WEBVTT 00:00:00.090 --> 00:00:02.430 The following content is provided under a Creative 00:00:02.430 --> 00:00:03.820 Commons license. 00:00:03.820 --> 00:00:06.030 Your support will help MIT OpenCourseWare 00:00:06.030 --> 00:00:10.120 continue to offer high quality educational resources for free. 00:00:10.120 --> 00:00:12.660 To make a donation or to view additional materials 00:00:12.660 --> 00:00:16.620 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:16.620 --> 00:00:17.850 at ocw.mit.edu. 00:00:21.910 --> 00:00:26.240 ROBERT FIELD: This is the first of two lectures on spectroscopy 00:00:26.240 --> 00:00:27.590 and dynamics. 00:00:27.590 --> 00:00:31.130 Now, I'm a spectroscopist, and so this is 00:00:31.130 --> 00:00:34.160 the core of what I really love. 00:00:34.160 --> 00:00:36.910 And there are a lot of questions about, 00:00:36.910 --> 00:00:38.990 well, what are we trying to do? 00:00:38.990 --> 00:00:43.430 And you heard two lectures from Professor Van Voorhis, 00:00:43.430 --> 00:00:48.270 and he talked about ab initio calculations-- 00:00:48.270 --> 00:00:50.480 electronic structure calculations-- 00:00:50.480 --> 00:00:55.210 where you can get really close to the exact answer. 00:00:55.210 --> 00:00:57.510 And it's really a powerful tool. 00:01:01.710 --> 00:01:04.140 And it gets you the truth. 00:01:04.140 --> 00:01:06.840 But it gets you so much truth that you 00:01:06.840 --> 00:01:09.280 don't know what to do with it. 00:01:09.280 --> 00:01:12.150 And the same thing is true for spectroscopy. 00:01:12.150 --> 00:01:13.150 You get a spectrum. 00:01:13.150 --> 00:01:16.450 It contains a huge amount of information. 00:01:16.450 --> 00:01:18.640 And you can take lots and lots of spectra. 00:01:18.640 --> 00:01:20.050 But what are we trying to do? 00:01:20.050 --> 00:01:24.880 When I was a graduate student, we 00:01:24.880 --> 00:01:29.290 were doing a unique super high resolution spectroscopy. 00:01:29.290 --> 00:01:33.970 And so we thought what we were generating 00:01:33.970 --> 00:01:39.630 was excellent tests for quantitative theory. 00:01:39.630 --> 00:01:44.880 And if, in those days, I went to a lecture by a theorist, 00:01:44.880 --> 00:01:48.420 they were saying, we're generating stuff 00:01:48.420 --> 00:01:52.690 to check against experiment. 00:01:52.690 --> 00:01:57.750 And it's a circle, and that's not what we're trying to do. 00:01:57.750 --> 00:02:02.370 The theory and the experiment are just the beginning. 00:02:02.370 --> 00:02:05.400 And you can start from either extreme. 00:02:05.400 --> 00:02:08.070 What we want is, how does it all work? 00:02:08.070 --> 00:02:10.710 What is going on? 00:02:10.710 --> 00:02:19.080 Can we build a picture, which is intuitive and checkable and 00:02:19.080 --> 00:02:22.680 predictive, so that we can say, oh, yeah. 00:02:22.680 --> 00:02:28.090 If we want to know something, we can get to it this way. 00:02:28.090 --> 00:02:29.890 And the purpose of this lecture is 00:02:29.890 --> 00:02:34.860 to not provide a textbook view of spectra, 00:02:34.860 --> 00:02:39.710 but to give you a sense of how you can get to stuff 00:02:39.710 --> 00:02:41.200 that challenge your intuition. 00:02:46.620 --> 00:02:53.440 What we want to do as scientists is to be surprised. 00:02:53.440 --> 00:02:58.590 We want to do a good experiment or do a good calculation. 00:02:58.590 --> 00:03:03.190 And we want to find that the result is not what we expected. 00:03:03.190 --> 00:03:07.210 And we can figure out why it's not what we expected. 00:03:07.210 --> 00:03:11.820 And that's never conveyed in any textbooks. 00:03:11.820 --> 00:03:17.520 Now, this lecture is based on my little book of lecture notes. 00:03:17.520 --> 00:03:20.760 And I have a number of copies of it. 00:03:20.760 --> 00:03:24.240 And if people have a strong wish to have a copy, 00:03:24.240 --> 00:03:24.960 you can have one. 00:03:24.960 --> 00:03:28.020 I can give it to you. 00:03:28.020 --> 00:03:32.430 And a lot of this lecture is based on the first chapter 00:03:32.430 --> 00:03:34.020 of this book. 00:03:34.020 --> 00:03:38.870 But many of the topics are developed throughout the book. 00:03:38.870 --> 00:03:39.370 OK. 00:03:39.370 --> 00:03:46.650 So this is a two lecture sequence, 00:03:46.650 --> 00:03:52.062 and the first half will cover this stuff on this board. 00:03:52.062 --> 00:03:53.520 And it's in the notes, so you don't 00:03:53.520 --> 00:03:54.970 have to copy all this stuff. 00:03:54.970 --> 00:03:57.700 I just want you to see where we're going. 00:03:57.700 --> 00:03:59.670 So I've already talked a little bit 00:03:59.670 --> 00:04:02.040 about experiment versus theory. 00:04:02.040 --> 00:04:04.090 They complement each other. 00:04:04.090 --> 00:04:07.410 We can use theory to devise an experiment, which 00:04:07.410 --> 00:04:10.200 is path breaking. 00:04:10.200 --> 00:04:13.410 Or we can use an experiment to challenge the theorist 00:04:13.410 --> 00:04:16.880 to calculate a new kind of thing. 00:04:16.880 --> 00:04:20.220 And I hope that some of these things, those ideas, 00:04:20.220 --> 00:04:23.360 present themselves in this lecture. 00:04:23.360 --> 00:04:28.070 And it's really important that if there's something 00:04:28.070 --> 00:04:32.300 that you don't understand or don't 00:04:32.300 --> 00:04:35.330 capture the importance of, you should ask me a question. 00:04:35.330 --> 00:04:39.560 I really want to talk about what it's all for. 00:04:39.560 --> 00:04:40.070 OK. 00:04:40.070 --> 00:04:42.470 So I'm going to talk about spectra, 00:04:42.470 --> 00:04:45.830 and it will be what kinds-- 00:04:45.830 --> 00:04:52.310 rotation, vibration, electronic, and other ramifications. 00:04:52.310 --> 00:04:55.730 And going from atom to diatomic to polyatomic 00:04:55.730 --> 00:04:56.690 to condensed phase. 00:04:59.370 --> 00:05:05.130 Each step along this path leads to new complexities 00:05:05.130 --> 00:05:08.410 and new insights. 00:05:08.410 --> 00:05:11.530 Then I'll talk about, OK, we got a spectrum. 00:05:11.530 --> 00:05:13.550 What do we expect to be in the spectrum? 00:05:13.550 --> 00:05:16.060 Well, one of the things that's important 00:05:16.060 --> 00:05:21.160 is the transition selection rules or transition rules. 00:05:21.160 --> 00:05:25.600 Selection rules correspond to an operator-- eigenvalues 00:05:25.600 --> 00:05:29.860 of an operator-- that commutes with the exact Hamiltonian. 00:05:29.860 --> 00:05:32.560 And those correspond to symmetries. 00:05:32.560 --> 00:05:35.740 And there are propensity rules like, OK, 00:05:35.740 --> 00:05:38.830 which transitions are going to be strong 00:05:38.830 --> 00:05:41.360 and which are going to be weak? 00:05:41.360 --> 00:05:44.960 And a beautiful example of propensity rules 00:05:44.960 --> 00:05:47.830 are based on the Franck-Condon principle. 00:05:47.830 --> 00:05:51.370 And the Franck-Condon principle is one of the first keys 00:05:51.370 --> 00:05:54.760 you use to unlock what's in a spectrum, 00:05:54.760 --> 00:05:56.500 or what a molecule is doing. 00:05:56.500 --> 00:06:02.350 Because it's the first level of complexity 00:06:02.350 --> 00:06:05.080 that is presented to you in the spectrum. 00:06:05.080 --> 00:06:08.690 What are the vibrational bands, and how do we assign them, 00:06:08.690 --> 00:06:09.940 and what are they telling us? 00:06:12.560 --> 00:06:16.700 There is a very different kind of information in an absorption 00:06:16.700 --> 00:06:21.770 spectrum, because it's always from the lowest 00:06:21.770 --> 00:06:25.150 electronic state, lowest vibrational level. 