very strong short range 00:16:11.340 --> 00:16:14.550 interactions, this sort of a problem 00:16:14.550 --> 00:16:19.310 is guaranteed to give stabilization of the ground 00:16:19.310 --> 00:16:21.095 state of the system. 00:16:23.700 --> 00:16:26.330 And that's really a surprise, because-- 00:16:26.330 --> 00:16:26.860 Yes? 00:16:26.860 --> 00:16:28.360 AUDIENCE: You're treating the system 00:16:28.360 --> 00:16:30.860 as though it has like a ladder of energies and everything 00:16:30.860 --> 00:16:32.960 is already at its lowest? 00:16:32.960 --> 00:16:33.920 PROFESSOR: Yes. 00:16:33.920 --> 00:16:38.390 Well, OK-- you know, I'm not allowed 00:16:38.390 --> 00:16:40.070 to talk about statistics. 00:16:40.070 --> 00:16:40.880 I can't, actually. 00:16:40.880 --> 00:16:41.870 Because we do it-- 00:16:41.870 --> 00:16:45.150 we now do some statistical mechanics in 560. 00:16:45.150 --> 00:16:46.550 So you know a little bit. 00:16:46.550 --> 00:16:53.300 And you know that the molecules like to be in the lowest energy 00:16:53.300 --> 00:16:53.940 levels. 00:16:53.940 --> 00:16:58.250 And so we have these ground state molecules in a gas. 00:16:58.250 --> 00:17:04.250 And we understand they're isolated properties, 00:17:04.250 --> 00:17:05.720 or we could understand. 00:17:05.720 --> 00:17:07.670 And what happens here? 00:17:07.670 --> 00:17:10.770 Somehow there's interactions between them. 00:17:10.770 --> 00:17:18.104 And now first order perturbation theory, 00:17:18.104 --> 00:17:19.354 you're allowed to have a sign. 00:17:23.119 --> 00:17:27.200 But you're allowed to have a sign for diagonal elements 00:17:27.200 --> 00:17:30.700 of the perturbation operator. 00:17:30.700 --> 00:17:32.600 OK, so we'll just continue with this. 00:17:38.600 --> 00:17:42.965 So we want to talk about Hamiltonian 00:17:42.965 --> 00:17:50.110 for the system, which is the Hamiltonian for particle 00:17:50.110 --> 00:17:52.900 A, the Hamiltonian for particle B, 00:17:52.900 --> 00:18:00.770 plus some interaction term between A and B. 00:18:00.770 --> 00:18:08.550 OK, now the typical interaction between two particles 00:18:08.550 --> 00:18:10.770 is a dipole-dipole interaction. 00:18:10.770 --> 00:18:13.560 If molecules have dipole moments, 00:18:13.560 --> 00:18:16.216 then we already know something about what they 00:18:16.216 --> 00:18:17.340 want to do with each other. 00:18:19.870 --> 00:18:22.960 And from electricity and magnetism-- 00:18:22.960 --> 00:18:25.100 we know this isn't quantum mechanics-- 00:18:25.100 --> 00:18:29.020 we know that we can write a general formula 00:18:29.020 --> 00:18:31.840 for the dipole-dipole interaction. 00:18:49.430 --> 00:18:50.890 So that's the general formula for 00:18:50.890 --> 00:18:53.904 the dipole-dipole interaction. 00:18:53.904 --> 00:18:55.320 And it's a little more complicated 00:18:55.320 --> 00:18:56.850 than you would like because there's two terms. 00:18:56.850 --> 00:18:58.530 And this one is kind of a puzzling term 00:18:58.530 --> 00:19:03.900 because you're projecting a dipole on the interatomic axis. 00:19:03.900 --> 00:19:08.190 And so we'd like to be able to simplify this. 00:19:08.190 --> 00:19:14.160 And we can simplify it by simply drawing 00:19:14.160 --> 00:19:17.500 some vectors corresponding to the dipole moment. 00:19:17.500 --> 00:19:23.970 And so we have the dipoles oriented like that. 00:19:23.970 --> 00:19:28.010 Or we have the dipoles oriented like this, 00:19:28.010 --> 00:19:35.430 or like this and like that. 00:19:35.430 --> 00:19:37.780 Now, this is a reduction. 00:19:37.780 --> 00:19:40.200 You know, these are the simplified terms 00:19:40.200 --> 00:19:45.450 where these guys are irrelevant or at least you 00:19:45.450 --> 00:19:48.540 don't have to worry about the projections. 00:19:48.540 --> 00:19:54.280 And we know that this is an attractive interaction, 00:19:54.280 --> 00:19:56.560 this is an attractive interaction. 00:19:56.560 --> 00:20:00.120 This is repulsive and this is repulsive. 