WEBVTT 00:00:00.090 --> 00:00:02.631 ANNOUNCER: The following content is provided under a Creative 00:00:02.631 --> 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 --> 00:00:17.250 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:17.250 --> 00:00:18.210 at ocw.mit.edu. 00:00:22.220 --> 00:00:23.810 ROBERT FIELD: When I was really young, 00:00:23.810 --> 00:00:27.350 I used to go to a television repair store 00:00:27.350 --> 00:00:32.780 as often as possible to take home one of the dead chassis. 00:00:32.780 --> 00:00:34.820 And then I would take it apart. 00:00:34.820 --> 00:00:36.770 And I don't know what I was looking for, 00:00:36.770 --> 00:00:40.670 but that was sort of the empirical stuff. 00:00:40.670 --> 00:00:43.610 I had no idea how a television worked, 00:00:43.610 --> 00:00:46.250 but I was really curious about maybe 00:00:46.250 --> 00:00:49.490 I could find it if I just did stuff. 00:00:49.490 --> 00:00:53.420 And what we've been talking about 00:00:53.420 --> 00:00:58.730 are ways in which we not just generate numbers, like 00:00:58.730 --> 00:01:03.630 parts of the television chassis, but insight. 00:01:03.630 --> 00:01:09.000 And there are we've talked about three ways so far. 00:01:09.000 --> 00:01:15.060 And one is Huckel theory, where Huckel theory is just 00:01:15.060 --> 00:01:20.130 a bunch of simple rules and simple ideas for how 00:01:20.130 --> 00:01:25.080 do you represent a large family of related molecules. 00:01:25.080 --> 00:01:29.730 And the Huckel theory is incredibly simple, 00:01:29.730 --> 00:01:33.990 but it enables you to make really sophisticated 00:01:33.990 --> 00:01:36.570 conclusions about how things work 00:01:36.570 --> 00:01:38.970 and what are the important factors. 00:01:38.970 --> 00:01:47.670 And so it's basically a procedure 00:01:47.670 --> 00:01:53.180 for distilling insight from random observations. 00:01:53.180 --> 00:01:54.870 Or maybe not random observations, 00:01:54.870 --> 00:01:59.100 but observations of many properties 00:01:59.100 --> 00:02:02.280 of many related molecules. 00:02:02.280 --> 00:02:03.675 Then you've seen LCAOMO. 00:02:06.590 --> 00:02:12.550 Based on the small variations all treatment of H2 plus, 00:02:12.550 --> 00:02:16.280 we got the idea of orbitals and what makes a bond. 00:02:16.280 --> 00:02:19.940 And then with the idea that we can describe 00:02:19.940 --> 00:02:23.720 the size of atomic orbitals by the energy 00:02:23.720 --> 00:02:26.840 below the ionization limit, we can 00:02:26.840 --> 00:02:31.430 make similar quantitative predictions based 00:02:31.430 --> 00:02:34.370 on this minimal basis set. 00:02:34.370 --> 00:02:37.700 So we draw molecular orbital diagrams. 00:02:37.700 --> 00:02:40.100 And these molecular orbital diagrams 00:02:40.100 --> 00:02:53.980 are especially valuable for isoelectronic and homologous 00:02:53.980 --> 00:02:55.180 comparisons. 00:02:55.180 --> 00:03:06.740 So isoelectronic would be, let's say, nitrogen, CO, BF. 00:03:10.250 --> 00:03:16.220 Molecules with the same number of electrons. 00:03:16.220 --> 00:03:23.414 And homologous would be something like CO, SIO, GEO, 00:03:23.414 --> 00:03:25.900 and so on. 00:03:25.900 --> 00:03:31.210 And with the concept of orbital size being related 00:03:31.210 --> 00:03:34.150 to the ionization energy of the atoms, 00:03:34.150 --> 00:03:37.660 we can make a lot of very useful comparisons. 00:03:37.660 --> 00:03:38.980 And so we develop insight. 00:03:41.810 --> 00:03:45.530 Now LCAOMO is a variation of method, 00:03:45.530 --> 00:03:49.130 and it can be a very large variation of method. 00:03:49.130 --> 00:03:51.720 And it could become of an issue. 00:03:51.720 --> 00:03:54.380 But it's usually not atomic orbitals. 00:03:54.380 --> 00:03:57.110 It's just Gaussian orbitals or something, 00:03:57.110 --> 00:03:59.390 which is computationally convenient. 00:03:59.390 --> 00:04:09.620 And so it can be a procedure to rationalize experience and make 00:04:09.620 --> 00:04:15.350 predictions as you go from the same number of electrons 00:04:15.350 --> 00:04:19.399 but increasing polarizability, polarity. 00:04:19.399 --> 00:04:23.510 And these-- this is where you develop chemical insight. 00:04:23.510 --> 00:04:24.927 And it's really exciting. 00:04:24.927 --> 00:04:26.510 And it's what was missing when I would 00:04:26.510 --> 00:04:31.760 take these chassis from the repair shop 00:04:31.760 --> 00:04:33.980 and chop them up into pieces. 00:04:33.980 --> 00:04:38.250 And oh yeah, there's a magnet in there and things like that. 00:04:38.250 --> 00:04:41.540 It's much deeper. 00:04:41.540 --> 00:04:45.260 And now perturbation theory is a kind of a different thing 00:04:45.260 --> 00:04:47.090 altogether. 00:04:47.090 --> 00:04:50.480 Perturbation theory says we do an experiment. 00:04:50.480 --> 00:04:53.720 We measure something, and we turn it 00:04:53.720 --> 00:04:55.730 into something we really wanted. 00:04:55.730 --> 00:04:59.090 The Rolling Stones are telling us 00:04:59.090 --> 00:05:03.740 that if you want to know how a particular property depends 00:05:03.740 --> 00:05:07.190 on internuclear distance, and you can't directly 00:05:07.190 --> 00:05:10.580 measure the property as a function of internuclear 00:05:10.580 --> 00:05:12.800 distance, well maybe you can measure it 00:05:12.800 --> 00:05:16.420 as a function of vibrational and rotational quantum numbers. 00:05:16.420 --> 00:05:18.570 And that's what perturbation theory does. 00:05:18.570 --> 00:05:21.260 It tells you how to get from what you observe to what 00:05:21.260 --> 00:05:22.370 you really want to know. 00:05:25.670 --> 00:05:34.010 And it can be horrible in terms of the algebraic exercises you 00:05:34.010 --> 00:05:39.120 have to go through in order to get what you really want 00:05:39.120 --> 00:05:41.110 or from what you-- 00:05:41.110 --> 00:05:41.610 yes. 00:05:44.280 --> 00:05:48.580 And today's lecture is going to be an example of not the worst 00:05:48.580 --> 00:05:50.710 thing you could ever do with perturbation theory, 00:05:50.710 --> 00:05:53.760 but pretty close to the worst. 00:05:53.760 --> 00:05:59.730 OK, but all three of the first things are related, 00:05:59.730 --> 00:06:04.320 are associated with getting insight 00:06:04.320 --> 00:06:11.310 from either a crude calculation refined against observations 00:06:11.310 --> 00:06:14.790 or just the observations and reducing those observations 00:06:14.790 --> 00:06:17.550 to something really neat about how things work. 00:06:20.470 --> 00:06:24.215 Then the next two lectures will be given by Professor Van Van 00:06:24.215 --> 00:06:25.690 Voorhis and it's-- 00:06:25.690 --> 00:06:31.360 they will be on ab initio theory, where basically you 00:06:31.360 --> 00:06:33.010 don't assume anything. 