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. 00:00:21.510 --> 00:00:25.530 ROBERT FIELD: Last time I talked about LCAO-MO 00:00:25.530 --> 00:00:27.690 for diatomic molecules. 00:00:27.690 --> 00:00:31.140 And I didn't finish, but the important point 00:00:31.140 --> 00:00:33.690 was that this is a toy model. 00:00:33.690 --> 00:00:36.840 This is based on a little bit of extension 00:00:36.840 --> 00:00:41.980 from something which is not really a toy model, H2 plus, 00:00:41.980 --> 00:00:44.830 to, basically, an interpretive framework 00:00:44.830 --> 00:00:47.140 that can be applied to, basically, 00:00:47.140 --> 00:00:50.200 all diatomic molecules. 00:00:50.200 --> 00:00:54.710 And the logic is relatively simple. 00:00:54.710 --> 00:00:56.470 We know this one. 00:00:56.470 --> 00:01:04.200 We get the molecular orbitals for H2 plus. 00:01:04.200 --> 00:01:05.470 There are basically two. 00:01:05.470 --> 00:01:08.230 There's a binding one and the anti-binding one. 00:01:08.230 --> 00:01:12.110 Of course, there's many more, but we don't care about them. 00:01:12.110 --> 00:01:14.200 And then we go to H2. 00:01:14.200 --> 00:01:17.560 And again, there's basically only two orbitals 00:01:17.560 --> 00:01:22.240 that we care about because the next higher principle quantum 00:01:22.240 --> 00:01:28.290 number has such high energy that we can just forget about them, 00:01:28.290 --> 00:01:31.620 because those states that derive from the higher principle 00:01:31.620 --> 00:01:36.300 quantum number are Rydberg states or complicated things, 00:01:36.300 --> 00:01:39.410 because they're at such high energy. 00:01:39.410 --> 00:01:46.200 And then it's a very small step to go from hydrogen to the AH 00:01:46.200 --> 00:01:49.530 molecules, because, well, we've got 00:01:49.530 --> 00:01:56.940 the electronic structure for the A atom, which is complicated-- 00:01:56.940 --> 00:02:01.350 more complicated than hydrogen. But because hydrogen only 00:02:01.350 --> 00:02:05.610 can make sigma bonds, because the p orbitals in hydrogen 00:02:05.610 --> 00:02:11.520 are so high that it's a simpler picture 00:02:11.520 --> 00:02:14.460 and can easily be understood. 00:02:14.460 --> 00:02:17.850 This is in all the textbooks, and it's 00:02:17.850 --> 00:02:22.150 more or less given to you as something to memorize. 00:02:22.150 --> 00:02:25.720 But there is a lot more to it than just memorization. 00:02:25.720 --> 00:02:29.890 And the important thing is that everything 00:02:29.890 --> 00:02:35.780 in LCAO-MO for diatomics is based on the periodic table. 00:02:35.780 --> 00:02:40.060 And the periodic table tells you about the ionization 00:02:40.060 --> 00:02:44.080 energies-- the periodicites of ionization energies. 00:02:44.080 --> 00:02:48.920 And so we can say for any non-integer-- 00:02:48.920 --> 00:02:51.190 Well, if we know what the ionization 00:02:51.190 --> 00:02:54.730 energy from a particular state is, 00:02:54.730 --> 00:02:57.310 we can use that ionization energy 00:02:57.310 --> 00:03:02.021 to derive a non-integer principle quantum number. 00:03:02.021 --> 00:03:05.370 This is all empirical. 00:03:05.370 --> 00:03:09.960 And then use that empirical principle quantum number 00:03:09.960 --> 00:03:12.500 to get the size of the orbital. 00:03:12.500 --> 00:03:16.290 And basically, the size is everything. 00:03:16.290 --> 00:03:18.450 Because everything is based on overlap, 00:03:18.450 --> 00:03:21.450 And the internuclear distance molecule 00:03:21.450 --> 00:03:26.790 is based on the relative sizes of the different atoms. 