00:20:00.120 --> 00:20:04.120 And we know also by plugging into the formulas, 00:20:04.120 --> 00:20:13.050 we know that the energy for this one is minus mu-A mu-B over two 00:20:13.050 --> 00:20:17.085 pi epsilon 0 R cubed. 00:20:19.700 --> 00:20:23.020 And this is minus the same stuff, 00:20:23.020 --> 00:20:26.870 except a four in the denominator. 00:20:26.870 --> 00:20:30.580 So that means it's less attractive than this one. 00:20:30.580 --> 00:20:31.990 Minus means attractive. 00:20:31.990 --> 00:20:35.110 And then we have two terms that resemble these 00:20:35.110 --> 00:20:37.440 except they're repulsive. 00:20:37.440 --> 00:20:38.670 So there's four terms. 00:20:42.900 --> 00:20:48.395 So there's two terms that are attractive 00:20:48.395 --> 00:20:50.960 and two terms that are repulsive. 00:20:50.960 --> 00:20:56.120 And they come in pairs, one positive and one negative. 00:20:56.120 --> 00:21:00.020 And so you might naively say well, you can't win. 00:21:00.020 --> 00:21:04.550 If the particles are randomly distributed and randomly 00:21:04.550 --> 00:21:09.570 oriented, there will be no stabilization, even 00:21:09.570 --> 00:21:12.200 with dipoles. 00:21:12.200 --> 00:21:16.100 But that's not true because the molecules 00:21:16.100 --> 00:21:23.840 try to find an orientation which is 00:21:23.840 --> 00:21:29.870 more stable than the sum of the energies without the dipoles. 00:21:29.870 --> 00:21:31.810 And so maybe we could draw a picture 00:21:31.810 --> 00:21:34.230 and try to figure out, well, how that would be. 00:21:39.730 --> 00:21:45.040 So does anybody want to give me guidance to a picture 00:21:45.040 --> 00:21:49.920 showing, let's say, four dipoles oriented 00:21:49.920 --> 00:21:53.800 in the corner of a square? 00:21:53.800 --> 00:21:55.630 And what would be the stable arrangement 00:21:55.630 --> 00:21:59.980 for four dipoles oriented on the corners of a square? 00:22:05.350 --> 00:22:11.099 OK, right? 00:22:16.480 --> 00:22:22.290 So we have two attractive interactions, 00:22:22.290 --> 00:22:24.630 two more attractive interactions, 00:22:24.630 --> 00:22:31.630 and then there are repulsive interactions this way. 00:22:31.630 --> 00:22:34.240 But we have nearest neighbors. 00:22:34.240 --> 00:22:37.660 And they're stronger than the repulsive interactions. 00:22:37.660 --> 00:22:40.880 And so for one layer, that can win. 00:22:40.880 --> 00:22:46.610 Now, we have another layer of four dipoles. 00:22:46.610 --> 00:22:48.770 And if that layer is-- 00:22:54.690 --> 00:22:58.854 so we have another layer, and there will be four of them. 00:22:58.854 --> 00:22:59.770 And what would happen? 00:22:59.770 --> 00:23:01.270 What would be the stable arrangement 00:23:01.270 --> 00:23:04.440 if I have a dipole above this one? 00:23:04.440 --> 00:23:06.630 Is it going to be this way? 00:23:06.630 --> 00:23:07.380 Right. 00:23:07.380 --> 00:23:11.370 And so the next layer, all the dipoles are reversed. 00:23:11.370 --> 00:23:15.300 And so you can conceive of an arrangement 00:23:15.300 --> 00:23:23.480 where the dipoles would adopt a more stable arrangement. 00:23:23.480 --> 00:23:26.990 And statistical mechanics, or whatever you want to call it, 00:23:26.990 --> 00:23:28.700 guides that. 00:23:28.700 --> 00:23:35.700 And so we can say, all right, the ensemble 00:23:35.700 --> 00:23:38.250 of dipoles interacting with each other 00:23:38.250 --> 00:23:44.070 can arrange to be energetically stabilizing. 00:23:44.070 --> 00:23:50.125 And so we can reduce the problem to just one picture, 00:23:50.125 --> 00:23:53.820 say this type or that dipole-dipole arrangement, 00:23:53.820 --> 00:23:57.090 and do second order perturbation theory. 00:23:57.090 --> 00:24:00.830 And that will be an overestimate of the stabilization. 00:24:00.830 --> 00:24:04.460 But it will be proportional to the stabilization. 00:24:04.460 --> 00:24:08.780 And so we can get what we need for a very 00:24:08.780 --> 00:24:13.480 complicated infinite number of particle problem. 00:24:13.480 --> 00:24:16.300 OK, another thing that I want to ask you, 00:24:16.300 --> 00:24:23.735 suppose you have two particles and they come together. 