00:06:33.010 --> 00:06:37.870 You don't do anything except solve for the exact energy 00:06:37.870 --> 00:06:41.780 levels and wave functions. 00:06:41.780 --> 00:06:44.700 Now this is not possible. 00:06:44.700 --> 00:06:49.260 Directly you can do this by making approximations, doing 00:06:49.260 --> 00:06:51.240 an enormous variational calculation. 00:06:53.930 --> 00:07:00.830 Now many people think, well why bother with approximate methods 00:07:00.830 --> 00:07:04.190 when you can get the truth? 00:07:04.190 --> 00:07:11.950 And the answer is the truth is no more valuable than the parts 00:07:11.950 --> 00:07:15.550 of a disabled television. 00:07:15.550 --> 00:07:17.030 It's-- they're stuff. 00:07:17.030 --> 00:07:19.340 But there's no insight there. 00:07:19.340 --> 00:07:24.290 And quantum chemists who do these calculations 00:07:24.290 --> 00:07:26.810 are not just generating numbers. 00:07:26.810 --> 00:07:31.710 They're trying to explain how things work. 00:07:31.710 --> 00:07:36.050 And it's the same business, it's just the tools are different. 00:07:36.050 --> 00:07:38.990 The goals are not just getting-- 00:07:38.990 --> 00:07:43.190 in spectroscopy we can measure things to 10 or 11 digits. 00:07:43.190 --> 00:07:44.990 And nobody cares. 00:07:44.990 --> 00:07:48.080 I mean, how many digits? 00:07:48.080 --> 00:07:52.670 It's a little bit challenging to remember telephone numbers. 00:07:52.670 --> 00:07:59.180 And having big tables of 10-digit numbers, so what? 00:07:59.180 --> 00:08:01.130 But what do the numbers tell you? 00:08:01.130 --> 00:08:03.800 And this quest-- the same question 00:08:03.800 --> 00:08:06.620 is asked by good quantum chemists 00:08:06.620 --> 00:08:09.650 by doing a series of calculations 00:08:09.650 --> 00:08:13.610 where they turn on and off certain terms 00:08:13.610 --> 00:08:14.990 in the Hamiltonian. 00:08:14.990 --> 00:08:16.940 So it's the same thing. 00:08:16.940 --> 00:08:23.120 I'm an experimentalist, but many people who are experimentalists 00:08:23.120 --> 00:08:26.510 think I'm a theorist because I do weird stuff. 00:08:26.510 --> 00:08:27.590 But I'm not a theorist. 00:08:27.590 --> 00:08:29.610 Troy is a theorist. 00:08:29.610 --> 00:08:34.309 And he does weird stuff too, but he's not an experimentalist. 00:08:34.309 --> 00:08:36.549 And we're both after insight. 00:08:36.549 --> 00:08:41.140 And OK, so let's just-- 00:08:44.500 --> 00:08:47.510 I guess I'll just launch it through the perturbation 00:08:47.510 --> 00:08:48.010 theory. 00:08:51.460 --> 00:08:51.960 OK. 00:08:56.800 --> 00:09:01.150 But you do want to understand the difference in how 00:09:01.150 --> 00:09:04.810 these different methods, these approximate methods, work, 00:09:04.810 --> 00:09:06.510 and what they're good for, OK? 00:09:11.090 --> 00:09:16.700 So the goal of perturbation theory 00:09:16.700 --> 00:09:28.370 is to go from molecular constants 00:09:28.370 --> 00:09:32.570 to structural constants, or structural parameters. 00:09:41.370 --> 00:09:44.360 Molecular constants are like rotational constant, 00:09:44.360 --> 00:09:48.500 vibrational constant, stuff that you get by fitting the energy 00:09:48.500 --> 00:09:53.600 levels you observe to a dumb empirical expression, a power 00:09:53.600 --> 00:09:56.870 series in quantum numbers. 00:09:56.870 --> 00:09:59.850 Now there's lots of dumb empirical expressions 00:09:59.850 --> 00:10:03.190 you could use, and some are better than others. 00:10:03.190 --> 00:10:06.180 And perturbation theory will often tell you 00:10:06.180 --> 00:10:09.070 what is the right way to do it. 00:10:09.070 --> 00:10:11.160 So you go from molecular constants 00:10:11.160 --> 00:10:14.710 to things in the potential. 00:10:14.710 --> 00:10:17.790 So this is the displacement from equilibrium, 00:10:17.790 --> 00:10:19.650 and we would like to know something 00:10:19.650 --> 00:10:21.900 about how a molecule works. 00:10:21.900 --> 00:10:25.270 And there also might be other constants, like spin orbit, 00:10:25.270 --> 00:10:28.920 and hyperfine, dipole moment. 00:10:28.920 --> 00:10:33.480 And so let's just say we have some observable, 00:10:33.480 --> 00:10:39.390 and it is also a function of coordinate. 00:10:39.390 --> 00:10:44.490 And we'd like to know what that function is. 00:10:44.490 --> 00:10:49.770 But what we are able to do is measure energy levels 00:10:49.770 --> 00:10:51.930 as a function of quantum numbers. 00:10:51.930 --> 00:10:55.800 And so the information that we really want, 00:10:55.800 --> 00:10:59.840 the potential or the internuclear 00:10:59.840 --> 00:11:03.860 distance dependence of some electronic property, that's 00:11:03.860 --> 00:11:07.610 all gotten from what we can observe 00:11:07.610 --> 00:11:09.690 via perturbation theory. 00:11:09.690 --> 00:11:11.960 It's a very powerful tool. 00:11:11.960 --> 00:11:17.570 And it's not pretty, but it always works. 00:11:17.570 --> 00:11:21.050 And it's a good basis for insight. 00:11:24.310 --> 00:11:31.460 So perturbation theory is a fit model. 00:11:36.200 --> 00:11:40.700 The other methods, one doesn't generally do a least-squares 00:11:40.700 --> 00:11:43.530 fit to the-- 00:11:43.530 --> 00:11:48.520 of the adjustable parameters in Huckel theory 00:11:48.520 --> 00:11:53.750 to determine the properties of a molecule. 00:11:53.750 --> 00:11:56.780 One just says, OK, I'm going to try these 00:11:56.780 --> 00:12:01.460 because I think that attaching an electro-negative atom 00:12:01.460 --> 00:12:07.640 to a carbon atom is going to do something that I can predict. 00:12:07.640 --> 00:12:11.360 And maybe it's going to tell me some surprises about how 00:12:11.360 --> 00:12:14.210 its influence is not just where it's attached, 00:12:14.210 --> 00:12:16.250 but other places in the molecule. 00:12:16.250 --> 00:12:19.160 You know this from your first organic courses. 00:12:19.160 --> 00:12:21.230 You know all sorts of tricks to be 00:12:21.230 --> 00:12:25.520 able to predict reactivity and things that are related. 00:12:25.520 --> 00:12:28.710 But Huckel theory is not a fit model. 00:12:28.710 --> 00:12:32.480 LCAOMO theory is not a fit model. 00:12:32.480 --> 00:12:34.130 These are experience-based. 00:12:34.130 --> 00:12:37.280 And you integrate all sorts of stuff 00:12:37.280 --> 00:12:41.440 that you learn from comparing molecules. 00:12:41.440 --> 00:12:43.520 And the comparing of molecules is 00:12:43.520 --> 00:12:51.220 what makes up for the deteriable lists of the approximations. 00:12:51.220 --> 00:12:53.830 And your job is to hone your insight. 