00:06:25.150 --> 00:06:27.160 And so there's a simplicity, because 00:06:27.160 --> 00:06:29.540 of a kind of state selection. 00:06:29.540 --> 00:06:32.100 And in emission, it's a very different ballgame. 00:06:32.100 --> 00:06:35.390 Because in the gas phase, the emission 00:06:35.390 --> 00:06:38.060 is from many different levels. 00:06:38.060 --> 00:06:40.920 In the condensed phase, it's not. 00:06:40.920 --> 00:06:43.260 Why? 00:06:43.260 --> 00:06:43.900 OK. 00:06:43.900 --> 00:06:48.280 And then we get to dynamics. 00:06:48.280 --> 00:06:51.610 And the main thing I want to do in these two lectures 00:06:51.610 --> 00:06:54.250 is to whet your appetite for dynamics. 00:06:54.250 --> 00:06:56.560 And there are many kinds of dynamics ranging 00:06:56.560 --> 00:07:02.310 from a simple two level quantum beat to intramolecular 00:07:02.310 --> 00:07:04.720 vibrational redistribution. 00:07:04.720 --> 00:07:08.940 Which you can understand by perturbation theory 00:07:08.940 --> 00:07:16.590 to the strange behavior of electronically excited states 00:07:16.590 --> 00:07:18.960 losing their ability to fluoresce 00:07:18.960 --> 00:07:24.030 not because the molecule breaks, but because the bright state-- 00:07:24.030 --> 00:07:25.930 that's an important concept-- 00:07:25.930 --> 00:07:27.180 the bright state is something. 00:07:27.180 --> 00:07:29.100 It's not an eigenstate. 00:07:29.100 --> 00:07:32.370 It's a special state that we understand well. 00:07:32.370 --> 00:07:35.730 It's one of the things that we build perturbation theory 00:07:35.730 --> 00:07:36.840 around. 00:07:36.840 --> 00:07:41.430 The bright state mixes into an enormous number of dark states. 00:07:41.430 --> 00:07:45.360 And the molecule forgets that it knows 00:07:45.360 --> 00:07:48.630 how to fluoresce because the different components, 00:07:48.630 --> 00:07:51.740 different eigenstates dephase. 00:07:51.740 --> 00:07:54.000 And that is a beautiful theory. 00:07:54.000 --> 00:07:55.620 And when I was a graduate student, 00:07:55.620 --> 00:07:58.050 this theory of radiationless transitions-- 00:07:58.050 --> 00:08:01.620 Bixon-Jortner theory-- was just created 00:08:01.620 --> 00:08:04.260 and many people didn't believe it. 00:08:04.260 --> 00:08:07.680 They thought molecules, big molecules, 00:08:07.680 --> 00:08:10.440 they're really easy to be quenched by collision 00:08:10.440 --> 00:08:12.900 and the loss of the ability to fluoresce 00:08:12.900 --> 00:08:15.270 or the absence of fluorescence was somehow 00:08:15.270 --> 00:08:20.340 collision related as opposed to a physical process. 00:08:20.340 --> 00:08:22.140 So there's lots of good stuff. 00:08:22.140 --> 00:08:27.710 And some of the good stuff is Ahmed Zewail's Nobel Prize 00:08:27.710 --> 00:08:30.350 where he claims-- 00:08:30.350 --> 00:08:32.659 and that's why he got the Nobel Prize, because people 00:08:32.659 --> 00:08:34.610 believed that claim. 00:08:34.610 --> 00:08:36.470 Now, I'm not saying it's wrong. 00:08:36.470 --> 00:08:40.760 But part of getting famous is to have 00:08:40.760 --> 00:08:44.070 a package, which you can sell. 00:08:44.070 --> 00:08:46.730 And he sold the daylights out of it. 00:08:46.730 --> 00:08:55.290 And he calls it clocking real dynamics in real time. 00:08:55.290 --> 00:08:58.380 And it's basically wave packets. 00:08:58.380 --> 00:09:01.770 But they're wave packets doing neat stuff. 00:09:01.770 --> 00:09:06.730 And for example, one way a molecule 00:09:06.730 --> 00:09:09.760 can lose the ability to fluoresce 00:09:09.760 --> 00:09:13.000 is because the molecule breaks. 00:09:13.000 --> 00:09:15.790 And what is the mechanism by which a molecule breaks? 00:09:15.790 --> 00:09:20.080 Does the bond just simply break or is there some motion 00:09:20.080 --> 00:09:22.200 that precedes that? 00:09:22.200 --> 00:09:26.310 And what Zewail did was to show what 00:09:26.310 --> 00:09:28.830 are the motions that lead the molecule 00:09:28.830 --> 00:09:34.740 into the region of state space where the bond breaks. 00:09:34.740 --> 00:09:37.200 And, of course, if you want to manipulate molecules, 00:09:37.200 --> 00:09:41.060 you either want to get to those regions or avoid them. 00:09:41.060 --> 00:09:45.268 And so there's all sorts of insight there. 00:09:45.268 --> 00:09:47.610 OK. 00:09:47.610 --> 00:09:52.390 Now this is what I believe. 00:09:52.390 --> 00:09:56.360 That if you understand small molecules, 00:09:56.360 --> 00:09:58.250 you will see examples of everything 00:09:58.250 --> 00:10:02.390 you need to know to deal with almost any dynamical process 00:10:02.390 --> 00:10:03.980 in chemistry. 00:10:03.980 --> 00:10:06.390 Now, this is certainly an exaggeration, 00:10:06.390 --> 00:10:10.490 but this has been my motto for years. 00:10:10.490 --> 00:10:15.680 And so I really stress the small molecules. 00:10:15.680 --> 00:10:20.210 And it's not that small molecules are really hard. 00:10:20.210 --> 00:10:22.400 They're really beautiful, and they 00:10:22.400 --> 00:10:26.480 do enough so that you can anticipate what you need 00:10:26.480 --> 00:10:29.840 to deal with bigger molecules. 00:10:29.840 --> 00:10:32.630 So let's begin. 00:10:38.380 --> 00:10:40.780 OK. 00:10:40.780 --> 00:10:41.980 So what is a molecule? 00:10:41.980 --> 00:10:45.170 As chemists, we would never think 00:10:45.170 --> 00:10:48.830 of a molecule as a bag of nuclei and electrons. 00:10:53.550 --> 00:10:57.680 We wouldn't think of it as a bag of atoms either. 00:10:57.680 --> 00:11:00.250 We believe in chemical bonds. 00:11:00.250 --> 00:11:02.160 This is an important thing. 00:11:02.160 --> 00:11:04.490 It's not a conserved quantity. 00:11:04.490 --> 00:11:06.260 Bonds can break. 00:11:06.260 --> 00:11:10.370 But we believe that bonds tell an important story. 00:11:16.520 --> 00:11:23.480 And so almost all of our pictures for complicated 00:11:23.480 --> 00:11:27.200 phenomena are based on the-- 00:11:27.200 --> 00:11:29.420 I hesitate to use the word sanctity-- 00:11:29.420 --> 00:11:34.990 but the importance of bonds. 00:11:34.990 --> 00:11:36.790 OK. 00:11:36.790 --> 00:11:39.160 We start-- I'm going to erase this, 00:11:39.160 --> 00:11:42.340 because I've made my point, and it's 00:11:42.340 --> 00:11:48.520 embarrassing to keep emphasizing my secret motto. 00:11:48.520 --> 00:11:49.460 But it is true. 00:11:49.460 --> 00:11:49.960 OK. 00:11:49.960 --> 00:11:51.970 We have the Born-Oppenheimer approximation. 00:11:56.550 --> 00:12:02.110 And this is very important, because we can't 00:12:02.110 --> 00:12:04.680 solve a three body problem. 00:12:04.680 --> 00:12:06.840 We can solve a two body problem. 00:12:06.840 --> 00:12:10.560 But we have molecules, which are consisting 00:12:10.560 --> 00:12:13.350 of nuclei and electrons. 00:12:13.350 --> 00:12:16.380 And this Born-Oppenheimer approximation 00:12:16.380 --> 00:12:20.310 enables us to separate the nuclear part of the problem 00:12:20.310 --> 00:12:22.560 from the electronic part of the problem. 00:12:22.560 --> 00:12:26.730 Because these two things move at very different velocities. 00:12:26.730 --> 00:12:29.710 And so it's a profound simplification. 00:12:29.710 --> 00:12:35.250 We get potential energy curves or potential energy surfaces. 