00:24:26.720 --> 00:24:31.140 And so they come together with some kinetic energy. 00:24:31.140 --> 00:24:33.980 And they can't get rid of that kinetic energy, 00:24:33.980 --> 00:24:36.080 so they come apart. 00:24:36.080 --> 00:24:38.130 Now, if you have three particles, 00:24:38.130 --> 00:24:44.160 if something drives them together by mutual attraction, 00:24:44.160 --> 00:24:47.070 then one particle can take away the excess energy. 00:24:47.070 --> 00:24:51.130 You can end up leaving the two particles together. 00:24:51.130 --> 00:24:53.690 That's how you make liquids. 00:24:53.690 --> 00:24:56.330 So the driving force for the attraction 00:24:56.330 --> 00:25:00.080 between atoms or molecules is somehow 00:25:00.080 --> 00:25:03.740 related to what we're going to drive here. 00:25:03.740 --> 00:25:07.580 And that is the only way you make liquids from gases. 00:25:13.100 --> 00:25:15.300 And it's really strange because we're 00:25:15.300 --> 00:25:19.140 going to be talking about energy levels and interactions 00:25:19.140 --> 00:25:23.520 between molecules, and somehow coming up 00:25:23.520 --> 00:25:31.830 with a picture that leads to universal attraction 00:25:31.830 --> 00:25:36.000 between molecules that are not charged, 00:25:36.000 --> 00:25:39.510 not reactive, not anything. 00:25:39.510 --> 00:25:43.110 And they will always attract each other. 00:25:43.110 --> 00:25:45.390 And that's the first step towards making liquids. 00:25:48.720 --> 00:25:52.070 OK, now the important thing in all of this 00:25:52.070 --> 00:25:54.830 is that the interactions are small 00:25:54.830 --> 00:25:59.070 compared to the energy levels of the individual molecules. 00:25:59.070 --> 00:26:01.370 And so that's why we're doing perturbation theory. 00:26:14.890 --> 00:26:21.920 OK, so in perturbation theory we have an H 0. 00:26:21.920 --> 00:26:23.090 And the H 0 is-- 00:26:27.300 --> 00:26:30.010 I've got to make sure I use consistent notation. 00:26:30.010 --> 00:26:33.110 Well, I already see I have inconsistent 00:26:33.110 --> 00:26:34.200 notation in my notes. 00:26:34.200 --> 00:26:41.810 But anyway, OK, so we have the Hamiltonian 00:26:41.810 --> 00:26:45.470 is the individual system Hamiltonians, 00:26:45.470 --> 00:26:47.000 the sum of the two. 00:26:47.000 --> 00:26:50.990 And the first order term is VAB. 00:26:55.820 --> 00:27:00.590 And so whenever we can separate a system, 00:27:00.590 --> 00:27:06.250 we know that the zero order wave functions 00:27:06.250 --> 00:27:09.980 are products of the eigenfunctions of HA and HB. 00:27:09.980 --> 00:27:12.500 And the energy levels of 0 [INAUDIBLE] energy levels 00:27:12.500 --> 00:27:16.340 are sum of the energy levels for HA and HB. 00:27:16.340 --> 00:27:27.660 So we know that we can write H 0 psi A alpha psi B alpha. 00:27:27.660 --> 00:27:29.880 So this guy-- 00:27:29.880 --> 00:27:35.520 HA operates on psi A and treats this as a constant. 00:27:35.520 --> 00:27:39.400 HB operates on psi B and treats this as a constant. 00:27:39.400 --> 00:27:43.170 And when you do that, what you discover is that we get-- 00:28:03.340 --> 00:28:07.160 OK, so these are our energy levels. 00:28:07.160 --> 00:28:09.240 And these are our basis functions. 00:28:09.240 --> 00:28:10.725 And we are off to the races. 00:28:16.470 --> 00:28:21.140 So this is E 0. 00:28:21.140 --> 00:28:25.860 And this is psi 0 for the system. 00:28:25.860 --> 00:28:36.490 OK, now the first thing we do in perturbation theory is we say, 00:28:36.490 --> 00:28:39.840 let's look for the first order correction of the energy. 00:28:39.840 --> 00:28:45.640 That's the diagonal matrix element of the H1 term, this. 00:28:52.530 --> 00:28:56.660 So suppose we do have dipole moments on both particle A 00:28:56.660 --> 00:28:59.640 and particle B 00:29:21.730 --> 00:29:23.260 So that's the first order energy. 00:29:30.120 --> 00:29:34.180 Well, this is just the dipole-dipole interaction. 00:29:34.180 --> 00:29:37.890 And so we can take the reduced formula. 00:29:37.890 --> 00:29:46.040 We're just going to regard VAB as mu A dot mu B over R cubed. 