00:12:53.830 --> 00:12:59.080 And so you have nothing to protect you except the truth. 00:12:59.080 --> 00:13:00.730 You can observe molecules, but you 00:13:00.730 --> 00:13:06.630 can't be guided to something which is true, because there 00:13:06.630 --> 00:13:08.530 is no calculation. 00:13:08.530 --> 00:13:12.350 There's nothing exact and there's nothing complete. 00:13:12.350 --> 00:13:16.640 But in quantum chemistry you can get really close to the truth. 00:13:16.640 --> 00:13:19.142 But you don't know anything about why. 00:13:19.142 --> 00:13:20.600 And so you're doing the same thing, 00:13:20.600 --> 00:13:22.400 but from the opposite ends. 00:13:22.400 --> 00:13:25.460 OK, but perturbation theory is special 00:13:25.460 --> 00:13:29.360 because you take a huge amount of highly 00:13:29.360 --> 00:13:33.380 accurate experimental data of various types and you fit it. 00:13:36.960 --> 00:13:40.140 And so I want to talk about this. 00:13:40.140 --> 00:13:42.650 So you have the Hamiltonian. 00:13:42.650 --> 00:13:44.720 It's a matrix, and it's expressed 00:13:44.720 --> 00:13:47.640 in terms of parameters. 00:13:47.640 --> 00:13:51.380 Let's just use notation p, p of i. 00:13:51.380 --> 00:13:53.270 A set of parameters. 00:13:56.310 --> 00:13:59.030 So this is not a quantum chemical, Hamiltonian. 00:13:59.030 --> 00:14:02.510 It's a thing where we say there are certain degrees of freedom, 00:14:02.510 --> 00:14:07.160 and each one is controlled by a number of parameters. 00:14:09.900 --> 00:14:11.150 And we have the energy levels. 00:14:14.790 --> 00:14:17.370 And so what you do is you say let's 00:14:17.370 --> 00:14:21.070 choose a set of these parameters, 00:14:21.070 --> 00:14:26.480 and calculate the energy levels, and compare them 00:14:26.480 --> 00:14:29.070 to the observed energy levels. 00:14:29.070 --> 00:14:31.970 And it's not-- the first try, it's not going to be good. 00:14:31.970 --> 00:14:35.960 And then you say, OK, now I've got to do a least-squares fit. 00:14:35.960 --> 00:14:41.680 I have to vary the parameters in this matrix Hamiltonian 00:14:41.680 --> 00:14:45.310 to match the energy levels and maybe 00:14:45.310 --> 00:14:49.710 to match other things that you can observe in the experiment. 00:14:49.710 --> 00:14:52.250 But it's a complicated least-squares fit, 00:14:52.250 --> 00:14:55.850 because you have a matrix, and you want to diagonalize it. 00:14:55.850 --> 00:14:58.760 How are the parameters related to the eigenvalues 00:14:58.760 --> 00:15:00.370 and eigenvectors? 00:15:00.370 --> 00:15:03.590 You don't know. 00:15:03.590 --> 00:15:06.840 But when you're close, then you can start fitting these things. 00:15:06.840 --> 00:15:08.670 Then you can say, yeah, I know. 00:15:08.670 --> 00:15:11.720 And then once you've done that, you have not just 00:15:11.720 --> 00:15:16.870 the energy levels, but you have the wave functions. 00:15:20.510 --> 00:15:23.930 Now again, you don't know that there is something 00:15:23.930 --> 00:15:26.750 missing in your Hamiltonian. 00:15:26.750 --> 00:15:28.700 But you know that from the data you 00:15:28.700 --> 00:15:33.674 input you can match all the energy levels. 00:15:33.674 --> 00:15:35.340 But you didn't put it in all the levels. 00:15:35.340 --> 00:15:37.890 You put in just the ones that you measured. 00:15:37.890 --> 00:15:39.390 But you are arrogant. 00:15:39.390 --> 00:15:45.200 And you think, well, maybe I measured more than I-- 00:15:45.200 --> 00:15:47.490 I determined more than I measured, 00:15:47.490 --> 00:15:51.300 that I can extend the measurements to other things. 00:15:51.300 --> 00:15:53.970 And I told you at the beginning of the course you cannot 00:15:53.970 --> 00:15:56.500 observe the wave function. 00:15:56.500 --> 00:16:01.260 But if you do this, if you do a least-squares fit and match 00:16:01.260 --> 00:16:05.640 the energy levels, you have a pretty darn good representation 00:16:05.640 --> 00:16:07.980 of the wave function. 00:16:07.980 --> 00:16:10.770 And you can use that to calculate other stuff, 00:16:10.770 --> 00:16:11.745 especially dynamics. 00:16:14.900 --> 00:16:18.200 Remember dynamics, at least if the Hamiltonian is 00:16:18.200 --> 00:16:21.410 independent of time, the dynamics is going 00:16:21.410 --> 00:16:26.660 to be a function of x and t. 00:16:26.660 --> 00:16:32.100 But you start with the initial preparation. 00:16:32.100 --> 00:16:34.800 If you know that, then you, because you know the energy 00:16:34.800 --> 00:16:43.300 levels, you can calculate the full x of t for all time. 00:16:43.300 --> 00:16:44.820 So that's pretty powerful. 00:16:44.820 --> 00:16:49.860 If you fitted enough stuff, if your Hamiltonian has 00:16:49.860 --> 00:16:52.620 the important things in it, then you 00:16:52.620 --> 00:16:57.050 are basically able to do anything you want, 00:16:57.050 --> 00:17:00.530 whether it's static or dynamic. 00:17:00.530 --> 00:17:04.460 And if the Hamiltonian is time dependent, you can do that too, 00:17:04.460 --> 00:17:05.930 I just haven't showed you how yet. 00:17:08.980 --> 00:17:11.650 OK, so I really like perturbation theory, 00:17:11.650 --> 00:17:17.089 because it directly that deals with whatever data you have. 00:17:17.089 --> 00:17:21.130 And out from that, you get this thing 00:17:21.130 --> 00:17:24.720 which tells you everything, unless there's something 00:17:24.720 --> 00:17:28.560 that you didn't sample, that you didn't know you didn't sample. 00:17:28.560 --> 00:17:32.070 And then you discover that your data and the predictions 00:17:32.070 --> 00:17:33.660 are not in agreement. 00:17:33.660 --> 00:17:34.930 And then that's when you-- 00:17:34.930 --> 00:17:36.940 you don't go home and say, oh, I screwed up. 00:17:36.940 --> 00:17:38.040 I got to go take a nap. 00:17:38.040 --> 00:17:40.500 I don't-- you go home and you say, let's celebrate, 00:17:40.500 --> 00:17:44.800 because I discovered something that was really missing. 00:17:44.800 --> 00:17:47.730 And that's what we want to do. 00:17:47.730 --> 00:17:54.150 OK, so I have to mess around in the mud here. 00:17:54.150 --> 00:17:56.790 Because the perturbation theory you've 00:17:56.790 --> 00:17:59.880 done so far has been, perhaps, ugly, 00:17:59.880 --> 00:18:02.820 but it's been kind of simple because it's 00:18:02.820 --> 00:18:05.580 basically vibration. 00:18:05.580 --> 00:18:08.200 And now we've got vibration and rotation. 00:18:08.200 --> 00:18:10.320 And we have to combine the two, and we 00:18:10.320 --> 00:18:14.250 have to see whether there's something special that we 00:18:14.250 --> 00:18:15.900 can get from the combination. 