00:12:35.250 --> 00:12:39.180 And that is the repository of essentially everything 00:12:39.180 --> 00:12:40.410 we want to know. 00:12:40.410 --> 00:12:43.140 If we know the potential surface, 00:12:43.140 --> 00:12:46.540 we can begin to do almost anything. 00:12:46.540 --> 00:12:49.450 And certainly for a big molecule, 00:12:49.450 --> 00:12:53.590 it's not just a simple curve like this. 00:12:53.590 --> 00:12:59.580 If you have N atoms, there's 3N minus 6 vibrational modes. 00:12:59.580 --> 00:13:03.210 And well, that sounds terrible. 00:13:03.210 --> 00:13:07.140 But even for this, you have essentially an infinite number 00:13:07.140 --> 00:13:10.580 of vibrational levels and an infinite number 00:13:10.580 --> 00:13:12.390 of rotational levels. 00:13:12.390 --> 00:13:15.830 And so if you have a polyatomic molecule, 00:13:15.830 --> 00:13:20.480 you have 3N minus 6 infinities of infinities. 00:13:20.480 --> 00:13:24.470 So you're not wanting to get everything. 00:13:24.470 --> 00:13:26.960 You want to generate enough information 00:13:26.960 --> 00:13:31.310 to be able to calculate anything you want. 00:13:31.310 --> 00:13:33.070 And sometimes, you make approximations 00:13:33.070 --> 00:13:36.176 and you're not sure that those approximations are good. 00:13:36.176 --> 00:13:37.300 And you want them to break. 00:13:37.300 --> 00:13:39.520 You want to discover something new. 00:13:39.520 --> 00:13:41.890 So the Born-Oppenheimer approximation, 00:13:41.890 --> 00:13:44.890 we go from clamped nuclei calculation 00:13:44.890 --> 00:13:47.560 where the-- since the nuclei moves slow 00:13:47.560 --> 00:13:51.600 compared to the electrons, well, let's not let them move at all. 00:13:51.600 --> 00:13:54.930 And then we build a perturbation theory picture 00:13:54.930 --> 00:13:56.550 where we let them move. 00:13:56.550 --> 00:13:58.530 And we can deal with that because we understand 00:13:58.530 --> 00:14:00.090 vibrations and rotations. 00:14:03.480 --> 00:14:04.150 OK. 00:14:04.150 --> 00:14:05.920 So we have a potential energy surface. 00:14:09.790 --> 00:14:13.440 And there are things that we can anticipate 00:14:13.440 --> 00:14:15.680 about a potential energy surface. 00:14:15.680 --> 00:14:21.840 And LCAO-MO theory enables you to say 00:14:21.840 --> 00:14:27.840 a lot of important things about the potential energy surface. 00:14:27.840 --> 00:14:33.450 So it provides a qualitative framework. 00:14:33.450 --> 00:14:37.880 And so from molecular orbital theory-- 00:14:37.880 --> 00:14:41.160 and this is not what Professor Van Voorhis talked about. 00:14:41.160 --> 00:14:44.590 This is the baby stuff. 00:14:44.590 --> 00:14:48.490 And we don't expect to get the exact answer. 00:14:48.490 --> 00:14:52.410 But we do expect to be able to explain trends. 00:14:52.410 --> 00:14:55.800 Trends within a molecule and between related molecules. 00:14:59.130 --> 00:15:04.760 So this provides a framework for expectations. 00:15:04.760 --> 00:15:08.270 And there are things that we get like bond order. 00:15:12.680 --> 00:15:23.270 And we talk about orbitals that are bonding, non-bonding, 00:15:23.270 --> 00:15:24.180 and antibonding. 00:15:27.170 --> 00:15:31.170 And this comes directly out of the simple ideas. 00:15:31.170 --> 00:15:35.970 Recall when we had atom with a hydrogen, 00:15:35.970 --> 00:15:38.330 the hydrogen doesn't make pi bonds. 00:15:38.330 --> 00:15:44.300 And so they're pi orbitals for the atom A, which have nothing 00:15:44.300 --> 00:15:45.620 to interact with. 00:15:45.620 --> 00:15:47.780 And they're usually non-bonding. 00:15:47.780 --> 00:15:50.660 So we have these sorts of things. 00:15:50.660 --> 00:15:56.030 We have spN hybridization. 00:16:01.270 --> 00:16:05.210 And this is just telling you if a molecule wants 00:16:05.210 --> 00:16:09.240 to make the maximum number of bonds, you do something. 00:16:09.240 --> 00:16:14.130 And if it's sp cubed, you have four 00:16:14.130 --> 00:16:17.490 tetrahedrally arranged bonds. 00:16:17.490 --> 00:16:23.370 And if it's sp cubed, sp squared is planar with 120 degrees. 00:16:23.370 --> 00:16:27.540 These things tell you something about geometric expectations. 00:16:27.540 --> 00:16:29.850 Now, molecules don't follow the rules exactly, 00:16:29.850 --> 00:16:31.900 but they come pretty close. 00:16:31.900 --> 00:16:34.410 And so if you have some reason to believe 00:16:34.410 --> 00:16:37.500 that a particular hybridization is appropriate, 00:16:37.500 --> 00:16:41.889 then you have certain expectations for the geometry 00:16:41.889 --> 00:16:44.180 and how that's going to present itself in the spectrum. 00:16:50.300 --> 00:16:52.990 So bond order's related to internuclear distance 00:16:52.990 --> 00:16:55.270 and vibrational frequencies. 00:16:55.270 --> 00:16:58.930 Sp hybridization has to do with geometry, 00:16:58.930 --> 00:17:01.190 and all of these things are really important. 00:17:01.190 --> 00:17:02.740 So we have a potential surface. 00:17:05.810 --> 00:17:08.710 And let's say this is a boldface thing, 00:17:08.710 --> 00:17:12.130 implying that there are 3N minus 6 different displacement 00:17:12.130 --> 00:17:13.149 coordinates. 00:17:17.020 --> 00:17:23.859 This potential encodes the normal modes. 00:17:23.859 --> 00:17:27.130 What's a normal mode? 00:17:27.130 --> 00:17:29.980 Well, it's a classical mechanical concept. 00:17:29.980 --> 00:17:35.290 And it basically corresponds to situations 00:17:35.290 --> 00:17:39.820 where all of the atoms move at the same frequency 00:17:39.820 --> 00:17:42.620 in each normal mode. 00:17:42.620 --> 00:17:47.200 And these normal modes each has an expected frequency 00:17:47.200 --> 00:17:50.890 and expected geometry. 00:17:50.890 --> 00:17:52.870 Because if it's a polyatomic molecule, 00:17:52.870 --> 00:17:55.870 you have not just two things moving. 00:17:55.870 --> 00:17:59.090 They'll always be moving at the same frequency. 00:17:59.090 --> 00:18:02.380 But you have three or four or 100. 00:18:02.380 --> 00:18:03.630 And OK. 00:18:03.630 --> 00:18:08.200 So you learn about the shape of the potential and the force 00:18:08.200 --> 00:18:09.240 constants and so on. 00:18:11.940 --> 00:18:14.976 Now, we have the rotational structure. 00:18:18.550 --> 00:18:22.060 Now, molecules are not rigid rotors. 00:18:22.060 --> 00:18:26.590 But it's useful to think about molecules as rigid rotors 00:18:26.590 --> 00:18:30.310 to develop a basis set for describing rotation. 00:18:30.310 --> 00:18:32.110 And perturbation theory enables us 00:18:32.110 --> 00:18:40.970 to describe the energy levels of a non-rigid vibrating rotor. 00:18:40.970 --> 00:18:42.770 And it's straightforward. 00:18:42.770 --> 00:18:47.960 It may be ugly, but it tells you how you take information 00:18:47.960 --> 00:18:50.780 from the spectrum and learn about, say, 00:18:50.780 --> 00:18:53.150 the internuclear distance dependence 00:18:53.150 --> 00:18:56.900 of molecular constants like a spin orbit constant. 00:18:56.900 --> 00:18:58.970 Or some hyperfine constant. 00:18:58.970 --> 00:19:02.130 Or just the rotational constant. 00:19:02.130 --> 00:19:03.890 And it's a simple thing. 00:19:03.890 --> 00:19:09.260 And I'm toying with the idea of using this sort of problem 00:19:09.260 --> 00:19:11.720 on the exam. 00:19:11.720 --> 00:19:15.120 And I'm not sure whether I did on the second exam. 00:19:15.120 --> 00:19:19.680 But if I did, you'll have a chance to redeem yourself. 