00:29:46.040 --> 00:30:01.110 And so the energy is minus one over two pi epsilon 0 R cubed. 00:30:01.110 --> 00:30:06.002 I left out the two pi epsilon 0 on this formula. 00:30:13.260 --> 00:30:15.490 And so skipping a couple of steps, 00:30:15.490 --> 00:30:26.650 we have mu A mu B, the expectation value of these. 00:30:26.650 --> 00:30:29.560 So if there's a dipole-dipole interaction, 00:30:29.560 --> 00:30:36.710 there will be a first order correction term. 00:30:36.710 --> 00:30:42.580 And keep in mind that these guys can have signs. 00:30:42.580 --> 00:30:45.950 And that sign does appear in the first order correction 00:30:45.950 --> 00:30:46.670 to the energy. 00:30:49.450 --> 00:30:52.900 But that's basically these pictures we drew here. 00:30:52.900 --> 00:30:54.530 We have a dipole pointing this way, 00:30:54.530 --> 00:30:56.230 we have a dipole pointing that way. 00:30:56.230 --> 00:30:58.120 That's repulsive. 00:30:58.120 --> 00:31:01.270 And that's all-- so there's no surprises here. 00:31:06.150 --> 00:31:09.680 OK, but the important thing is that the energy-- 00:31:09.680 --> 00:31:12.220 If we have two molecules with dipoles, 00:31:12.220 --> 00:31:17.080 we get an interaction energy that we can calculate. 00:31:17.080 --> 00:31:18.670 But the interesting thing is going 00:31:18.670 --> 00:31:22.210 to be when we have molecules where at least one 00:31:22.210 --> 00:31:24.070 of the molecules doesn't have a dipole. 00:31:27.140 --> 00:31:29.330 Then we're getting into territory 00:31:29.330 --> 00:31:31.657 that we didn't think we understood. 00:31:41.610 --> 00:31:50.000 But we're continuing with the problem 00:31:50.000 --> 00:31:51.100 where we do have dipoles. 00:31:56.950 --> 00:31:58.390 I just want to make sure that-- 00:32:04.210 --> 00:32:05.760 yeah, OK. 00:32:05.760 --> 00:32:14.280 So for the second order correction to the lowest energy 00:32:14.280 --> 00:32:21.470 level, we have the sum over m, which is two quantum numbers. 00:32:21.470 --> 00:32:24.740 And m is not equal to the 0-0 state. 00:32:24.740 --> 00:32:28.550 Alpha is not 0, beta is not 0 sum over everything 00:32:28.550 --> 00:32:31.140 except the ground state. 00:32:31.140 --> 00:32:47.660 And what we have is an integral of psi m 0 star VAB psi 0 0 00:32:47.660 --> 00:33:01.940 d tau squared over then E 0 0 minus E 0 m. 00:33:01.940 --> 00:33:05.870 Standard formula for second order variation theory. 00:33:05.870 --> 00:33:07.320 This is the ground state. 00:33:07.320 --> 00:33:08.850 This is some excited state. 00:33:08.850 --> 00:33:10.079 This is negative. 00:33:15.460 --> 00:33:18.002 This is squared. 00:33:18.002 --> 00:33:18.835 And this is squared. 00:33:21.480 --> 00:33:24.604 So this overall correction is negative. 00:33:29.450 --> 00:33:31.930 Now we can unpack all this and we 00:33:31.930 --> 00:33:38.450 can write the various terms in more instructive form. 00:33:38.450 --> 00:33:44.990 And what we'll have in the denominator will be E 0-- 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. 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 00:34:24.719 --> 00:34:30.400 alpha plus E B beta. 00:34:30.400 --> 00:34:33.909 So we have the ground state energy and the known excited 00:34:33.909 --> 00:34:34.780 state energies. 00:34:34.780 --> 00:34:37.179 And there is the difference between these. 00:34:37.179 --> 00:34:40.330 So that part is simple. 00:34:40.330 --> 00:34:48.870 And the matrix element also simplifies to the product 00:34:48.870 --> 00:34:50.075 of the dipole moment. 00:34:54.170 --> 00:34:58.210 OK, now suppose-- 00:35:06.200 --> 00:35:07.640 I just want to make sure that I'm 00:35:07.640 --> 00:35:10.520 doing this in the order it appears in the notes. 00:35:19.860 --> 00:35:28.400 Suppose mu A is not 0 and mu B is equal to 0. 00:35:31.580 --> 00:35:38.510 OK, well then all these formulas go out the window 00:35:38.510 --> 00:35:41.860 because nominally we think well, this B 00:35:41.860 --> 00:35:43.070 doesn't have a dipole moment. 00:35:45.690 --> 00:35:52.250 But suppose we have a particle B in an electric field. 