00:18:15.900 --> 00:18:17.880 And you bet there is. 00:18:17.880 --> 00:18:22.170 OK, so we're going to look at the energy 00:18:22.170 --> 00:18:27.180 levels of a non-rigid, non-harmonic, or anharmonic 00:18:27.180 --> 00:18:28.260 oscillator. 00:18:28.260 --> 00:18:31.260 And we're going to find out how we generate 00:18:31.260 --> 00:18:35.550 from an expression of the potential energy 00:18:35.550 --> 00:18:37.980 surface or potential energy curve, 00:18:37.980 --> 00:18:41.460 because what I'm going to be talking about is diatomics. 00:18:41.460 --> 00:18:44.700 But it all extends to polyatomics. 00:18:44.700 --> 00:18:50.520 And we're going to calculate the relationships 00:18:50.520 --> 00:18:53.980 between what you observed and what you want to know. 00:18:53.980 --> 00:18:59.010 So for a diatonic molecule, we have a dumb representation 00:18:59.010 --> 00:19:03.090 of the quantum numbers. 00:19:03.090 --> 00:19:04.820 And we can write it. 00:19:04.820 --> 00:19:09.030 And I'm just really a stubborn person 00:19:09.030 --> 00:19:16.280 about spectroscopic notation, so I include this Hc convert 00:19:16.280 --> 00:19:19.790 wave number unit, the reciprocal centimeter units to energy. 00:19:19.790 --> 00:19:23.220 But you'll never catch me doing it in real life. 00:19:23.220 --> 00:19:26.210 But that leads to all sorts of algebraic errors 00:19:26.210 --> 00:19:28.760 that I don't know I'm making because I'm 00:19:28.760 --> 00:19:30.460 so unused to doing this. 00:19:30.460 --> 00:19:37.670 OK, so we have an expression for the vibrational energy levels. 00:19:37.670 --> 00:19:39.770 And I can't apolo-- 00:19:39.770 --> 00:19:44.060 I can't explain why we use four letters 00:19:44.060 --> 00:19:49.350 to represent one symbol, but that's the traditional thing. 00:19:49.350 --> 00:19:51.130 And for polyatomic molecules, which 00:19:51.130 --> 00:19:53.500 came after diatomic molecules, we 00:19:53.500 --> 00:19:55.270 start using only two letters. 00:19:55.270 --> 00:19:58.990 Instead of omega XE, we just use XE. 00:19:58.990 --> 00:20:03.910 But since I'm a diatomician, I'm going to do this sort of thing. 00:20:03.910 --> 00:20:07.780 And that's v plus 1/2 squared. 00:20:07.780 --> 00:20:14.560 And then the next term is omega YE v plus a 1/2 cubed. 00:20:17.610 --> 00:20:24.630 OK, and so then that's the vibrational part. 00:20:24.630 --> 00:20:26.730 And then we have the rotational part. 00:20:26.730 --> 00:20:31.170 The equilibrium internuclear distant rotational constant 00:20:31.170 --> 00:20:35.880 minus alpha e times v plus 1/2. 00:20:40.680 --> 00:20:42.370 Times JJ plus 1. 00:20:45.530 --> 00:20:49.340 And then minus DE, the centrifugal distortion 00:20:49.340 --> 00:20:57.410 constant, times JJ plus 1 quantity squared. 00:20:57.410 --> 00:20:59.750 And we can have more constants here. 00:20:59.750 --> 00:21:01.520 We can have additional constants there. 00:21:01.520 --> 00:21:03.920 But basically we have constants, which 00:21:03.920 --> 00:21:07.850 are known by multiplying a particular combination 00:21:07.850 --> 00:21:13.900 of vibrational and rotational constants, quantum numbers. 00:21:13.900 --> 00:21:16.630 And so these are the things we can determine, 00:21:16.630 --> 00:21:18.640 or sometimes we determine some of these 00:21:18.640 --> 00:21:21.640 and we want to get a prediction of the others 00:21:21.640 --> 00:21:24.436 because they're outside of our observation. 00:21:29.300 --> 00:21:38.210 And the potential is going to be dependent on the displacement 00:21:38.210 --> 00:21:42.540 from equilibrium and the rotational constant. 00:21:42.540 --> 00:21:46.050 And so there will be terms in the potential, 00:21:46.050 --> 00:21:57.690 like 1/2 KQ squared plus 1/6 AQ cubed, et cetera. 00:21:57.690 --> 00:21:59.400 So we have another dumb power series. 00:22:01.980 --> 00:22:05.880 And fortunately, we know that as we go up 00:22:05.880 --> 00:22:09.640 in the displace-- in this displacement coordinate, 00:22:09.640 --> 00:22:12.700 the coefficients get to be really small. 00:22:12.700 --> 00:22:17.210 And so we don't need to have a lot of these parameters. 00:22:17.210 --> 00:22:22.420 But we should have at least two, the harmonic and the cubic one. 00:22:22.420 --> 00:22:27.670 And we know for molecules the cubic parameter gives you 00:22:27.670 --> 00:22:32.450 a potential curve, which resembles reality. 00:22:32.450 --> 00:22:34.960 It's hard wall here, soft wall there. 00:22:34.960 --> 00:22:36.790 Molecules break. 00:22:36.790 --> 00:22:42.380 Now this has the unfortunate property of doing that. 00:22:42.380 --> 00:22:46.880 And so it's got death built into it. 00:22:46.880 --> 00:22:50.320 But it doesn't really matter, because you basically 00:22:50.320 --> 00:22:54.190 are looking at increasingly wide regions of the potential. 00:22:54.190 --> 00:22:56.410 And the fact that it does something terrible 00:22:56.410 --> 00:22:59.960 in your simple representation is almost irrelevant. 00:22:59.960 --> 00:23:02.380 But you have to be aware that you 00:23:02.380 --> 00:23:04.400 can tunnel through barriers. 00:23:04.400 --> 00:23:07.470 And when you start being able to tunnel through barriers, 00:23:07.470 --> 00:23:10.480 and it depends on how close you are to the top of the barrier 00:23:10.480 --> 00:23:14.230 and how wide it is, then you start seeing effects. 00:23:14.230 --> 00:23:18.970 And then this term is going to give nonsense. 00:23:18.970 --> 00:23:21.000 But there is a domain of goodness 00:23:21.000 --> 00:23:23.190 where you don't have to worry about that. 00:23:26.670 --> 00:23:29.100 So we have this, and then we have 00:23:29.100 --> 00:23:33.330 the rotational constant kind of contributing 00:23:33.330 --> 00:23:43.270 to the effect of potential, which is HCV of R JJ plus 1. 00:23:43.270 --> 00:23:45.010 Now here I've got-- 00:23:45.010 --> 00:23:47.560 here I have coordinates. 00:23:47.560 --> 00:23:49.810 Here I have quantum numbers. 00:23:49.810 --> 00:23:51.205 How can I do that? 00:23:51.205 --> 00:23:58.960 It's because central force problems can be represented 00:23:58.960 --> 00:24:00.880 in a universal form. 00:24:00.880 --> 00:24:04.150 And so we can just integrate over the angular part 00:24:04.150 --> 00:24:06.790 of the problem. 00:24:06.790 --> 00:24:09.400 And so now we have a mixed representation. 00:24:09.400 --> 00:24:17.200 But this guy is still a function of R or Q. And so this v of R 00:24:17.200 --> 00:24:19.090 is an operator. 00:24:19.090 --> 00:24:20.740 It's not just a constant. 00:24:20.740 --> 00:24:24.521 Some people call it the rotational constant operator. 00:24:24.521 --> 00:24:25.145 Kind of stupid. 00:24:28.950 --> 00:24:33.500 And so our job is now to figure out 00:24:33.500 --> 00:24:38.330 how to get from this parameter rise potential 00:24:38.330 --> 00:24:39.270 to the energy levels. 