00:19:19.680 --> 00:19:20.180 OK. 00:19:25.041 --> 00:19:25.540 OK. 00:19:25.540 --> 00:19:33.620 Then in the gas phase, nothing much happens. 00:19:33.620 --> 00:19:36.710 You can have collisions, but the time between collisions 00:19:36.710 --> 00:19:40.800 can be controlled by what the pressure you use. 00:19:40.800 --> 00:19:43.160 And so you can sort of think about the gas 00:19:43.160 --> 00:19:45.950 phase as something where the molecules are isolated. 00:19:48.990 --> 00:19:51.725 And another way of saying that is the expectation 00:19:51.725 --> 00:20:01.845 value of the Hamiltonian in any state is time independent. 00:20:01.845 --> 00:20:04.950 The Hamiltonian is energy. 00:20:04.950 --> 00:20:06.650 Energy is conserved. 00:20:06.650 --> 00:20:11.350 And unless there are collisions, energy will be conserved. 00:20:11.350 --> 00:20:12.010 And so. 00:20:16.930 --> 00:20:20.930 But in the condensed phase, you have lot of collisions. 00:20:20.930 --> 00:20:24.210 They're very fast. 00:20:24.210 --> 00:20:28.190 And so one big difference between the gas phase 00:20:28.190 --> 00:20:32.440 and the condensed phase is energy is not conserved. 00:20:32.440 --> 00:20:35.230 And it's not conserved at different rates 00:20:35.230 --> 00:20:37.660 for different kinds of motions. 00:20:37.660 --> 00:20:41.350 And you want to understand that. 00:20:41.350 --> 00:20:41.890 OK. 00:20:41.890 --> 00:20:43.764 So now let's talk about the kinds of spectra. 00:20:50.700 --> 00:20:52.110 We have rotational spectra. 00:20:55.920 --> 00:21:00.870 And that usually is in the micro region of the spectrum. 00:21:00.870 --> 00:21:05.440 And it requires that the electric dipole 00:21:05.440 --> 00:21:07.510 moment be not equal to zero. 00:21:12.830 --> 00:21:14.790 I'll talk about this some more in a minute. 00:21:14.790 --> 00:21:15.720 We have vibration. 00:21:18.640 --> 00:21:22.010 And vibrational spectrum is in the infrared. 00:21:22.010 --> 00:21:30.350 And the requirement for vibration 00:21:30.350 --> 00:21:33.240 is that the dipole moment-- 00:21:33.240 --> 00:21:34.970 which is a vector quantity if you don't 00:21:34.970 --> 00:21:38.420 have a diatomic molecule-- 00:21:38.420 --> 00:21:43.580 changes with displacements, each of the normal modes. 00:21:43.580 --> 00:21:48.020 And so we have a molecule like CO2. 00:21:48.020 --> 00:21:51.590 CO2 does not have a dipole moment at equilibrium. 00:21:51.590 --> 00:21:54.340 It's a linear molecule, symmetric. 00:21:54.340 --> 00:22:02.760 But it can do this and that is a change in dipole moment. 00:22:02.760 --> 00:22:07.780 It can do this and that produces a new dipole moment. 00:22:07.780 --> 00:22:09.620 Or produces a dipole moment. 00:22:09.620 --> 00:22:11.510 And it can do this, which does not. 00:22:11.510 --> 00:22:14.120 So you have different modes, which are infrared 00:22:14.120 --> 00:22:17.270 active and infrared inactive. 00:22:17.270 --> 00:22:19.750 And it's related to this dipole moment. 00:22:19.750 --> 00:22:23.700 Now notice, I put a vector sign on it. 00:22:23.700 --> 00:22:29.610 It corresponds to motion, directions in the body frame 00:22:29.610 --> 00:22:31.140 where the dipole moment changes. 00:22:31.140 --> 00:22:33.570 When you do this, the dipole moment 00:22:33.570 --> 00:22:35.250 is perpendicular to the axis. 00:22:35.250 --> 00:22:37.740 When you do this, it's along the axis. 00:22:37.740 --> 00:22:39.690 And so there is stuff-- 00:22:39.690 --> 00:22:41.760 really a lot of stuff-- just looking 00:22:41.760 --> 00:22:44.560 at what vibrational modes are active. 00:22:44.560 --> 00:22:46.553 And then there's electronic. 00:22:50.810 --> 00:22:53.100 And that's mostly in the visible and UV. 00:22:53.100 --> 00:22:55.570 But the electronic spectrum could be in the X-ray region 00:22:55.570 --> 00:22:56.070 even. 00:22:56.070 --> 00:23:00.480 But mostly, molecules break when they get outside 00:23:00.480 --> 00:23:02.490 of the ordinary UV region. 00:23:02.490 --> 00:23:04.050 And so there's not much there. 00:23:06.751 --> 00:23:07.250 OK. 00:23:07.250 --> 00:23:08.720 What's needed. 00:23:08.720 --> 00:23:14.500 Well, what's needed is the electronic transition moment. 00:23:14.500 --> 00:23:17.450 Let's call this e1, e2. 00:23:17.450 --> 00:23:19.850 Going from two different electronic states, 00:23:19.850 --> 00:23:22.730 there is an electric dipole transition moment, 00:23:22.730 --> 00:23:25.070 which is not equal to zero. 00:23:25.070 --> 00:23:30.120 So H2, which has no vibrational spectrum 00:23:30.120 --> 00:23:34.540 and no rotational spectrum has an electronic spectrum. 00:23:34.540 --> 00:23:36.460 Everything has an electronic spectrum. 00:23:40.350 --> 00:23:40.870 OK. 00:23:40.870 --> 00:23:44.050 Now, a diatomic molecule. 00:23:48.970 --> 00:23:50.950 Here is sort of a template for everything 00:23:50.950 --> 00:23:53.890 a diatomic molecule can do. 00:23:53.890 --> 00:23:55.300 Now, they're cleverer than this. 00:23:55.300 --> 00:23:57.850 But this is sort of in preparation 00:23:57.850 --> 00:24:04.700 for dealing with greater complexity. 00:24:04.700 --> 00:24:06.830 And then we can have up here. 00:24:10.051 --> 00:24:10.550 OK. 00:24:10.550 --> 00:24:12.980 So this is the electronic ground state. 00:24:12.980 --> 00:24:17.300 And normally, if you have a diatomic molecule, 00:24:17.300 --> 00:24:21.080 you can predict what is the electronic ground state 00:24:21.080 --> 00:24:23.074 and how is it going to look-- 00:24:23.074 --> 00:24:24.740 how is its potential curve going to look 00:24:24.740 --> 00:24:28.560 relative to the excited states. 00:24:28.560 --> 00:24:30.300 That's something you should be able to do 00:24:30.300 --> 00:24:32.550 using LCAO-MO theory. 00:24:35.380 --> 00:24:37.580 OK. 00:24:37.580 --> 00:24:39.080 So this is the ground state. 00:24:43.310 --> 00:24:44.550 This is the repulsive state. 00:24:50.240 --> 00:24:52.970 And usually, the ground state correlates 00:24:52.970 --> 00:24:58.920 with ground state of the atoms. 00:24:58.920 --> 00:25:01.010 And so here we have a repulsive state, 00:25:01.010 --> 00:25:03.570 which also correlates with the ground state of the atom. 00:25:03.570 --> 00:25:06.920 So this is an excited state, and that correlates with an excited 00:25:06.920 --> 00:25:09.990 state of the atoms. 00:25:09.990 --> 00:25:14.915 And this is AB plus an electron. 00:25:18.070 --> 00:25:21.810 So that's one way the molecule breaks. 00:25:21.810 --> 00:25:25.140 And this dotted curve represents Rydberg 00:25:25.140 --> 00:25:30.220 states They're Rydberg states converging 00:25:30.220 --> 00:25:34.780 to every rotation vibration level of this excited state. 00:25:34.780 --> 00:25:36.110 And it can be complicated. 00:25:36.110 --> 00:25:37.810 But it's beautiful, because I know 00:25:37.810 --> 00:25:42.490 the magic decoder for how do you deal with Rydberg states. 00:25:42.490 --> 00:25:45.370 Because there's a lot of them, but they're 00:25:45.370 --> 00:25:47.090 closely related to each other. 00:25:47.090 --> 00:25:49.360 And we can exploit that relationship 00:25:49.360 --> 00:25:52.160 in guiding an experiment. 00:25:52.160 --> 00:25:55.510 And so here, now we have a curve crossing 00:25:55.510 --> 00:26:00.180 between a repulsive state and a bound state. 00:26:00.180 --> 00:26:03.088 And that leads to what we call predissociation. 