00:35:52.250 --> 00:35:56.465 The electric field will polarize B. 00:35:56.465 --> 00:36:03.810 And if we have molecule A, which has a dipole moment, near B, 00:36:03.810 --> 00:36:07.320 it will also polarize B. And second order perturbation 00:36:07.320 --> 00:36:09.510 theory tells us how to handle that. 00:36:13.550 --> 00:36:22.850 And so we get a formula, a second order 00:36:22.850 --> 00:36:28.220 correction to the energy B. And we call this B comma induction. 00:36:32.020 --> 00:36:45.860 And that's going to be mu A squared over four pi 00:36:45.860 --> 00:36:52.340 squared epsilon 0 squared R to the six times 00:36:52.340 --> 00:36:59.220 sum beta not equal to 0 of the squared matrix element 00:36:59.220 --> 00:37:20.480 integral psi B star beta mu B psi B 0 d tau B squared 00:37:20.480 --> 00:37:27.290 over E B 0 minus E B beta. 00:37:27.290 --> 00:37:30.860 Now, I just told you that the dipole moment-- 00:37:30.860 --> 00:37:34.400 the permanent dipole moment for particle B is 0. 00:37:34.400 --> 00:37:39.260 But this is not the expectation value of the dipole moment. 00:37:39.260 --> 00:37:44.610 This is an off-diagonal matrix element of the dipole moment. 00:37:44.610 --> 00:37:46.860 This is the transition dipole. 00:37:46.860 --> 00:37:49.230 Transition between v equals 0-- 00:37:49.230 --> 00:37:53.100 well, between the ground state of B 00:37:53.100 --> 00:37:57.800 and some excited state, the beta, excited state of B. Well, 00:37:57.800 --> 00:38:02.730 we can write something like that because we 00:38:02.730 --> 00:38:08.220 know for all molecules there are transitions, which 00:38:08.220 --> 00:38:11.240 are electric dipole allowed. 00:38:11.240 --> 00:38:16.290 Now they could be between the ground state 00:38:16.290 --> 00:38:19.020 and an electronically excited state or the ground 00:38:19.020 --> 00:38:22.770 state and a vibrationally excited state. 00:38:22.770 --> 00:38:24.800 So we know something about these: 00:38:24.800 --> 00:38:26.930 we know the rules for when they're not 0. 00:38:26.930 --> 00:38:29.030 And this is a perfectly reasonable thing 00:38:29.030 --> 00:38:32.970 because it says this is related to the transition of-- 00:38:32.970 --> 00:38:36.640 the intensity of the observable transition. 00:38:36.640 --> 00:38:38.620 And this should really start to make 00:38:38.620 --> 00:38:43.450 you a little bit appreciative of perturbation theory. 00:38:43.450 --> 00:38:48.020 Because we're taking something which you can observe, 00:38:48.020 --> 00:38:51.220 a transition strength, and using it 00:38:51.220 --> 00:38:57.850 to describe an interaction between two molecules. 00:38:57.850 --> 00:39:03.400 So it's a relationship between two apparently unrelated 00:39:03.400 --> 00:39:05.260 phenomena. 00:39:05.260 --> 00:39:07.810 Physical chemistry is full of that. 00:39:07.810 --> 00:39:08.930 You build a model. 00:39:08.930 --> 00:39:11.950 And the model is telling you that certain things 00:39:11.950 --> 00:39:13.150 are related. 00:39:13.150 --> 00:39:14.380 And how are they related? 00:39:14.380 --> 00:39:17.560 Well, they're related by one over R to the sixth. 00:39:17.560 --> 00:39:20.830 They're related by the strength of the dipole on one, 00:39:20.830 --> 00:39:23.860 the permanent dipole on one, and some transition 00:39:23.860 --> 00:39:26.500 intensity-derived quantity on the other. 00:39:26.500 --> 00:39:29.290 Isn't that neat? 00:39:29.290 --> 00:39:30.520 And since the dipole-- 00:39:33.520 --> 00:39:37.070 we say well, this is a dipole-dipole interaction. 00:39:37.070 --> 00:39:43.990 And the mu A, the dipole on A, is inducing a dipole on B. 00:39:43.990 --> 00:39:45.670 It's not inducing anything. 00:39:45.670 --> 00:39:48.460 It's using transition dipoles. 00:39:48.460 --> 00:39:52.960 But it's creating something that looks like a permanent dipole. 00:39:52.960 --> 00:39:55.270 And you have dipole-dipole interaction. 00:39:55.270 --> 00:39:59.080 But the only difference is that it's one over R to the sixth 00:39:59.080 --> 00:40:00.445 because it's an induced dipole. 00:40:05.200 --> 00:40:09.500 OK, so we're using transition-- 00:40:09.500 --> 00:40:14.050 well, electric dipole transition moments. 