00:03:26.790 --> 00:03:31.826 And for different electronic states of the atoms, 00:03:31.826 --> 00:03:33.450 these sizes are different, because they 00:03:33.450 --> 00:03:40.050 have the ionization energy from that level implicitly 00:03:40.050 --> 00:03:41.640 expressed. 00:03:41.640 --> 00:03:45.390 So because you know about orbital sizes, 00:03:45.390 --> 00:03:49.830 and that the different atomic orbitals have different sizes, 00:03:49.830 --> 00:03:52.470 you can do an enormous amount as far 00:03:52.470 --> 00:03:54.870 as understanding the electronic structure 00:03:54.870 --> 00:03:57.920 of all diatomic molecules. 00:03:57.920 --> 00:04:06.060 Now, this-- when you have two states which have different 00:04:06.060 --> 00:04:07.170 energies-- 00:04:07.170 --> 00:04:15.204 We do something like this to describe the molecular orbitals 00:04:15.204 --> 00:04:16.079 that arise from them. 00:04:16.079 --> 00:04:17.450 That that's perturbation theory. 00:04:20.810 --> 00:04:24.490 So the solid line is the dominant character. 00:04:24.490 --> 00:04:29.260 The dotted line is the admix character of the other orbital. 00:04:29.260 --> 00:04:31.630 And this is all qualitative, but you 00:04:31.630 --> 00:04:35.500 have these different atomic orbital energies. 00:04:35.500 --> 00:04:38.600 You know them from the periodic table, basically. 00:04:38.600 --> 00:04:41.701 And so you can say, yes. 00:04:41.701 --> 00:04:43.450 There's going to be two orbitals arriving, 00:04:43.450 --> 00:04:46.720 derived from these two states. 00:04:46.720 --> 00:04:49.050 And one is polarized towards this atom. 00:04:49.050 --> 00:04:51.550 The other is polarized towards that atom. 00:04:51.550 --> 00:04:58.050 And so you get the shape of the orbital. 00:04:58.050 --> 00:05:03.360 And it's actually useful for saying, well, 00:05:03.360 --> 00:05:07.920 if I were to do chemistry, which end of this is electronegative, 00:05:07.920 --> 00:05:09.540 and which end is positive? 00:05:09.540 --> 00:05:15.370 Or which end of this would attach to a metal surface? 00:05:15.370 --> 00:05:21.080 Would it attach pointing into the surface or lying down? 00:05:21.080 --> 00:05:22.510 And there's all sorts of insights, 00:05:22.510 --> 00:05:25.240 if you can draw these sorts of pictures. 00:05:25.240 --> 00:05:28.330 And this is what we do as physical chemists. 00:05:28.330 --> 00:05:33.380 We take the crudest model, and we say, OK. 00:05:33.380 --> 00:05:36.140 We understand the important features. 00:05:36.140 --> 00:05:39.190 And as we discover new, important features, 00:05:39.190 --> 00:05:42.230 we build them in, or we forget about them 00:05:42.230 --> 00:05:45.440 because they're too subtle. 00:05:45.440 --> 00:05:49.550 And we're always looking for something where 00:05:49.550 --> 00:05:51.950 the crude picture doesn't work. 00:05:51.950 --> 00:05:58.610 And we then find the important thing that is needed. 00:05:58.610 --> 00:06:01.640 And for example, if you were to look 00:06:01.640 --> 00:06:06.670 at the molecular orbital diagram for C2, which 00:06:06.670 --> 00:06:09.689 is a perfectly legitimate gaseous molecule, 00:06:09.689 --> 00:06:11.980 you'll see that there's a little bit of ambiguity about 00:06:11.980 --> 00:06:14.330 which is the ground state. 00:06:14.330 --> 00:06:16.030 And this is an important thing. 00:06:16.030 --> 00:06:18.700 And there was a big controversy about this that 00:06:18.700 --> 00:06:20.690 was settled by spectroscopy. 00:06:20.690 --> 00:06:21.190 OK. 00:06:21.190 --> 00:06:25.750 So if you can build an intuitive picture 00:06:25.750 --> 00:06:30.850 for all diatomic molecules, you can also 00:06:30.850 --> 00:06:35.100 build an intuitive picture for what 00:06:35.100 --> 00:06:37.006 we could call chromophores. 