00:26:07.870 --> 00:26:10.630 So the vibrational level of this state-- 00:26:10.630 --> 00:26:15.100 which would normally be bound above the curve crossing-- 00:26:15.100 --> 00:26:17.420 are not bound. 00:26:17.420 --> 00:26:20.540 And that's encoded in the spectrum too. 00:26:20.540 --> 00:26:24.930 And so for more complicated molecules, 00:26:24.930 --> 00:26:28.590 they're going to be these curve crossings or surface crossings. 00:26:28.590 --> 00:26:31.110 And we want to know how do we deal with them, 00:26:31.110 --> 00:26:36.190 and what is a diatomic-like way of dealing with them. 00:26:36.190 --> 00:26:43.240 And the important thing is at that internuclear distance 00:26:43.240 --> 00:26:50.030 where the curves cross, then you could be at a level-- 00:26:50.030 --> 00:26:52.730 starting at a level on this state-- 00:26:52.730 --> 00:26:54.890 the bound state. 00:26:54.890 --> 00:26:58.400 And it will have the same momentum at the crossing radius 00:26:58.400 --> 00:27:00.950 as the repulsive state. 00:27:00.950 --> 00:27:02.010 And that's where it goes. 00:27:02.010 --> 00:27:05.100 That's where it leaks out. 00:27:05.100 --> 00:27:10.350 I mean, we're normally used to thinking about processes-- 00:27:10.350 --> 00:27:13.620 non-radiative processes, all kinds of processes-- 00:27:13.620 --> 00:27:15.240 as an integral overall space. 00:27:18.192 --> 00:27:23.610 But because the momenta are the same 00:27:23.610 --> 00:27:26.990 on the two curves at this radius, 00:27:26.990 --> 00:27:29.090 they can go freely from one to the other. 00:27:29.090 --> 00:27:31.370 The molecules can go freely from one to the other. 00:27:31.370 --> 00:27:36.010 We get a tremendous simplification. 00:27:36.010 --> 00:27:39.880 And this is something that is always ignored in textbooks 00:27:39.880 --> 00:27:42.820 and is a profound insight. 00:27:42.820 --> 00:27:46.540 Because you know exactly where things happen and why 00:27:46.540 --> 00:27:48.480 they happen. 00:27:48.480 --> 00:27:51.990 And so you can arrange the information 00:27:51.990 --> 00:27:54.940 to describe what's going on at this point. 00:27:54.940 --> 00:27:58.110 And this is where semi-classical theory is really valuable. 00:27:58.110 --> 00:28:02.280 Because not only do you know that you have stationary phase 00:28:02.280 --> 00:28:02.910 at this point. 00:28:02.910 --> 00:28:07.800 But you know what the spatial oscillation frequency is. 00:28:07.800 --> 00:28:11.310 Because the spatial oscillation frequency is H-- 00:28:11.310 --> 00:28:14.270 or the wavelength-- is h over p. 00:28:17.830 --> 00:28:20.320 And you know what the momentum is at this point. 00:28:20.320 --> 00:28:23.140 And so you know where the nodes are. 00:28:23.140 --> 00:28:26.980 How far the nodes are apart and what the amplitude is here. 00:28:26.980 --> 00:28:28.630 And so it tells you exactly what you 00:28:28.630 --> 00:28:30.970 want to know in order to describe 00:28:30.970 --> 00:28:34.600 this non-radiative process. 00:28:34.600 --> 00:28:39.820 And my belief is that almost all of the complex things that 00:28:39.820 --> 00:28:43.570 molecules do happen at a predictable region 00:28:43.570 --> 00:28:45.490 in coordinated space. 00:28:45.490 --> 00:28:49.600 And you can get the information you need to understand them, 00:28:49.600 --> 00:28:51.700 because all of a sudden, the molecule 00:28:51.700 --> 00:28:55.750 is behaving in a kind of classical way. 00:28:55.750 --> 00:28:58.735 And we're entitled to think locally rather than globally. 00:29:02.160 --> 00:29:03.140 OK. 00:29:03.140 --> 00:29:05.210 I'm going very slowly. 00:29:05.210 --> 00:29:07.880 We may get through most of what I planned to talk about today. 00:29:07.880 --> 00:29:10.952 I have 11 pages of notes, and this is-- 00:29:10.952 --> 00:29:12.950 I'm on page three. 00:29:12.950 --> 00:29:16.160 So maybe we'll take off. 00:29:16.160 --> 00:29:17.450 OK. 00:29:17.450 --> 00:29:19.700 So how do you do spectra? 00:29:19.700 --> 00:29:23.780 Well, in the old days, you had some light source 00:29:23.780 --> 00:29:28.020 like a candle, and you had a lens, 00:29:28.020 --> 00:29:31.410 and you had an absorption cell. 00:29:31.410 --> 00:29:33.870 And there would be some sort of a spectrometer here. 00:29:36.840 --> 00:29:37.660 Could be a grading. 00:29:37.660 --> 00:29:38.610 It could be a prism. 00:29:38.610 --> 00:29:39.930 It could be anything. 00:29:39.930 --> 00:29:42.600 But it's something that says, OK, I 00:29:42.600 --> 00:29:45.870 looked at the selected wavelengths 00:29:45.870 --> 00:29:49.620 at which the gas in this cell removed light 00:29:49.620 --> 00:29:51.840 from the continuum. 00:29:51.840 --> 00:29:54.780 And then you have a detector, which in the old days 00:29:54.780 --> 00:29:57.660 was a photographic plate. 00:29:57.660 --> 00:29:59.730 But it could be a photo multiplier, 00:29:59.730 --> 00:30:03.570 and you're looking at the spectrum 00:30:03.570 --> 00:30:08.720 by scanning the grading or something like that. 00:30:08.720 --> 00:30:11.950 And so what you would get is some kind 00:30:11.950 --> 00:30:21.030 of a record where you have dark regions corresponding to where 00:30:21.030 --> 00:30:26.710 there has been no absorption and, well, actually it 00:30:26.710 --> 00:30:27.960 would be the other way around. 00:30:27.960 --> 00:30:30.480 There'd be bright regions, because there'd 00:30:30.480 --> 00:30:34.740 be no exposure of the emulsion on the plate and dark regions 00:30:34.740 --> 00:30:37.320 where there-- 00:30:37.320 --> 00:30:39.221 yes-- where the light hits. 00:30:39.221 --> 00:30:39.720 OK. 00:30:39.720 --> 00:30:42.450 So but we're much cleverer than this. 00:30:42.450 --> 00:30:45.640 And we can do all sorts of wonderful things. 00:30:45.640 --> 00:30:49.620 And again, I've been around for a long time 00:30:49.620 --> 00:30:52.710 and lasers were just beginning to be used 00:30:52.710 --> 00:30:54.960 when I was a graduate student. 00:30:54.960 --> 00:30:58.440 And I was one of the first small molecules 00:30:58.440 --> 00:31:00.460 spectroscopists to use lasers. 00:31:00.460 --> 00:31:02.220 But not as a graduate student. 00:31:02.220 --> 00:31:05.400 I wanted them, but we had such-- 00:31:05.400 --> 00:31:14.610 lasers were so terrible in the region between 1965 and 1971 00:31:14.610 --> 00:31:16.290 when I was a graduate student. 00:31:16.290 --> 00:31:20.950 And so lasers were things to be admired, but hardly to be used. 00:31:20.950 --> 00:31:23.870 But one of the crucial things was dye lasers. 00:31:28.540 --> 00:31:31.610 Because these guys are monochromatic, 00:31:31.610 --> 00:31:36.740 and they can be tuned and tuned continuously over a wide region 00:31:36.740 --> 00:31:38.350 of the spectrum. 00:31:38.350 --> 00:31:43.100 And so that's way better than a candle or a light bulb. 00:31:43.100 --> 00:31:45.292 Because it's monochromatic, and you're asking one 00:31:45.292 --> 00:31:47.000 question at a time as you tune the laser. 00:31:52.050 --> 00:31:57.380 Now, lasers enable you to do many kinds of experiments. 00:31:57.380 --> 00:32:09.140 You can simply tune the laser through a series 00:32:09.140 --> 00:32:17.040 of transitions, and you get fluorescence 00:32:17.040 --> 00:32:19.870 every time the laser tunes through a transition. 00:32:22.500 --> 00:32:26.990 And if one laser is good, two lasers are better. 