00:40:14.050 --> 00:40:18.786 (There used to be two erasers!) 00:40:33.930 --> 00:40:38.290 So these electric dipole transition moments 00:40:38.290 --> 00:40:42.194 play the role of producing a dipole that can 00:40:42.194 --> 00:40:43.485 interact with the other dipole. 00:40:49.530 --> 00:40:53.390 And so-- well, do we need this? 00:40:53.390 --> 00:40:55.330 Well, we don't really. 00:40:55.330 --> 00:40:58.030 If you had dipoles, then the interaction 00:40:58.030 --> 00:41:00.550 would be one over R cubed. 00:41:00.550 --> 00:41:04.000 But if one of the dipoles is 0, then the interaction 00:41:04.000 --> 00:41:07.360 is one over R to the sixth. 00:41:07.360 --> 00:41:09.170 And it's one over R to the sixth times 00:41:09.170 --> 00:41:13.670 the permanent dipole of one and this thing from perturbation 00:41:13.670 --> 00:41:16.310 theory, which is the induced dipole of the other. 00:41:25.840 --> 00:41:31.270 Suppose mu A equals 0 and mu B equals 0. 00:41:34.740 --> 00:41:36.610 Will the particles interact with each other? 00:41:36.610 --> 00:41:38.170 Will they attract each other? 00:41:38.170 --> 00:41:40.360 And the answer is yes. 00:41:40.360 --> 00:41:42.960 And it's going to be an induced dipole-induced dipole 00:41:42.960 --> 00:41:44.810 interaction. 00:41:44.810 --> 00:41:51.840 Which, now since most people who write for general science have 00:41:51.840 --> 00:41:54.390 no clue about what quantum mechanics is, 00:41:54.390 --> 00:41:57.870 they will talk about all sorts of fanciful things-- 00:41:57.870 --> 00:42:01.230 you get an institute-- 00:42:01.230 --> 00:42:05.910 an instantaneous fluctuation on one particle which 00:42:05.910 --> 00:42:08.350 creates an instantaneous fluctuation 00:42:08.350 --> 00:42:11.220 on the other particle, and bam, you've got stabilization. 00:42:11.220 --> 00:42:11.970 Well, yeah, maybe. 00:42:11.970 --> 00:42:16.070 But you know, this is a simple perturbation theory. 00:42:16.070 --> 00:42:20.770 And so I don't really need to actually write out everything. 00:42:20.770 --> 00:42:27.730 Because one, you do that and you include the second order 00:42:27.730 --> 00:42:33.010 perturbation expression for particle A and particle B, 00:42:33.010 --> 00:42:35.740 you get a term that looks just like this. 00:42:35.740 --> 00:42:38.900 It has one over R to the sixth. 00:42:38.900 --> 00:42:43.250 But it now involves transition moments, particle A 00:42:43.250 --> 00:42:44.990 and particle B-- 00:42:44.990 --> 00:42:45.710 nothing new. 00:42:45.710 --> 00:42:48.650 You don't have to induce anything more. 00:42:48.650 --> 00:42:52.702 You don't get another factor of one over R cubed. 00:42:52.702 --> 00:42:54.660 You've already got the one over R to the sixth. 00:42:57.576 --> 00:43:07.090 So I'm going to skip a lot of steps 00:43:07.090 --> 00:43:09.250 because the notes are really clear in there. 00:43:09.250 --> 00:43:11.630 And we don't have a whole lot of time. 00:43:11.630 --> 00:43:15.430 And so I'm just going to write down the result. 00:43:15.430 --> 00:43:27.900 And now the name of this term is called dispersion, or London 00:43:27.900 --> 00:43:32.220 because a fellow by the name of London invented this. 00:43:32.220 --> 00:43:38.400 And the energy level formula for this correction term 00:43:38.400 --> 00:43:43.410 is one over four pi squared epsilon 0 squared 00:43:43.410 --> 00:43:45.990 R to the sixth. 00:43:45.990 --> 00:43:54.990 And then this sum of alpha equals one beta equals one 00:43:54.990 --> 00:43:56.350 to infinity. 00:43:56.350 --> 00:44:02.080 In other words, not alpha equals 0, not beta equals 0. 00:44:02.080 --> 00:44:02.610 And then-- 00:44:29.720 --> 00:44:30.430 Sorry about that. 00:44:33.870 --> 00:44:38.520 So we have a numerator, which is the squared transition 00:44:38.520 --> 00:44:44.320 moment between the 0 level of particle A and the alpha level 00:44:44.320 --> 00:44:48.040 and the 0 level of B and the beta level. 00:44:48.040 --> 00:44:50.090 We have the energy of the ground state 00:44:50.090 --> 00:44:53.170 and we have the energy of these excited states. 