00:06:40.270 --> 00:06:42.820 So there are a lot of molecules that are 00:06:42.820 --> 00:06:46.640 larger than diatomic molecules. 00:06:46.640 --> 00:06:51.650 But the electronic structure is mostly nothing, 00:06:51.650 --> 00:06:53.900 except a few atoms that are close to each other, where 00:06:53.900 --> 00:06:56.960 there's a double bond or there's something special. 00:06:56.960 --> 00:07:03.620 And so if you can do LCAO-MO for diatomic molecules, 00:07:03.620 --> 00:07:07.250 you can also address electronic structure 00:07:07.250 --> 00:07:09.740 in much larger molecules, because it's 00:07:09.740 --> 00:07:13.400 really due to a few important atoms 00:07:13.400 --> 00:07:17.210 or several different groups of important atoms. 00:07:17.210 --> 00:07:19.700 And so the idea is, in here, enable 00:07:19.700 --> 00:07:23.495 you to talk about the electronic structure of almost anything. 00:07:27.350 --> 00:07:31.010 And one of the things that we care about is spectroscopy. 00:07:35.270 --> 00:07:38.600 How do we learn about the structure of a molecule? 00:07:38.600 --> 00:07:42.760 And mostly, you learn about it from doing 00:07:42.760 --> 00:07:44.200 electronic spectroscopy. 00:07:44.200 --> 00:07:47.290 You do transitions between the ground state and some higher 00:07:47.290 --> 00:07:48.640 states. 00:07:48.640 --> 00:07:51.800 And you want to be able to predict 00:07:51.800 --> 00:07:53.670 what you are going to see. 00:07:53.670 --> 00:07:57.320 And for example-- and I think you may hear about this 00:07:57.320 --> 00:07:59.840 a little bit more-- 00:07:59.840 --> 00:08:03.280 if you know the spectrum of nitrogen, 00:08:03.280 --> 00:08:08.140 it doesn't start absorbing until far into the vacuum UV. 00:08:08.140 --> 00:08:10.480 And then go one atom over-- 00:08:10.480 --> 00:08:15.730 the spectrum of oxygen. Well, that defines the vacuum UV. 00:08:15.730 --> 00:08:18.640 It starts to absorb around 200 nanometers. 00:08:18.640 --> 00:08:25.520 And 2 and 1/2 billion years ago, oxygen 00:08:25.520 --> 00:08:29.060 started to enter into the atmosphere and profoundly 00:08:29.060 --> 00:08:30.890 changed life on Earth. 00:08:30.890 --> 00:08:33.890 Because with the vacuum UV radiation, 00:08:33.890 --> 00:08:36.950 nothing could live on the surface of land, 00:08:36.950 --> 00:08:39.820 because it would all be killed by this hard UV. 00:08:42.500 --> 00:08:49.110 And so everything was underwater at the bottom of the ocean. 00:08:49.110 --> 00:08:54.370 But all of a sudden, when oxygen appeared, there was protection. 00:08:54.370 --> 00:08:58.600 And then there's additional protection farther out 00:08:58.600 --> 00:09:02.310 towards the visible from ozone. 00:09:02.310 --> 00:09:09.210 It's not such a good shield, but it absorbs between 200 and 350 00:09:09.210 --> 00:09:10.380 nanometers. 00:09:10.380 --> 00:09:14.100 And it protects us, too. 00:09:14.100 --> 00:09:17.340 But the big thing is oxygen. And why should oxygen 00:09:17.340 --> 00:09:20.107 be so different from nitrogen? 00:09:20.107 --> 00:09:21.690 And maybe you should think about that. 00:09:24.260 --> 00:09:24.760 OK. 00:09:24.760 --> 00:09:28.870 I should have mentioned the schedule up there. 00:09:28.870 --> 00:09:31.630 I'm going to give another lecture on perturbation 00:09:31.630 --> 00:09:34.360 theory-- one of my favorites, which I thought I wasn't 00:09:34.360 --> 00:09:37.930 going to get to give on Friday. 00:09:37.930 --> 00:09:40.570 And then Troy is going to give two lectures 00:09:40.570 --> 00:09:41.710 on quantum chemistry. 