00:24:42.309 --> 00:24:44.100 Of course, we start with the energy levels. 00:24:44.100 --> 00:24:46.410 And we're going to want to determine the parameters 00:24:46.410 --> 00:24:48.480 in the potential. 00:24:48.480 --> 00:24:53.450 But you've got to have that connection. 00:24:53.450 --> 00:25:00.220 So this rotational operator has some constants in front of it. 00:25:09.560 --> 00:25:11.870 And this is in wave number of units, 00:25:11.870 --> 00:25:14.000 because I'm a spectroscopist. 00:25:14.000 --> 00:25:21.070 And we have the vibrational frequency, 1 over 2 pi C. K 00:25:21.070 --> 00:25:24.620 over mu square root. 00:25:24.620 --> 00:25:26.970 Also in wave number of units. 00:25:26.970 --> 00:25:31.520 Now the first problem is we don't like R, 00:25:31.520 --> 00:25:34.370 because harmonic oscillators are expressed 00:25:34.370 --> 00:25:37.620 in terms of displacement from equilibrium. 00:25:37.620 --> 00:25:43.841 And so we know that Q is equal to R minus RE, 00:25:43.841 --> 00:25:49.790 or R is equal to Q plus RE. 00:25:49.790 --> 00:25:53.990 So we want to replace this operator R by this, 00:25:53.990 --> 00:25:55.250 an operator plus a constant. 00:25:58.050 --> 00:26:02.800 So 1 over R squared can be expressed as 1 00:26:02.800 --> 00:26:08.930 over RE squared, the equivalent internucleus, plus 1 over Q 00:26:08.930 --> 00:26:09.690 over RE. 00:26:13.338 --> 00:26:15.650 Plus [INAUDIBLE] squared. 00:26:18.580 --> 00:26:20.680 So this is the power series expansion, 00:26:20.680 --> 00:26:23.662 but we only keep the first term generally. 00:26:26.260 --> 00:26:28.540 And so now we're ready to go to work. 00:26:28.540 --> 00:26:32.830 And we can say we have B as a function of Q 00:26:32.830 --> 00:26:39.670 is equal to B of E, the equilibrium value 1 minus 2Q 00:26:39.670 --> 00:26:47.940 or RE plus 3Q squared over RE squared. 00:26:47.940 --> 00:26:51.490 And there's more terms, but this is enough. 00:26:51.490 --> 00:26:53.566 You bet it's enough. 00:26:53.566 --> 00:26:54.940 By the time this lecture is over, 00:26:54.940 --> 00:26:58.250 you're going to want not ever to see this stuff again. 00:26:58.250 --> 00:27:02.140 OK, so we have now a representation 00:27:02.140 --> 00:27:04.720 of the rotational operator in terms 00:27:04.720 --> 00:27:09.390 of a constant term, a linear term, and a squared term in Q. 00:27:09.390 --> 00:27:13.880 And we know how to do matrix elements of Qs, right? 00:27:13.880 --> 00:27:15.660 So we know we're in business. 00:27:15.660 --> 00:27:20.880 And we have the potential as another power series in Qs. 00:27:20.880 --> 00:27:22.770 So everything is going to come together. 00:27:22.770 --> 00:27:28.050 It's just going to be will the migraine wipe out your ability 00:27:28.050 --> 00:27:30.900 to pay any attention to this after we're done. 00:27:30.900 --> 00:27:36.810 OK, so I don't want to cover that up yet. 00:27:36.810 --> 00:27:42.390 So we know that this displacement operator 00:27:42.390 --> 00:27:47.430 can be replaced by the creation and annihilation operators. 00:27:47.430 --> 00:27:56.820 And so we have 4 pi C, mu omega E square root. 00:27:56.820 --> 00:27:59.190 A plus A dagger are friends. 00:27:59.190 --> 00:28:03.140 These guys, we hardly need to take a deep breath 00:28:03.140 --> 00:28:04.710 to know what to do with those. 00:28:08.660 --> 00:28:14.730 And so we're going to want to express all of the-- 00:28:14.730 --> 00:28:18.930 all of the actors here, these guys and those guys, 00:28:18.930 --> 00:28:24.660 in terms of expressions in the creation and annihilation 00:28:24.660 --> 00:28:26.350 operators. 00:28:26.350 --> 00:28:29.040 So for the rotational constant expansion, 00:28:29.040 --> 00:28:34.370 we're going to need something like 2Q over RE. 00:28:34.370 --> 00:28:36.870 And that, after a little bit of algebra, 00:28:36.870 --> 00:28:48.300 is 4BE over omega E square root times A plus A dagger. 00:28:48.300 --> 00:28:54.250 And we need 3Q squared over RE squared. 00:28:54.250 --> 00:29:05.030 And that's going to be 3BE over omega E, A plus A dagger 00:29:05.030 --> 00:29:07.640 squared. 00:29:07.640 --> 00:29:11.450 So all of the terms can be expressed, 00:29:11.450 --> 00:29:15.500 can be reduced to constants that we care about, 00:29:15.500 --> 00:29:19.340 or that are easily measured, and these powers of As 00:29:19.340 --> 00:29:21.500 and a daggers. 00:29:21.500 --> 00:29:25.340 Now this is-- you have notes, printed notes. 00:29:25.340 --> 00:29:27.200 And I'm following them pretty well, 00:29:27.200 --> 00:29:34.480 although there are a few typos, which I hope to correct. 00:29:34.480 --> 00:29:43.810 And so we do a similar thing for the terms here. 00:29:43.810 --> 00:29:45.476 And we're only going to keep-- 00:29:45.476 --> 00:29:46.850 we're only going to look at this. 00:29:46.850 --> 00:29:48.730 We're not going to go to Q 1/4. 00:29:48.730 --> 00:29:51.540 But this is bad enough. 00:29:51.540 --> 00:29:56.030 OK, and this is part of the zero order Hamiltonian. 00:29:56.030 --> 00:30:00.700 So we're going to have another term, 00:30:00.700 --> 00:30:08.960 and that's going to be 1/6 little a Q cubed. 00:30:08.960 --> 00:30:13.065 OK, so that's-- little a is the anharmonicity parameter. 00:30:13.065 --> 00:30:13.940 And what is its sign? 00:30:18.090 --> 00:30:22.560 Can we know its sign from experiment? 00:30:22.560 --> 00:30:23.960 You said yes. 00:30:23.960 --> 00:30:25.816 Why do you say yes? 00:30:25.816 --> 00:30:27.001 AUDIENCE: [INAUDIBLE] 00:30:27.001 --> 00:30:28.000 ROBERT FIELD: I'm sorry? 00:30:28.000 --> 00:30:29.788 AUDIENCE: It's cubed, so it's odd. 00:30:29.788 --> 00:30:32.860 So there's [INAUDIBLE]. 00:30:32.860 --> 00:30:34.810 ROBERT FIELD: The only way you know 00:30:34.810 --> 00:30:40.730 the sign of an off diagonal matrix element is-- 00:30:40.730 --> 00:30:45.950 I'm sorry, is if there is a diagonal element of it that 00:30:45.950 --> 00:30:51.030 can be put into the E, the first order correction to the energy. 00:30:51.030 --> 00:30:52.590 As soon as you have to square it when 00:30:52.590 --> 00:30:56.160 you do second order perturbation theory, you've lost the sign. 00:30:56.160 --> 00:31:00.780 But here we know the sign because of physical insight. 00:31:00.780 --> 00:31:03.840 We know that this kind of a potential would be nonsense. 00:31:06.750 --> 00:31:13.045 So we know that A has a sign and it's negative. 00:31:13.045 --> 00:31:23.180 OK, so we have a over 6 times our favorite parameter 00:31:23.180 --> 00:31:30.420 here to the 3/2 power times A plus A dagger cubed. 