00:44:53.170 --> 00:44:56.340 All of this stuff, you know or at least you 00:44:56.340 --> 00:44:59.520 can parametrize it in terms of this strength 00:44:59.520 --> 00:45:01.040 of our transition. 00:45:01.040 --> 00:45:04.590 And there is an explicit formula that relates the dipole moment 00:45:04.590 --> 00:45:09.180 to the intensity depending on what kind of experiment you do. 00:45:09.180 --> 00:45:11.610 So I'm not going to take you through that. 00:45:11.610 --> 00:45:15.870 But you know you can measure the intensity, the molar 00:45:15.870 --> 00:45:22.260 absorptivity, or whatever you want that is related to this. 00:45:22.260 --> 00:45:24.510 So these are experimentally observable. 00:45:24.510 --> 00:45:26.490 These are experimentally observable. 00:45:26.490 --> 00:45:32.970 And usually this infinite sum is dominated 00:45:32.970 --> 00:45:37.630 by the lowest allowed transition from the ground state. 00:45:37.630 --> 00:45:40.260 And that can be an electronic transition 00:45:40.260 --> 00:45:42.660 or a vibrational transition. 00:45:42.660 --> 00:45:45.270 So for our homonuclear molecule, there 00:45:45.270 --> 00:45:48.000 isn't going to be an allowed vibrational transition. 00:45:48.000 --> 00:45:52.620 So nitrogen and oxygen cannot talk to each other through 00:45:52.620 --> 00:45:53.820 transition dipole-- 00:45:53.820 --> 00:45:57.870 vibrational transition dipoles because the dipole moment 00:45:57.870 --> 00:46:00.390 is not dependent on [INAUDIBLE] but they 00:46:00.390 --> 00:46:02.620 have electronic transitions. 00:46:02.620 --> 00:46:04.980 And so there is this electronic-- as 00:46:04.980 --> 00:46:08.200 opposed to a vibrational polarizability of the two. 00:46:08.200 --> 00:46:15.540 And that's what gives rise to attraction between N2 and O2, 00:46:15.540 --> 00:46:16.290 or N2 and N2. 00:46:20.320 --> 00:46:23.500 So everything is knowable and predictable. 00:46:23.500 --> 00:46:26.910 And even though the perturbation's sum is infinite, 00:46:26.910 --> 00:46:30.500 and even though we've made a gross approximation and say, 00:46:30.500 --> 00:46:34.270 well, we're just interested in this sort of an interaction 00:46:34.270 --> 00:46:39.730 or this sort of an interaction, and we 00:46:39.730 --> 00:46:42.850 know that the true interaction will 00:46:42.850 --> 00:46:47.290 be less because there will be an average over all 00:46:47.290 --> 00:46:48.670 of the random orientations. 00:46:48.670 --> 00:46:51.610 But there's going to be a lowest energy picture. 00:46:51.610 --> 00:46:54.310 And you can get that from statistical mechanics. 00:46:54.310 --> 00:46:56.740 And so you can do the configurational average 00:46:56.740 --> 00:47:00.220 over all possible orientations of all of the interacting 00:47:00.220 --> 00:47:01.120 particles. 00:47:01.120 --> 00:47:04.100 And you can calculate this. 00:47:04.100 --> 00:47:07.490 So that's beyond the level of this course. 00:47:07.490 --> 00:47:10.480 But the engine of the calculation is this formula. 00:47:26.900 --> 00:47:31.040 So molecules are polarizable. 00:47:31.040 --> 00:47:35.020 They react to an external electric field. 00:47:35.020 --> 00:47:37.660 And there are several ways you can describe 00:47:37.660 --> 00:47:39.600 the defined polarizability. 00:47:39.600 --> 00:47:42.600 One is to say well the polarizability-- 00:47:42.600 --> 00:47:44.430 let's call it alpha-- 00:47:44.430 --> 00:47:47.700 is equal to the second derivative 00:47:47.700 --> 00:47:53.610 of the energy with respect to the electric field. 00:47:57.350 --> 00:47:59.860 But the polarizability, you know, 00:47:59.860 --> 00:48:04.180 you also get from perturbation theory. 00:48:04.180 --> 00:48:08.310 And so each of these terms or the energy 00:48:08.310 --> 00:48:11.110 is associated with the interaction 00:48:11.110 --> 00:48:13.300 of a dipole with an electric field 00:48:13.300 --> 00:48:15.770 is the electric field times the dipole. 00:48:15.770 --> 00:48:17.470 And so we have electric field squared. 