00:32:26.990 --> 00:32:29.180 And so you can do all sorts of things 00:32:29.180 --> 00:32:36.610 like suppose this spectrum is really complicated. 00:32:36.610 --> 00:32:39.420 And you want to be able to simplify it. 00:32:39.420 --> 00:32:41.760 And so you can do a double resonance experiment 00:32:41.760 --> 00:32:43.770 where you tune this laser to one line 00:32:43.770 --> 00:32:47.700 and then you tune this laser through a series of transition. 00:32:47.700 --> 00:32:49.594 That spectrum is going to be simple, 00:32:49.594 --> 00:32:51.510 and it's going to be telling you who this was. 00:32:54.070 --> 00:32:58.090 And there's just no end of tricks. 00:32:58.090 --> 00:33:05.860 And often, instead of detecting the fluorescence, 00:33:05.860 --> 00:33:08.080 you tune the laser. 00:33:08.080 --> 00:33:09.370 Let's do this. 00:33:09.370 --> 00:33:13.930 And so starting here is some kind of a continuum, ionization 00:33:13.930 --> 00:33:15.140 continuum. 00:33:15.140 --> 00:33:21.670 And so you have this photon being used twice. 00:33:21.670 --> 00:33:24.730 One here and one to take it above the ionization limit. 00:33:24.730 --> 00:33:29.080 And so you have an excitation, which you do 00:33:29.080 --> 00:33:30.970 want to know how strong it is. 00:33:30.970 --> 00:33:33.070 And so you might monitor the fluorescence. 00:33:33.070 --> 00:33:35.380 But you don't know who this is, and you find out 00:33:35.380 --> 00:33:36.640 by tuning this. 00:33:36.640 --> 00:33:41.290 But you would detect the excitation here 00:33:41.290 --> 00:33:43.630 by subsequent ionization. 00:33:43.630 --> 00:33:46.860 It's easy to collect ions. 00:33:46.860 --> 00:33:49.890 Every ion you produce, you can detect. 00:33:49.890 --> 00:33:51.900 Every photon you produce, you can't detect. 00:33:51.900 --> 00:33:55.230 Because you have a solid angle consideration and photo 00:33:55.230 --> 00:33:57.120 multipliers are not perfect. 00:33:57.120 --> 00:34:01.066 And so ionization detection is way more sensitive. 00:34:01.066 --> 00:34:02.440 So you can do that kind of thing. 00:34:08.280 --> 00:34:12.989 You can also do-- 00:34:12.989 --> 00:34:16.230 this is a kind of sequential excitation. 00:34:16.230 --> 00:34:18.989 You could imagine doing an experiment where 00:34:18.989 --> 00:34:24.460 you have an energy level here, and you have a laser, 00:34:24.460 --> 00:34:26.920 which is not on resonance. 00:34:26.920 --> 00:34:28.520 That's a coherent process. 00:34:28.520 --> 00:34:34.449 It uses the oscilltor strength at this level to get to here. 00:34:34.449 --> 00:34:39.230 And that's related to many other kinds of current experiments. 00:34:39.230 --> 00:34:39.940 And that's neat. 00:34:50.659 --> 00:34:53.810 Now, we're recording spectra, and we need 00:34:53.810 --> 00:34:57.460 to know what the rules are. 00:34:57.460 --> 00:35:02.060 And so there are certain transitions that are allowed 00:35:02.060 --> 00:35:05.050 and certain transitions that are forbidden. 00:35:05.050 --> 00:35:11.500 Now, I talked about the transition requirements 00:35:11.500 --> 00:35:15.930 for rotation, vibration, and electronic. 00:35:15.930 --> 00:35:18.190 But let's just talk about the electronic spectrum, 00:35:18.190 --> 00:35:21.060 because the other two are simple. 00:35:21.060 --> 00:35:29.570 The transition operator is equal to the sum over electrons 00:35:29.570 --> 00:35:35.550 of e times r sub i. 00:35:35.550 --> 00:35:38.840 It's a one electron operator. 00:35:38.840 --> 00:35:40.910 And that means if we have wave functions which 00:35:40.910 --> 00:35:53.000 are Slater determinates of spin orbitals like 2s alpha. 00:35:57.390 --> 00:36:01.560 This one electron operator can only 00:36:01.560 --> 00:36:06.030 have a non-zero matrix element if the two states 00:36:06.030 --> 00:36:09.760 differ by one spin orbital. 00:36:09.760 --> 00:36:12.780 That's a big simplification. 00:36:12.780 --> 00:36:17.960 And so one can actually use this to selectively access 00:36:17.960 --> 00:36:21.140 different kinds of states by designing an experiment. 00:36:21.140 --> 00:36:24.800 But the important thing is that for electronic transitions, 00:36:24.800 --> 00:36:30.180 we have the selection rule delta so is equal to 1. 00:36:30.180 --> 00:36:31.680 Not 2. 00:36:31.680 --> 00:36:34.540 Not 0. 00:36:34.540 --> 00:36:42.150 And now, the operator doesn't have any spin involved with it. 00:36:42.150 --> 00:36:47.060 And so that means delta s equals 0. 00:36:47.060 --> 00:36:50.350 You did not change, you did not go from a singlet state 00:36:50.350 --> 00:36:52.780 to a triplet state. 00:36:52.780 --> 00:36:56.830 And the only way you can get from a singlet state 00:36:56.830 --> 00:36:59.620 to a triplet state is if the triplet state is 00:36:59.620 --> 00:37:01.270 perturbed by a singlet state. 00:37:03.930 --> 00:37:08.330 So this picture I drew where I had a repulsive state crossing 00:37:08.330 --> 00:37:11.450 through a bound state, it might have been that one of those 00:37:11.450 --> 00:37:13.010 was a triplet. 00:37:13.010 --> 00:37:16.940 And as a result, and the other is a singlet, 00:37:16.940 --> 00:37:19.610 and you have spin orbit interactions, 00:37:19.610 --> 00:37:23.900 and you get extra states, extra lines appearing. 00:37:23.900 --> 00:37:29.750 But you get this wonderful selection rule. 00:37:29.750 --> 00:37:36.075 There is also for the electric dipole 00:37:36.075 --> 00:37:42.980 that we have plus to minus parity. 00:37:42.980 --> 00:37:44.970 Now what's parity? 00:37:44.970 --> 00:37:46.710 I don't like talking about parity, 00:37:46.710 --> 00:37:51.970 because the useful definition leads to complexity. 00:37:51.970 --> 00:37:54.450 But basically, parity corresponds 00:37:54.450 --> 00:38:03.060 to the symmetry, the inversion symmetry in the laboratory 00:38:03.060 --> 00:38:04.130 frame. 00:38:04.130 --> 00:38:06.960 Now, you say, well, a molecule doesn't have any inversion 00:38:06.960 --> 00:38:07.470 symmetry. 00:38:07.470 --> 00:38:10.180 But space is isotopic. 00:38:10.180 --> 00:38:12.510 And so you can go from a left handed to a right handed 00:38:12.510 --> 00:38:13.760 coordinate system. 00:38:13.760 --> 00:38:16.720 And that's what happens when you invert space. 00:38:16.720 --> 00:38:19.740 And so you can classify levels according 00:38:19.740 --> 00:38:22.200 to whether they're odd or even with respect 00:38:22.200 --> 00:38:23.520 to space inversion. 00:38:23.520 --> 00:38:25.010 This is close to the truth. 00:38:25.010 --> 00:38:28.500 This is close to all you need to know unless you're actually 00:38:28.500 --> 00:38:32.190 going to do stuff with the parity operator. 00:38:32.190 --> 00:38:34.360 But it's a useful way of saying, OK, 00:38:34.360 --> 00:38:35.760 I put parity labels on things. 00:38:35.760 --> 00:38:38.280 I learned how to do that, and that's enough. 00:38:42.880 --> 00:38:50.290 Now, you follow selection rules where good quantum numbers 00:38:50.290 --> 00:38:51.940 are conserved. 00:38:51.940 --> 00:38:54.940 Or they change in a way that you predict 00:38:54.940 --> 00:38:57.960 based on the way you did the experiment. 00:38:57.960 --> 00:39:01.300 A good quantum number, I remind you, 00:39:01.300 --> 00:39:03.940 is the eigenvalue of an operator that commutes 00:39:03.940 --> 00:39:05.494 with the exact Hamiltonian. 00:39:13.780 --> 00:39:16.910 There are very few rigorously good quantum numbers. 