00:09:41.710 --> 00:09:46.690 And there's going to be significant lab experience. 00:09:46.690 --> 00:09:49.270 And you'll hear about that from the TAs. 00:09:49.270 --> 00:09:53.380 And so you will do real calculations. 00:09:53.380 --> 00:10:01.300 So LCAO-MO could be the organization structure 00:10:01.300 --> 00:10:04.000 for quantitative calculations. 00:10:04.000 --> 00:10:08.740 But quantitative calculations are usually huge basis sets, 00:10:08.740 --> 00:10:12.130 and you lose the LCAO completely. 00:10:12.130 --> 00:10:15.820 One could take the results of such a calculation, 00:10:15.820 --> 00:10:20.637 and then project it onto a LCAO-MO picture. 00:10:20.637 --> 00:10:21.970 And that's generally what we do. 00:10:21.970 --> 00:10:26.410 We reduce experiments, we reduce theory, to something 00:10:26.410 --> 00:10:28.730 that we can intuit. 00:10:28.730 --> 00:10:31.601 And it's really important to have intuition. 00:10:31.601 --> 00:10:32.100 OK. 00:10:32.100 --> 00:10:42.860 Huckel theory-- Huckel theory is another kind 00:10:42.860 --> 00:10:49.570 of non-rigorous theory. 00:10:49.570 --> 00:10:54.070 In fact, it's laughable in its simplicity. 00:10:54.070 --> 00:10:59.770 And the idea is that even such a crude theory can make sense out 00:10:59.770 --> 00:11:03.790 of huge families of molecules and enable 00:11:03.790 --> 00:11:06.910 you to be quantitative about things without having 00:11:06.910 --> 00:11:09.550 to do a huge calculation. 00:11:09.550 --> 00:11:15.090 Now, you almost always have to diagonize a matrix. 00:11:15.090 --> 00:11:19.050 And because usually, in Huckel theory, 00:11:19.050 --> 00:11:25.420 you're dealing with more than two or three relevant orbitals, 00:11:25.420 --> 00:11:27.110 it's a big deal. 00:11:27.110 --> 00:11:30.070 I don't know whether you have experience 00:11:30.070 --> 00:11:32.200 with matrix diagonalization routines, 00:11:32.200 --> 00:11:33.670 but there are many of them. 00:11:33.670 --> 00:11:35.620 And you're going to have experience with them. 00:11:39.610 --> 00:11:43.300 But the things you put into them have absolutely nothing 00:11:43.300 --> 00:11:45.730 to do with a real Hamiltonian. 00:11:45.730 --> 00:11:48.130 It's a make up picture. 00:11:48.130 --> 00:11:51.610 It's a picture based on, basically, two parameters-- 00:11:51.610 --> 00:11:52.810 alpha and beta. 00:11:52.810 --> 00:11:55.000 And I'll talk about this. 00:11:55.000 --> 00:11:59.590 And these are parameters that are agreed upon 00:11:59.590 --> 00:12:01.450 by international committee-- 00:12:01.450 --> 00:12:02.650 It's not a committee. 00:12:02.650 --> 00:12:05.020 But people say, OK. 00:12:05.020 --> 00:12:06.850 This is the value of alpha and beta 00:12:06.850 --> 00:12:09.805 that we're going to use to describe many problems. 00:12:12.830 --> 00:12:15.700 But I just want to make sure you understand 00:12:15.700 --> 00:12:19.270 that Huckel theory is only a little bit more 00:12:19.270 --> 00:12:25.250 ridiculous than LCAO-MO theory. 00:12:25.250 --> 00:12:32.120 Because in LCAO-MO, you really have some semi-empirical sense 00:12:32.120 --> 00:12:33.360 of how big orbitals are. 00:12:33.360 --> 00:12:34.984 And once you know how big orbitals are, 00:12:34.984 --> 00:12:36.900 you know what's going on. 00:12:36.900 --> 00:12:40.840 In this, you just have a couple of parameters that says, OK. 00:12:40.840 --> 00:12:41.910 These are the rules. 00:12:41.910 --> 00:12:45.360 Let's now apply that. 00:12:45.360 --> 00:12:45.860 OK. 00:12:48.788 --> 00:12:50.730 I'm carrying around too much stuff. 