00:31:30.420 --> 00:31:32.270 Now this is something-- 00:31:32.270 --> 00:31:35.690 you do the operator algebra before you launch 00:31:35.690 --> 00:31:37.740 into a horrible calculation. 00:31:37.740 --> 00:31:40.220 And so we know how to do this in principle 00:31:40.220 --> 00:31:44.900 to reduce this to a simple expression. 00:31:44.900 --> 00:31:48.080 And now we take this part, this thing, 00:31:48.080 --> 00:31:52.040 and we're just going to call it capital A. Because we want it-- 00:31:52.040 --> 00:31:54.320 we don't want to clutter up what is going 00:31:54.320 --> 00:31:57.600 to be a terrible thing anyway. 00:31:57.600 --> 00:32:02.960 And we can always convert back to the little a at the end 00:32:02.960 --> 00:32:05.850 if we need to. 00:32:05.850 --> 00:32:11.450 OK, so now I've set the stage. 00:32:11.450 --> 00:32:12.940 I'll leave this for a while. 00:32:16.390 --> 00:32:19.420 That really is-- as you know, I really love revision theory, 00:32:19.420 --> 00:32:22.750 because it's the tool that I use all the time. 00:32:22.750 --> 00:32:29.440 And it's a psychological condition 00:32:29.440 --> 00:32:33.100 that is not curable, all right? 00:32:33.100 --> 00:32:35.440 So what do we want? 00:32:35.440 --> 00:32:41.480 We want E0 as a function of E and J. 00:32:41.480 --> 00:32:48.880 And that's just the J H0 VJ. 00:32:48.880 --> 00:33:00.540 And we know that HC will make EB plus 1/2 plus HCBEJJ plus 1. 00:33:03.290 --> 00:33:06.311 OK, well this goes in the energy denominators. 00:33:09.750 --> 00:33:11.800 And so that's good, because we need 00:33:11.800 --> 00:33:17.380 energy denominators as well as the zero order energies. 00:33:17.380 --> 00:33:23.970 Right now we need to know the bad stuff, the thing that's 00:33:23.970 --> 00:33:25.920 outside of the zero order Hamiltonian, 00:33:25.920 --> 00:33:28.170 which is everything. 00:33:28.170 --> 00:33:38.730 And so we have this is HC times BE JJ plus 1 times-- 00:33:41.900 --> 00:33:43.930 I'm sorry, I have to erase this. 00:33:47.480 --> 00:33:54.116 Minus BE over 4BE over a big E square 00:33:54.116 --> 00:34:03.470 root times A plus A dagger plus 3 BE. 00:34:07.320 --> 00:34:18.010 3BE over omega E times A plus A dagger quality squared. 00:34:18.010 --> 00:34:20.639 So that's the rotational part. 00:34:20.639 --> 00:34:22.840 And then we have the vibrational power, 00:34:22.840 --> 00:34:27.330 which we have A, A plus A dagger cubed. 00:34:30.460 --> 00:34:32.920 OK, so now we look at this thing and we say, 00:34:32.920 --> 00:34:38.679 oh well, this has matrix elements delta V equals 00:34:38.679 --> 00:34:40.239 plus or minus 1. 00:34:40.239 --> 00:34:47.889 And this has delta V equals plus or minus 2 and 0. 00:34:47.889 --> 00:34:55.711 And this has delta V equals plus and minus 3 plus and minus 1. 00:34:55.711 --> 00:34:58.660 We always want to sort things according to the selection 00:34:58.660 --> 00:35:01.960 rules, because we always combine the things 00:35:01.960 --> 00:35:04.830 with the same selection rules. 00:35:04.830 --> 00:35:09.040 And it's best to do that at the beginning 00:35:09.040 --> 00:35:11.050 rather than somehow trying to do it at the end. 00:35:11.050 --> 00:35:14.020 Because if you have the same selection rule, 00:35:14.020 --> 00:35:15.190 you have cross terms. 00:35:18.560 --> 00:35:19.950 And that's really important. 00:35:19.950 --> 00:35:24.210 The cross terms are where a lot of good stuff happens. 00:35:24.210 --> 00:35:32.970 OK, so we know E0. 00:35:32.970 --> 00:35:37.740 And we would know E1 if there are any off the-- 00:35:37.740 --> 00:35:42.930 there are any diagonal matrix elements of this operator. 00:35:42.930 --> 00:35:45.430 This should be a plus as well. 00:35:45.430 --> 00:35:50.660 So this is H1. 00:35:50.660 --> 00:35:54.740 And this doesn't have any diagonal elements. 00:35:54.740 --> 00:35:59.750 And this doesn't have any diagonal elements. 00:35:59.750 --> 00:36:02.480 But this one does. 00:36:02.480 --> 00:36:04.780 And it's actually something you encountered, 00:36:04.780 --> 00:36:06.010 I think, on exam two. 00:36:06.010 --> 00:36:08.380 I'm not sure, but you've certainly encountered 00:36:08.380 --> 00:36:11.470 the diagonal element of this. 00:36:11.470 --> 00:36:18.880 And so if we look at A plus A dagger squared, 00:36:18.880 --> 00:36:22.840 we get A squared plus A dagger squared 00:36:22.840 --> 00:36:30.110 plus 2 number operator plus 1. 00:36:30.110 --> 00:36:33.130 And that's diagonal. 00:36:33.130 --> 00:36:36.520 So we have a diagonal element, and that gives us E1. 00:36:47.670 --> 00:37:03.640 So E1 of E and J is HC6BE squared over omega E times J, J 00:37:03.640 --> 00:37:05.930 plus 1, B plus 1/2. 00:37:10.460 --> 00:37:16.110 OK, well that looks pretty good, because this 00:37:16.110 --> 00:37:18.240 is the coefficient of-- 00:37:18.240 --> 00:37:20.250 we have an expression for the energy levels, 00:37:20.250 --> 00:37:28.110 which includes the E minus alpha E, B 00:37:28.110 --> 00:37:35.020 plus 1/2 of times JJ plus 1. 00:37:35.020 --> 00:37:39.360 So we have a term that involves B plus 1/2 and JJ plus 1. 00:37:39.360 --> 00:37:40.650 And it has a name. 00:37:40.650 --> 00:37:42.450 Alpha minus alpha. 00:37:42.450 --> 00:37:46.840 And here we have a term which has those, 00:37:46.840 --> 00:37:48.700 that quantum number dependents. 00:37:48.700 --> 00:37:51.810 And it has a value. 00:37:51.810 --> 00:37:56.660 And so it's telling us that alpha E, 00:37:56.660 --> 00:38:00.370 and I'll put on this harmonic oscillator, 00:38:00.370 --> 00:38:06.250 is equal to minus HC6BE v squared over omega 00:38:06.250 --> 00:38:08.830 E. One constant. 00:38:12.360 --> 00:38:26.220 Now it's also telling you that B of E increases with V. 00:38:26.220 --> 00:38:29.970 And for a harmonic oscillator, you 00:38:29.970 --> 00:38:35.290 have an equal lobe at each turning point, the largest 00:38:35.290 --> 00:38:35.790 lobe. 00:38:35.790 --> 00:38:39.020 But they're equal in magnitude. 00:38:39.020 --> 00:38:41.770 So one can ask, at the inner turning 00:38:41.770 --> 00:38:48.340 point is the change in the rotational operator, 00:38:48.340 --> 00:38:52.120 or is it larger-- 00:38:52.120 --> 00:38:58.210 is this change in BE larger relative to the equilibrium 00:38:58.210 --> 00:39:00.760 value, or is this larger? 00:39:00.760 --> 00:39:05.980 And the answer is we're talking about 1 over R squared. 00:39:05.980 --> 00:39:10.830 And 1 over R squared gets really large at small r as opposed 00:39:10.830 --> 00:39:15.450 to getting smaller at large R. And the amount of change 00:39:15.450 --> 00:39:18.040 is much greater at small r. 00:39:18.040 --> 00:39:23.610 And so that causes the effective rotational 00:39:23.610 --> 00:39:26.280 constant to increase. 