00:48:17.470 --> 00:48:19.700 We take the second derivative of that. 00:48:19.700 --> 00:48:21.070 And we've got back to this. 00:48:25.420 --> 00:48:28.037 So now here we come with intuition. 00:48:31.620 --> 00:48:36.830 If I asked you to tell me the van der Waals 00:48:36.830 --> 00:48:40.910 or the dispersion interaction between helium and helium, 00:48:40.910 --> 00:48:43.872 and compare that to xenon to xenon, 00:48:43.872 --> 00:48:45.080 would you be able to tell me? 00:48:48.550 --> 00:48:49.970 So which one is stronger? 00:48:49.970 --> 00:48:57.064 Which one-- which pair of inert gases is stickier? 00:48:57.064 --> 00:48:57.880 AUDIENCE: Xenon. 00:48:57.880 --> 00:48:58.880 PROFESSOR: Xenon, right. 00:48:58.880 --> 00:48:59.650 Why? 00:48:59.650 --> 00:49:04.690 Because the energy denominator for the first excited state 00:49:04.690 --> 00:49:06.880 is much lower for xenon than for helium. 00:49:09.410 --> 00:49:17.444 OK, what about large molecule versus small molecule? 00:49:25.360 --> 00:49:29.890 Do I need to provide a hint? 00:49:29.890 --> 00:49:31.060 3n minus six. 00:49:33.970 --> 00:49:37.170 The more vibrational modes, the more transitions 00:49:37.170 --> 00:49:40.230 are available for polarization. 00:49:40.230 --> 00:49:43.897 And so again, now we're talking about molecules 00:49:43.897 --> 00:49:44.730 which can be floppy. 00:49:44.730 --> 00:49:48.620 And the floppier the molecule-- 00:49:48.620 --> 00:49:52.190 that means they have low frequency vibrations-- 00:49:52.190 --> 00:49:55.930 you can polarize the daylights out of them. 00:49:55.930 --> 00:50:00.510 And so large molecules are really sticky, stickier 00:50:00.510 --> 00:50:02.580 than inert gases. 00:50:02.580 --> 00:50:05.640 Because you have this dispersion interaction 00:50:05.640 --> 00:50:07.120 that brings them together. 00:50:07.120 --> 00:50:10.050 And once they're together, then they can do magic. 00:50:10.050 --> 00:50:13.230 But the thing that drives the condensation, or the formation 00:50:13.230 --> 00:50:17.300 of droplets, or whatever, is this dispersion interaction. 00:50:17.300 --> 00:50:18.190 And it's universal. 00:50:18.190 --> 00:50:23.790 Every pair of molecules or atoms is going to attract each other. 00:50:23.790 --> 00:50:26.940 And the strength is something that you 00:50:26.940 --> 00:50:29.430 can rate in terms of what you know 00:50:29.430 --> 00:50:32.670 about atomic and molecular properties. 00:50:32.670 --> 00:50:34.560 And isn't that neat? 00:50:34.560 --> 00:50:38.190 Because you're saying the viscosity, 00:50:38.190 --> 00:50:44.120 or whatever it is, the property of a gas, all of those things 00:50:44.120 --> 00:50:49.250 are related to the properties of the individual atoms 00:50:49.250 --> 00:50:53.161 or molecules embedded in perturbation theory. 00:50:53.161 --> 00:50:54.910 This is why I think perturbation theory is 00:50:54.910 --> 00:51:00.550 so wonderful, because it enables you to relate different things. 00:51:00.550 --> 00:51:02.530 And even if you're not going to evaluate 00:51:02.530 --> 00:51:04.780 the magnitudes of the terms, you can 00:51:04.780 --> 00:51:07.530 make comparisons which are always going to be right. 00:51:10.680 --> 00:51:11.950 And that's kind of wonderful. 00:51:11.950 --> 00:51:15.970 I mean, you don't have to use the equations. 00:51:15.970 --> 00:51:19.660 Or you just know that these particles will attract 00:51:19.660 --> 00:51:22.960 each other and the driving force is either 00:51:22.960 --> 00:51:25.330 a permanent or an induced dipole. 00:51:25.330 --> 00:51:29.380 And the induced dipole will be strong if you 00:51:29.380 --> 00:51:34.060 have a strong transition to a low-lying state, 00:51:34.060 --> 00:51:36.700 or a lot of strong transitions to a lot of low-lying states 00:51:36.700 --> 00:51:38.020 as you have in big molecules. 00:51:41.170 --> 00:51:45.650 OK, so I'm not positive what I'm going to talk about on Friday. 00:51:45.650 --> 00:51:48.030 It's probably going to be photochemistry. 00:51:48.030 --> 00:51:53.510 But I may panic and do something else. 00:51:53.510 --> 00:51:56.500 OK, I'll see you Friday.