00:39:16.910 --> 00:39:19.160 But if a molecule has any symmetry, 00:39:19.160 --> 00:39:23.120 group theory tells you a bunch of things 00:39:23.120 --> 00:39:25.580 that commute with the Hamiltonian, 00:39:25.580 --> 00:39:28.160 and it gives you symmetry labels. 00:39:28.160 --> 00:39:30.470 And that's very important in inorganic chemistry 00:39:30.470 --> 00:39:34.940 where you have either molecules with symmetry or molecules 00:39:34.940 --> 00:39:40.460 with atoms with ligands in a symmetric arrangement. 00:39:40.460 --> 00:39:43.820 And since the transition is on the center atom 00:39:43.820 --> 00:39:48.770 usually that you can classify them using group theory as 00:39:48.770 --> 00:39:50.200 allowed or forbidden. 00:39:56.900 --> 00:40:05.550 So if we have an electronic transition, 00:40:05.550 --> 00:40:11.690 the easiest thing to observe is vibrational bands. 00:40:11.690 --> 00:40:14.870 If you have a relatively low resolution spectrum, 00:40:14.870 --> 00:40:17.150 you're going to see vibrational bands. 00:40:17.150 --> 00:40:20.210 You have to work harder to see the rotational transitions 00:40:20.210 --> 00:40:25.190 in each vibrational band, but you get an enormous amount 00:40:25.190 --> 00:40:28.040 of qualitative information just looking 00:40:28.040 --> 00:40:29.810 at the vibrational bands. 00:40:29.810 --> 00:40:43.060 Because the vibrational bands encode the difference 00:40:43.060 --> 00:40:49.070 between the ground state potential and the excited state 00:40:49.070 --> 00:40:50.060 potential. 00:40:50.060 --> 00:40:52.370 Now, this is a universal notation. 00:40:52.370 --> 00:40:54.560 Ground state is always double prime. 00:40:54.560 --> 00:40:56.570 Upper state is always single prime. 00:40:56.570 --> 00:40:59.480 Very strange, but that's the way it is. 00:40:59.480 --> 00:41:02.980 And so if these potentials are different, 00:41:02.980 --> 00:41:05.941 the vibrational bands encode the difference. 00:41:10.860 --> 00:41:13.740 And this comes from the Franck-Condon principle, which 00:41:13.740 --> 00:41:19.120 says nuclei move slowly, electrons move fast. 00:41:19.120 --> 00:41:21.780 The transition is an instantaneous process, 00:41:21.780 --> 00:41:23.670 as far as the nuclei are concerned. 00:41:23.670 --> 00:41:28.500 And so there's no change in nuclear coordinates, 00:41:28.500 --> 00:41:32.280 and there is no change in nuclear momentum. 00:41:32.280 --> 00:41:34.146 This is what's in all the textbooks, 00:41:34.146 --> 00:41:35.520 and nobody ever talks about this, 00:41:35.520 --> 00:41:43.420 because we don't really normally know or think about momentum. 00:41:43.420 --> 00:41:45.370 But we do know what it is. 00:41:45.370 --> 00:41:47.170 We do know what the operator is. 00:41:47.170 --> 00:41:51.040 And we know that kinetic energy is 00:41:51.040 --> 00:41:55.240 related to the momentum squared over 2 times the mass. 00:41:58.660 --> 00:41:59.361 So what is this? 00:41:59.361 --> 00:42:00.860 This means transitions are vertical. 00:42:03.525 --> 00:42:09.970 In other words, if we have a pair of electronic states, 00:42:09.970 --> 00:42:12.880 we draw these vertical lines. 00:42:12.880 --> 00:42:15.760 Not slanting lines. 00:42:15.760 --> 00:42:19.340 This means momentum is conserved. 00:42:19.340 --> 00:42:24.110 And this, here at this vertical point, 00:42:24.110 --> 00:42:25.980 we have this much momentum. 00:42:25.980 --> 00:42:30.080 And, well, there's the same amount here. 00:42:30.080 --> 00:42:32.660 Now usually, this just means-- 00:42:32.660 --> 00:42:35.000 the delta p is equal to 0-- 00:42:35.000 --> 00:42:38.300 means that of all the strong transitions, 00:42:38.300 --> 00:42:43.570 turning point to turning point transitions are the strongest. 00:42:43.570 --> 00:42:49.450 Because at a turning point, the vibrational amplitude is large. 00:42:49.450 --> 00:42:51.100 But there's more to it than that, 00:42:51.100 --> 00:42:53.730 because there are secondary maxima 00:42:53.730 --> 00:42:55.780 in the vibrational transition intensities. 00:42:55.780 --> 00:43:00.760 These correspond to stationary phase between the initial state 00:43:00.760 --> 00:43:02.230 and the final state. 00:43:02.230 --> 00:43:05.640 And so in addition to the strongest transitions, 00:43:05.640 --> 00:43:10.980 you get other transitions that you can explain by this delta p 00:43:10.980 --> 00:43:11.610 equals 0. 00:43:14.810 --> 00:43:19.255 Now, you don't know anything when you start. 00:43:19.255 --> 00:43:22.130 You know something maybe about the ground state. 00:43:22.130 --> 00:43:24.955 And this could be a polyatomic molecule 00:43:24.955 --> 00:43:27.490 if we're just looking at one mode. 00:43:27.490 --> 00:43:31.820 And suppose we have an excited state where 00:43:31.820 --> 00:43:36.160 the vibrational frequency is the same. 00:43:36.160 --> 00:43:40.850 In other words, there is no change in bonding character. 00:43:40.850 --> 00:43:45.950 And so what you end up getting is just 00:43:45.950 --> 00:43:49.740 the zero to zero transition in absorption. 00:43:49.740 --> 00:43:54.080 Or if you have many vibrational levels up here, you see v to v, 00:43:54.080 --> 00:43:55.280 delta v equals 0. 00:44:01.490 --> 00:44:05.170 Now, you could have a bound state, 00:44:05.170 --> 00:44:10.470 and it's usually true that the excited state is less bound. 00:44:10.470 --> 00:44:13.780 And so you would have-- 00:44:17.250 --> 00:44:20.850 the Franck-Condon active region corresponds 00:44:20.850 --> 00:44:23.820 to turning point to turning point in the lower state. 00:44:23.820 --> 00:44:25.390 So I shouldn't draw this. 00:44:25.390 --> 00:44:31.580 Let's draw a vertical transition. 00:44:31.580 --> 00:44:33.180 Well, I missed. 00:44:33.180 --> 00:44:34.950 Let me just start again. 00:44:34.950 --> 00:44:39.150 So we have an excited state, and we have a ground state. 00:44:39.150 --> 00:44:40.920 Ground state is bound. 00:44:40.920 --> 00:44:42.640 And this is v equal 0. 00:44:42.640 --> 00:44:45.890 And so we go from here to there. 00:44:50.270 --> 00:44:54.540 So if the excited state is less bound, 00:44:54.540 --> 00:44:58.460 it's a larger inner nucleus and smaller vibrational frequency. 00:44:58.460 --> 00:45:02.480 We have many vibrational levels accessed 00:45:02.480 --> 00:45:04.552 by the Franck-Condon principle. 00:45:07.930 --> 00:45:10.810 And if we have, in the other sense, 00:45:10.810 --> 00:45:14.080 we have an excited state, which is more bound than the ground 00:45:14.080 --> 00:45:23.165 state then the Franck-Condon region is narrower. 00:45:27.620 --> 00:45:30.290 Because this wall is nearly vertical. 00:45:30.290 --> 00:45:35.000 You have more transitions when it's displaced this way. 00:45:35.000 --> 00:45:39.230 And this branch of the potential is nearly much flatter, 00:45:39.230 --> 00:45:42.560 and you have fewer transitions here. 00:45:42.560 --> 00:45:47.420 So the vibrational pattern tells you qualitatively from the 00:45:47.420 --> 00:45:51.031 get go whether you're going to a more bound or a less bound 00:45:51.031 --> 00:45:51.530 state. 00:45:54.050 --> 00:45:56.017 That's very useful information. 00:46:06.090 --> 00:46:09.590 Now, you want to go further. 00:46:09.590 --> 00:46:11.860 You want to say, oh, well, I'm observing 00:46:11.860 --> 00:46:15.840 a bunch of vibrational levels in the excited state. 00:46:15.840 --> 00:46:18.900 What are their quantum numbers? 