00:12:53.580 --> 00:13:00.420 So organic chemists are really wonderful, 00:13:00.420 --> 00:13:04.740 because they give you an abbreviated way of drawing 00:13:04.740 --> 00:13:06.660 a molecular structure. 00:13:06.660 --> 00:13:09.570 And almost everyone in this room is 00:13:09.570 --> 00:13:11.880 gifted in being able to visualize 00:13:11.880 --> 00:13:13.860 three-dimensional structures. 00:13:13.860 --> 00:13:16.800 Physical chemists tend not to be so gifted. 00:13:16.800 --> 00:13:20.100 But here is something where we don't have any hydrogens-- 00:13:20.100 --> 00:13:22.590 I mean, there are hydrogens, but they're implied. 00:13:22.590 --> 00:13:24.720 And we don't put carbons here. 00:13:24.720 --> 00:13:28.780 We just say, at every vertex, there is a carbon atom. 00:13:28.780 --> 00:13:31.140 And so we consider these things. 00:13:31.140 --> 00:13:36.070 And every organic chemist knows, if I draw something like this, 00:13:36.070 --> 00:13:39.390 that either I was stupid and I drew 00:13:39.390 --> 00:13:44.110 something that was impossible or what the structure is. 00:13:44.110 --> 00:13:44.610 OK. 00:13:44.610 --> 00:13:46.770 Now, this is a conjugated system. 00:13:46.770 --> 00:13:49.630 We have double bonds and single bonds. 00:13:49.630 --> 00:13:55.980 And that kind of thing is known to be unusually stable. 00:13:55.980 --> 00:14:00.870 And in order for it to work, it has to be planar. 00:14:00.870 --> 00:14:04.890 If you have a conjugated system which is not planar, 00:14:04.890 --> 00:14:09.180 then it's not as stable. 00:14:09.180 --> 00:14:17.280 So Huckel theory is based on the existence of unusual stability 00:14:17.280 --> 00:14:20.600 of conjugated systems. 00:14:20.600 --> 00:14:28.730 And it can be extended to non-conjugated systems-- 00:14:28.730 --> 00:14:32.870 to non-planar systems-- and we have rules for how to do that. 00:14:32.870 --> 00:14:41.220 But the simple rules are, you start with a picture, 00:14:41.220 --> 00:14:44.660 and you can write down a Hamiltonian. 00:14:44.660 --> 00:14:48.130 And it's a toy model, but it's still something 00:14:48.130 --> 00:14:49.990 that the computer has to diagonalize. 00:14:58.440 --> 00:15:02.990 So when we're doing a variational calculation 00:15:02.990 --> 00:15:04.790 expressed in matrix language-- 00:15:04.790 --> 00:15:06.740 We have a Hamiltonian matrix. 00:15:06.740 --> 00:15:15.210 We have eigenvectors, and we have eigenvalues. 00:15:15.210 --> 00:15:18.600 And we have the overlap matrix. 00:15:18.600 --> 00:15:26.580 And we have something where this is the alpha-- 00:15:26.580 --> 00:15:32.580 where, here, we can have non-orthonormal basis 00:15:32.580 --> 00:15:34.050 functions. 00:15:34.050 --> 00:15:37.640 And this makes them normalized. 00:15:37.640 --> 00:15:43.290 So these are the orthonormalized basis functions or basis 00:15:43.290 --> 00:15:44.470 vectors. 00:15:44.470 --> 00:15:47.010 And this is the kind of equation we have to solve. 00:15:47.010 --> 00:15:50.340 It's the generalized eigenvalue equation. 00:15:50.340 --> 00:15:54.420 And we don't like this, because it 00:15:54.420 --> 00:15:57.930 doesn't have the simplicity of just a Hamiltonian, where 00:15:57.930 --> 00:16:02.040 we have matrix elements along the diagonal and somewhere 00:16:02.040 --> 00:16:03.090 else. 00:16:03.090 --> 00:16:08.790 And the idea is, we want to take the secular determinate 00:16:08.790 --> 00:16:14.400 and make it 0 by adjusting values of the energy 00:16:14.400 --> 00:16:16.830 differences along the diagonal. 00:16:16.830 --> 00:16:20.770 But when we have this overlap matrix, it's complicated. 