00:39:26.280 --> 00:39:30.610 But we know for a harmonica-- for an anharmonic oscillator, 00:39:30.610 --> 00:39:34.580 we have a small lobe here and a big lobe here. 00:39:34.580 --> 00:39:37.100 And so we expect that there's going 00:39:37.100 --> 00:39:41.870 to be a battle between the harmonic contribution to alpha 00:39:41.870 --> 00:39:44.490 and the anharmonic contribution. 00:39:44.490 --> 00:39:47.980 And we expect that this is going to win. 00:39:47.980 --> 00:39:48.710 Why? 00:39:48.710 --> 00:39:53.180 Because every time anybody measures alpha, 00:39:53.180 --> 00:39:54.570 it's a positive number. 00:39:54.570 --> 00:40:00.830 And that's why it was expressed in the formula, which I've 00:40:00.830 --> 00:40:07.220 concealed, with a negative sign, to take into account that alpha 00:40:07.220 --> 00:40:08.460 is always-- 00:40:08.460 --> 00:40:10.580 the contribution is always negative, 00:40:10.580 --> 00:40:13.584 and so alpha is always positive. 00:40:13.584 --> 00:40:14.750 Yeah, and these things are-- 00:40:17.960 --> 00:40:19.150 it's historical. 00:40:19.150 --> 00:40:22.400 But anyway, so we have a contribution 00:40:22.400 --> 00:40:24.380 which has the wrong sign. 00:40:24.380 --> 00:40:27.170 And we know we're going to get another contribution 00:40:27.170 --> 00:40:31.350 from the interaction between the rotation and the vibration, 00:40:31.350 --> 00:40:35.360 and that this is going to make things right OK. 00:40:35.360 --> 00:40:38.950 And it comes from a term. 00:40:38.950 --> 00:40:39.450 OK. 00:40:46.380 --> 00:40:48.475 All right, there's no way I can make this simpler. 00:40:53.400 --> 00:40:55.860 Just have to bear with me and I will-- 00:40:59.480 --> 00:41:06.170 so we have delta V equals plus and minus 1 matrix 00:41:06.170 --> 00:41:16.080 elements from both the A plus A dagger and the A 00:41:16.080 --> 00:41:19.170 plus A dagger cubed terms. 00:41:22.500 --> 00:41:30.240 And we have-- we have this in both the anharmonic expression 00:41:30.240 --> 00:41:33.570 and in the rotational constant operator. 00:41:33.570 --> 00:41:35.970 We know what these-- 00:41:35.970 --> 00:41:37.810 how these things work out. 00:41:37.810 --> 00:41:48.160 And so delta V equals plus and minus 1 terms from the A 00:41:48.160 --> 00:42:00.270 plus A dagger cubed V plus 1 and V, V minus 1. 00:42:00.270 --> 00:42:04.130 So we can work these things out. 00:42:04.130 --> 00:42:10.720 And this one is 3V plus 1/2 to the 3/2. 00:42:10.720 --> 00:42:17.170 And this is 3V plus V to the 3/2. 00:42:17.170 --> 00:42:19.440 So all of this stuff, you want to simplify it 00:42:19.440 --> 00:42:21.780 as you go but it. 00:42:21.780 --> 00:42:29.002 And these come from the A term, the A, 00:42:29.002 --> 00:42:33.820 A plus A dagger cubed' term. 00:42:33.820 --> 00:42:37.390 We don't have a term that's linear at A plus A dagger 00:42:37.390 --> 00:42:41.470 from the anharmonicity, but we do-- 00:42:41.470 --> 00:42:44.500 and we do have a squared term, but that's 00:42:44.500 --> 00:42:45.620 the harmonic correction. 00:42:45.620 --> 00:42:48.130 So we start here, with A plus A dagger cubed. 00:42:52.610 --> 00:42:54.800 And then we're going to want to look 00:42:54.800 --> 00:42:59.120 at the delta V equals plus or minus 2, and delta V 00:42:59.120 --> 00:43:03.290 equals plus and minus 3 terms. 00:43:03.290 --> 00:43:05.340 And there's lots of algebra. 00:43:05.340 --> 00:43:06.580 You can do all these things. 00:43:06.580 --> 00:43:08.580 There's nothing challenging here. 00:43:08.580 --> 00:43:11.600 It's just how do you keep this stuff that you derived 00:43:11.600 --> 00:43:13.760 that you're going to need in a place 00:43:13.760 --> 00:43:16.760 that you can find it again, because the pages just 00:43:16.760 --> 00:43:18.290 get filled with garbage. 00:43:18.290 --> 00:43:18.790 OK. 00:43:22.150 --> 00:43:26.620 So the best way for me to do this is to write the results 00:43:26.620 --> 00:43:31.150 in the next-to-the-final step, and then in the final step, 00:43:31.150 --> 00:43:33.670 and show what goes to what. 00:43:40.990 --> 00:43:45.620 So the second order correction is a function of EJ. 00:43:49.180 --> 00:44:00.070 We're going to have terms that involve the delta V equals 00:44:00.070 --> 00:44:01.810 plus or minus 1. 00:44:01.810 --> 00:44:06.540 And we're going to get that from the cubic correction 00:44:06.540 --> 00:44:10.270 through the harmonic oscillator and the linear correction 00:44:10.270 --> 00:44:12.354 to the rotational concept. 00:44:12.354 --> 00:44:14.020 So we're going to get an expression that 00:44:14.020 --> 00:44:14.710 looks like this. 00:44:14.710 --> 00:44:23.170 B squared minus 2AB plus A squared. 00:44:26.330 --> 00:44:27.310 So this is a B term. 00:44:27.310 --> 00:44:29.150 This is the anharmonic term. 00:44:29.150 --> 00:44:30.130 We have a cross term. 00:44:30.130 --> 00:44:33.540 There are going to be three terms in this expression. 00:44:33.540 --> 00:44:35.450 OK, and so let's do that. 00:44:39.555 --> 00:44:40.430 How much time I have? 00:44:40.430 --> 00:44:41.390 Not very much time. 00:44:41.390 --> 00:44:43.056 Well that's good, because you won't have 00:44:43.056 --> 00:44:44.560 to watch much more of this. 00:44:44.560 --> 00:44:51.020 OK, so we have the HcBe JJ plus 1. 00:44:53.960 --> 00:44:56.180 And this guy is squared, because we're 00:44:56.180 --> 00:44:58.160 doing second-order perturbation theory. 00:44:58.160 --> 00:45:01.880 This is just the B term squared. 00:45:01.880 --> 00:45:08.000 And then we have 4BE be over omega E. 00:45:08.000 --> 00:45:15.380 And then we have the delta V of 1 and minus 1 matrix elements. 00:45:15.380 --> 00:45:25.520 V plus 1 over minus HC omega, and V plus V over HC omega. 00:45:25.520 --> 00:45:27.980 We have the energy denominators, two energy denominators 00:45:27.980 --> 00:45:31.400 of opposite side and similar identical magnitude. 00:45:31.400 --> 00:45:35.230 And we always do that because then the algebra is simple. 00:45:35.230 --> 00:45:38.360 If we do-- we fail to do that, you might as well just 00:45:38.360 --> 00:45:40.610 go to a booby hatch right away, because there's just 00:45:40.610 --> 00:45:42.950 no way you're going to retain your sanity if you 00:45:42.950 --> 00:45:48.150 don't combine the terms of delta V plus 1 and delta V minus 1. 00:45:50.970 --> 00:45:54.100 OK, and this is the first term. 00:45:54.100 --> 00:45:59.230 And there is the next term in this sum, which is the AB term. 00:45:59.230 --> 00:46:06.190 And so we have a minus 2 times HCV JJ plus 1. 