00:46:18.900 --> 00:46:21.810 Are we seeing the v equals zero level? 00:46:21.810 --> 00:46:27.130 How do we know what vibrational level we're observing? 00:46:27.130 --> 00:46:29.380 Or levels? 00:46:29.380 --> 00:46:31.260 And we can calculate Franck-Condon factors. 00:46:35.160 --> 00:46:38.400 But you can do many things. 00:46:38.400 --> 00:46:42.550 So I mean, if you have a situation like this, 00:46:42.550 --> 00:46:45.630 you might not get to the lowest vibrational level. 00:46:45.630 --> 00:46:47.260 But if you have a situation like this, 00:46:47.260 --> 00:46:49.740 you probably will get to the lowest vibrational level. 00:46:49.740 --> 00:46:52.050 So one way to get vibrational numbering 00:46:52.050 --> 00:46:56.100 is seeing the vibrational pattern terminate. 00:46:56.100 --> 00:46:58.500 That's a good sign it's v equals 0. 00:46:58.500 --> 00:47:00.370 If it terminates abruptly, it's v equal 0. 00:47:00.370 --> 00:47:04.010 But if it terminates slowly, you're not sure. 00:47:04.010 --> 00:47:08.150 But there are other really wonderful things about this. 00:47:08.150 --> 00:47:12.240 And you can do isotope separation. 00:47:12.240 --> 00:47:14.940 Isotope shifts. 00:47:18.220 --> 00:47:20.740 Since the vibrational frequency is 00:47:20.740 --> 00:47:27.490 the square root of k over mu, we can change and reduce mass. 00:47:27.490 --> 00:47:31.180 That changes which vibrational bands you observe, 00:47:31.180 --> 00:47:33.250 and it changes it in a quantitative way, 00:47:33.250 --> 00:47:37.510 and so you can often use that to tell. 00:47:37.510 --> 00:47:43.330 Another thing you can do is to now. 00:47:43.330 --> 00:47:47.480 If you have an excited state, the vibrational wave function 00:47:47.480 --> 00:47:51.030 is going to look like this. 00:47:51.030 --> 00:47:53.020 And there's a node here, and a node here, 00:47:53.020 --> 00:47:55.230 and node here, node here, node here. 00:47:55.230 --> 00:47:59.010 And so if we're looking at the vibrational progression 00:47:59.010 --> 00:48:02.760 observed from such a level, the intensities 00:48:02.760 --> 00:48:07.920 will have minima corresponding to how many nodes there 00:48:07.920 --> 00:48:09.150 are in the excited state. 00:48:09.150 --> 00:48:12.270 And the number of nodes is the vibrational quantum number. 00:48:15.570 --> 00:48:18.450 So there are lots of ways of doing it. 00:48:18.450 --> 00:48:21.200 And then there is something else. 00:48:21.200 --> 00:48:24.320 If you observe at moderately low resolution 00:48:24.320 --> 00:48:28.700 a vibrational band in an electronic spectrum, 00:48:28.700 --> 00:48:40.250 it will have the peculiar shape like that or like that. 00:48:40.250 --> 00:48:42.950 This is called a band head. 00:48:42.950 --> 00:48:45.290 And it corresponds to the fact that because 00:48:45.290 --> 00:48:48.560 the rotational constants in the upper and lower state 00:48:48.560 --> 00:48:51.980 are different, one of the branches-- 00:48:51.980 --> 00:48:58.460 you have delta J equals plus or minus 1, 0. 00:48:58.460 --> 00:49:04.340 And the delta J plus 1 is called the R branch and minus 1 00:49:04.340 --> 00:49:06.550 is called the P branch. 00:49:06.550 --> 00:49:09.160 And delta J of 0 is called the Q branch. 00:49:09.160 --> 00:49:12.830 Now, these two guys are such that depending 00:49:12.830 --> 00:49:15.780 on the sign of the difference in B values, 00:49:15.780 --> 00:49:21.290 you get ahead on the high frequency side 00:49:21.290 --> 00:49:25.380 or ahead on the low frequency side. 00:49:25.380 --> 00:49:27.350 Well, actually, it's the other way around. 00:49:27.350 --> 00:49:31.800 But it tells you then the rotational constant, 00:49:31.800 --> 00:49:35.360 which is also a signal of how bound the state is. 00:49:35.360 --> 00:49:40.130 The rotational constant-- the shading of these band heads-- 00:49:40.130 --> 00:49:43.430 confirms your vibrational assignment. 00:49:43.430 --> 00:49:48.000 And typically, if you have a smaller vibrational frequency, 00:49:48.000 --> 00:49:52.120 you'll also have a smaller rotational constant. 00:49:52.120 --> 00:49:53.240 It's not always true. 00:49:53.240 --> 00:49:55.990 And it's always interesting when it doesn't happen. 00:49:55.990 --> 00:49:59.850 And so the vibrational bands have this asymmetric shape 00:49:59.850 --> 00:50:07.480 unless the rotational constant upstairs and downstairs 00:50:07.480 --> 00:50:08.380 is the same. 00:50:08.380 --> 00:50:11.170 And then you have a branch going this way and a branch 00:50:11.170 --> 00:50:11.800 going this way. 00:50:11.800 --> 00:50:13.633 And a gap in the middle that might be filled 00:50:13.633 --> 00:50:15.790 with some Q branch lines. 00:50:15.790 --> 00:50:18.460 And the Q branch lines tell you, oh, yeah, that 00:50:18.460 --> 00:50:26.470 was a transition where the lambda, the projection of L 00:50:26.470 --> 00:50:29.970 on the internuclear axis changed. 00:50:29.970 --> 00:50:32.600 And if it doesn't have this Q branch, 00:50:32.600 --> 00:50:35.270 it'll tell you this delta lambda equals 0. 00:50:35.270 --> 00:50:37.620 All sorts of stuff. 00:50:37.620 --> 00:50:42.790 And if you have no sign of band heads, 00:50:42.790 --> 00:50:45.040 but just this double hump structure, 00:50:45.040 --> 00:50:48.500 it tells you that the rotational constant is about the same as 00:50:48.500 --> 00:50:49.520 in the ground state. 00:50:49.520 --> 00:50:52.160 And it tells you also that the vibrational frequency is 00:50:52.160 --> 00:50:53.570 expected to be about the same. 00:50:58.750 --> 00:51:00.940 I better stop. 00:51:00.940 --> 00:51:05.120 So as soon as you go to polyatomic molecules. 00:51:05.120 --> 00:51:10.430 Instead of having just one vibration, you have 3n minus 6. 00:51:10.430 --> 00:51:14.090 And so 3n minus 6 downstairs, 3n minus 6 upstairs, 00:51:14.090 --> 00:51:16.590 that looks bad. 00:51:16.590 --> 00:51:21.550 But only some of the vibrational modes 00:51:21.550 --> 00:51:24.070 correspond to a distortion from-- 00:51:24.070 --> 00:51:26.010 a difference in geometry between the ground 00:51:26.010 --> 00:51:27.860 state and the excited state. 00:51:27.860 --> 00:51:33.160 And so most of the vibrational modes 00:51:33.160 --> 00:51:35.650 correspond to identical frequencies. 00:51:35.650 --> 00:51:38.830 And what we call that is Franck-Condon dark. 00:51:38.830 --> 00:51:42.520 Because the only transitions that are allowed are delta v 00:51:42.520 --> 00:51:45.070 equals 0 for that mode. 00:51:45.070 --> 00:51:47.710 And so only the modes that correspond 00:51:47.710 --> 00:51:53.180 to a change in structure appear as long progressions. 00:51:53.180 --> 00:51:55.370 And so there are only a few, and so the spectrum 00:51:55.370 --> 00:51:59.240 of a polyatomic molecule is not too much different 00:51:59.240 --> 00:52:01.220 from a diatomic molecule, because 00:52:01.220 --> 00:52:05.550 of the small number of Franck-Condon active modes. 00:52:05.550 --> 00:52:09.960 But you look closer and you see big differences. 00:52:09.960 --> 00:52:10.460 OK. 00:52:10.460 --> 00:52:16.400 So well, we'll talk more about this on Friday. 00:52:16.400 --> 00:52:19.340 And we might go to three lectures 00:52:19.340 --> 00:52:20.910 on this, because this is really-- 00:52:20.910 --> 00:52:22.700 you know the whole point of this course 00:52:22.700 --> 00:52:27.760 is to be able to understand how molecules talk to us.