00:16:20.770 --> 00:16:24.360 Now, there's a way of dealing with the generalized eigenvalue 00:16:24.360 --> 00:16:31.890 equations, but one way to deal with it is to say s 00:16:31.890 --> 00:16:34.065 is equal to the unit matrix. 00:16:39.910 --> 00:16:41.900 You can do anything you want. 00:16:41.900 --> 00:16:47.620 And it's basically saying, there is no overlap between orbitals 00:16:47.620 --> 00:16:49.902 on adjacent atoms. 00:16:49.902 --> 00:16:51.610 We're going to neglect it, and then we're 00:16:51.610 --> 00:16:54.520 going to bring it back if we need it. 00:16:54.520 --> 00:16:56.500 But it's a wonderful simplification, 00:16:56.500 --> 00:17:01.840 because it enables you to write a simple, effective 00:17:01.840 --> 00:17:09.230 Hamiltonian, which looks just like h c alpha is 00:17:09.230 --> 00:17:13.800 equal to e alpha c alpha. 00:17:13.800 --> 00:17:14.300 OK. 00:17:14.300 --> 00:17:17.530 This, we know how to solve. 00:17:17.530 --> 00:17:19.890 And we can use the same procedure. 00:17:26.420 --> 00:17:27.940 OK. 00:17:27.940 --> 00:17:28.960 So these are the rules. 00:17:39.550 --> 00:17:45.760 If we have a planar molecule, we can say there are p orbitals-- 00:17:45.760 --> 00:17:49.030 one on each carbon atom or one on each atom that's 00:17:49.030 --> 00:17:54.980 not hydrogen, which are perpendicular to the plane 00:17:54.980 --> 00:17:57.260 of the molecule. 00:17:57.260 --> 00:18:00.730 So easy orbitals. 00:18:00.730 --> 00:18:04.330 And those are special. 00:18:04.330 --> 00:18:07.510 They give rise to pi bonds. 00:18:07.510 --> 00:18:14.470 Pi bonds are bonds where there is one plane of symmetry 00:18:14.470 --> 00:18:25.510 containing the bond, And then there are p x, and p y, and s. 00:18:25.510 --> 00:18:27.255 And these give rise to sigma bonds. 00:18:29.850 --> 00:18:35.210 So we have pi bonds and sigma bonds. 00:18:35.210 --> 00:18:38.554 Never the twain shall meet. 00:18:38.554 --> 00:18:40.220 And we don't care about the sigma bonds, 00:18:40.220 --> 00:18:42.530 because anybody can make sigma bonds. 00:18:42.530 --> 00:18:48.950 But only the special, perpendicular-to-the-plane 00:18:48.950 --> 00:18:52.280 guys, which are responsible for the fact that the molecule 00:18:52.280 --> 00:18:53.690 likes to be planar. 00:18:53.690 --> 00:18:58.280 And so we're only going to consider these p z orbitals-- 00:18:58.280 --> 00:18:59.530 one on each atom. 00:19:04.580 --> 00:19:12.080 So we don't care about no pi-sigma interactions. 00:19:15.540 --> 00:19:18.160 And we're going to neglect the sigma orbitals, because they 00:19:18.160 --> 00:19:19.571 take care of themselves. 00:19:22.277 --> 00:19:25.610 And we can do anything we want. 00:19:25.610 --> 00:19:28.130 It's just a question of, if we make 00:19:28.130 --> 00:19:31.700 too many ridiculous assumptions, we'll get ridiculous results. 00:19:31.700 --> 00:19:34.880 And this has been time tested, and it 00:19:34.880 --> 00:19:36.830 gives pretty useful stuff. 00:19:36.830 --> 00:19:39.980 And it provides a framework for making arguments 00:19:39.980 --> 00:19:44.030 about molecular structure and molecular reactivity. 00:19:44.030 --> 00:19:48.060 In organic chemistry you learn about resonance forms. 00:19:48.060 --> 00:19:52.340 And this is compatible with generating the resonance forms 00:19:52.340 --> 00:19:54.650 and saying, what is the relative importance? 00:19:54.650 --> 00:19:56.930 And what is the charge distribution, 00:19:56.930 --> 00:19:59.360 and bond strength, and everything like that? 00:19:59.360 --> 00:20:04.080 So it's really useful using the most primitive tools 00:20:04.080 --> 00:20:08.240 that organic chemists introduce at an early stage 00:20:08.240 --> 00:20:09.829 in your education. 