00:46:09.280 --> 00:46:14.820 And it's not squared, because it's one B and one A term. 00:46:14.820 --> 00:46:26.630 And so we get 4BE over omega E square root. 00:46:26.630 --> 00:46:27.700 And then the A part. 00:46:30.470 --> 00:46:38.200 And we get 3B plus 1/2 to the 3/2 times 00:46:38.200 --> 00:46:48.500 V plus 1/2 to the 1/2, or minus HC omega E. 00:46:48.500 --> 00:46:55.220 And then the other term, which is 3V to the 3/2, V to 1/2, 00:46:55.220 --> 00:46:59.100 over HC omega E. 00:46:59.100 --> 00:47:01.010 We have to combine these two terms. 00:47:01.010 --> 00:47:02.720 And the combination is easy. 00:47:02.720 --> 00:47:03.590 And then there is-- 00:47:03.590 --> 00:47:05.990 so this term number two. 00:47:05.990 --> 00:47:10.580 And number three is the squared anharmonic term. 00:47:10.580 --> 00:47:14.570 So I'm going to skip it, since I'm basically done. 00:47:14.570 --> 00:47:17.070 There's just a lot of garbage like this. 00:47:17.070 --> 00:47:23.840 And at the end, we have an expression, term by term, 00:47:23.840 --> 00:47:25.670 which simplifies. 00:47:25.670 --> 00:47:32.480 And we get the second-order correction to the energy, VJ, 00:47:32.480 --> 00:47:42.520 is equal to minus HcBe cubed over HC omega. 00:47:45.100 --> 00:47:46.200 No, not HC. 00:47:46.200 --> 00:47:58.960 Just over omega E squared times JJ plus 1 squared. 00:47:58.960 --> 00:48:02.290 Well this is nice, because we have a term that just involves 00:48:02.290 --> 00:48:08.530 J, and it's negative, that says as a molecule rotates, 00:48:08.530 --> 00:48:15.050 the molecule stretches, the rotational constant decreases. 00:48:15.050 --> 00:48:17.440 This is called centrifugal distortion. 00:48:17.440 --> 00:48:19.960 And this is a very famous expression 00:48:19.960 --> 00:48:32.440 that DE is equal to 4BE cubed over omega squared. 00:48:32.440 --> 00:48:35.500 I left out a 4. 00:48:35.500 --> 00:48:37.150 So this is the Kratzer relationship. 00:48:37.150 --> 00:48:40.930 And that turns out to be of the most valuable ones, 00:48:40.930 --> 00:48:45.590 because the centrifugal distortion 00:48:45.590 --> 00:48:50.760 constant is extremely sensitive to contagion. 00:48:50.760 --> 00:48:54.200 If there is a perturbation, if there is something missing, 00:48:54.200 --> 00:48:57.280 this centrifugal distortion constant 00:48:57.280 --> 00:49:01.050 will have a crazy value. 00:49:01.050 --> 00:49:03.750 And if it doesn't ever-- if there's nothing wrong, 00:49:03.750 --> 00:49:06.000 then it will say, well, OK, I can-- 00:49:06.000 --> 00:49:09.600 if I know something about B and omega, 00:49:09.600 --> 00:49:15.720 I can determine how the rotational energy levels get 00:49:15.720 --> 00:49:18.960 closer together as you go up in V. 00:49:18.960 --> 00:49:22.080 And if you have only a pure rotation spectrum, 00:49:22.080 --> 00:49:23.910 then you actually-- 00:49:23.910 --> 00:49:25.530 where you're not sampling anything 00:49:25.530 --> 00:49:28.050 having to do with vibration, you actually 00:49:28.050 --> 00:49:32.250 can determine the vibrational constant 00:49:32.250 --> 00:49:35.790 from the centrifugal distortion. 00:49:35.790 --> 00:49:38.380 And I should tell you that spectroscopists come 00:49:38.380 --> 00:49:41.110 in families, or they used to. 00:49:41.110 --> 00:49:45.670 Pure rotation, infrared vibration rotation, 00:49:45.670 --> 00:49:48.400 electronic vibration rotation, electronic. 00:49:48.400 --> 00:49:51.910 And so the microwavers, who really have only rotation 00:49:51.910 --> 00:49:54.610 to look at, are kind of deprived. 00:49:54.610 --> 00:49:57.970 And they just don't know anything 00:49:57.970 --> 00:49:59.470 about electronic degrees of freedom. 00:49:59.470 --> 00:50:02.530 But they know how to get the vibrational frequency 00:50:02.530 --> 00:50:07.530 from the rotational spectrum, which is kind of nice. 00:50:07.530 --> 00:50:09.210 Gives them something to do. 00:50:09.210 --> 00:50:10.930 And there's more terms. 00:50:10.930 --> 00:50:13.800 And we're really done, but it's all in the notes. 00:50:13.800 --> 00:50:24.700 And so what we get is for the constant OAEXE, and center 00:50:24.700 --> 00:50:29.190 of distortion, and alpha E and beta, which 00:50:29.190 --> 00:50:34.290 is the vibrational distortion of the centrifugal distortion 00:50:34.290 --> 00:50:40.800 constant, we get all these in terms of omega and B. 00:50:40.800 --> 00:50:47.050 So it's a nice closed set where you get the-- 00:50:47.050 --> 00:50:51.780 you're able to predict how certain constants depend 00:50:51.780 --> 00:50:56.370 on the most fundamental ones, omega E and BE. 00:50:56.370 --> 00:50:57.900 That's kind of neat. 00:50:57.900 --> 00:51:03.310 And it's both a way of extending your observations 00:51:03.310 --> 00:51:05.230 and telling well, there's something missing, 00:51:05.230 --> 00:51:09.330 because these constants aren't coming out right. 00:51:09.330 --> 00:51:12.210 It's also a subject of great consternation. 00:51:12.210 --> 00:51:16.860 Because if you're a quantum chemist, what you do 00:51:16.860 --> 00:51:24.520 is you determine a lot of stuff by derivatives at equilibrium. 00:51:24.520 --> 00:51:27.320 The derivatives, the potential at equilibrium, 00:51:27.320 --> 00:51:29.810 are basically telling you the shape of the potential 00:51:29.810 --> 00:51:33.500 at moderately low energy. 00:51:33.500 --> 00:51:35.940 And it also turns out to be something with modern quantum 00:51:35.940 --> 00:51:36.440 chemistry. 00:51:36.440 --> 00:51:39.195 You can get all these derivatives pretty accurately. 00:51:42.050 --> 00:51:45.990 The trouble is measuring derivatives here 00:51:45.990 --> 00:51:49.160 and using the energy formulas gives you 00:51:49.160 --> 00:51:51.050 slightly different values from what you 00:51:51.050 --> 00:51:52.590 get from perturbation theory. 00:51:52.590 --> 00:51:55.100 So what we have is two communities 00:51:55.100 --> 00:51:59.410 who are very sophisticated using the same names 00:51:59.410 --> 00:52:02.370 for different quantities. 00:52:02.370 --> 00:52:04.360 It's dangerous. 00:52:04.360 --> 00:52:09.790 And it's also not understood by the two communities. 00:52:09.790 --> 00:52:11.650 And so people are struggling and struggling 00:52:11.650 --> 00:52:14.850 to get agreement between experiment and theory. 00:52:14.850 --> 00:52:18.340 And it's apples and oranges, but it's a subtle thing. 00:52:18.340 --> 00:52:22.180 OK, so that's basically all I want to say 00:52:22.180 --> 00:52:23.771 about perturbation theory. 00:52:23.771 --> 00:52:26.020 I will have one more lecture dealing with perturbation 00:52:26.020 --> 00:52:29.020 theory, and that lecture is on Van der Waals 00:52:29.020 --> 00:52:33.200 interactions between molecules and why we get liquids. 00:52:33.200 --> 00:52:34.750 OK.