00:20:09.829 --> 00:20:11.870 And that's one of the reasons why a lot of people 00:20:11.870 --> 00:20:14.395 become organic chemists, because it's so beautiful. 00:20:17.310 --> 00:20:18.990 OK. 00:20:18.990 --> 00:20:25.270 So our h i j matrix-- 00:20:25.270 --> 00:20:29.550 So we have a bunch of matrix elements, and we say, OK. 00:20:29.550 --> 00:20:34.080 h i i is equal to alpha. 00:20:34.080 --> 00:20:41.950 So every carbon atom has its alpha value. 00:20:41.950 --> 00:20:44.760 It's the same for all carbon atoms, 00:20:44.760 --> 00:20:46.395 regardless of who is nearby. 00:20:49.180 --> 00:20:56.970 And we have h I i plus or minus 1 adjacent atoms, 00:20:56.970 --> 00:20:59.240 and it's beta. 00:20:59.240 --> 00:20:59.850 That's it. 00:20:59.850 --> 00:21:03.380 That's the whole ball game. 00:21:03.380 --> 00:21:06.670 And it's really a simple-- 00:21:06.670 --> 00:21:10.750 There's far less here than in LCAO-MO. 00:21:13.830 --> 00:21:15.390 But it's still a toy. 00:21:15.390 --> 00:21:19.380 Both are toy models, and they're both very useful 00:21:19.380 --> 00:21:21.880 OK. 00:21:21.880 --> 00:21:31.875 And everything else which is not diagonal or off diagonal by 1-- 00:21:31.875 --> 00:21:32.375 0. 00:21:36.070 --> 00:21:38.080 That's really convenient. 00:21:38.080 --> 00:21:41.830 And so you can draw the h matrix, 00:21:41.830 --> 00:21:46.210 regardless of what it looks like, as alpha, alpha, alpha, 00:21:46.210 --> 00:21:49.120 alpha, et cetera, along the diagonal, 00:21:49.120 --> 00:21:56.430 and beta, beta, beta, et cetera, beta, along the near diagonal. 00:21:56.430 --> 00:22:00.240 And if it's a ring, you have a beta here, here, and here. 00:22:03.740 --> 00:22:05.060 So that's pretty simple. 00:22:05.060 --> 00:22:08.200 So we have 0s, 0s-- 00:22:08.200 --> 00:22:12.410 so that it has a tri-diagonal structure with something 00:22:12.410 --> 00:22:14.360 up here and there. 00:22:14.360 --> 00:22:17.880 Never forget this thing here and there, 00:22:17.880 --> 00:22:20.280 which is present when you have a ring, 00:22:20.280 --> 00:22:23.440 and it's not present when you don't. 00:22:23.440 --> 00:22:23.940 That's it. 00:22:23.940 --> 00:22:25.680 That's Huckel theory. 00:22:25.680 --> 00:22:27.710 It's just there. 00:22:27.710 --> 00:22:28.290 OK. 00:22:28.290 --> 00:22:34.960 And so now, things can be more complicated. 00:22:41.420 --> 00:22:43.150 So if we're not content with what 00:22:43.150 --> 00:22:45.370 we get from the really primitive theory, 00:22:45.370 --> 00:22:48.370 we can do something like saying, well, we 00:22:48.370 --> 00:22:53.450 can make beta be dependent on internuclear distance. 00:22:53.450 --> 00:22:55.240 If the molecule, for some reason, 00:22:55.240 --> 00:22:58.550 is constrained to have not equal bonds lengths. 00:22:58.550 --> 00:23:02.690 So we can add an additional parameter-- some kind of reason 00:23:02.690 --> 00:23:06.500 for this beta to be dependent on r. 00:23:06.500 --> 00:23:10.230 But that's already getting sophisticated. 00:23:10.230 --> 00:23:16.770 And heteroatoms-- In other words, 00:23:16.770 --> 00:23:19.970 if you have a nitrogen instead of a carbon in a benzene-type 00:23:19.970 --> 00:23:22.700 ring, you can have-- 00:23:22.700 --> 00:23:27.550 So, well, nitrogen is different from carbon. 00:23:27.550 --> 00:23:29.465 It has a different-- 00:23:29.465 --> 00:23:32.740 In LCAO theory, the distance-- 00:23:32.740 --> 00:23:35.110 The ionization energy for nitrogen 00:23:35.110 --> 00:23:36.520 is different from carbon. 00:23:36.520 --> 00:23:38.550 It's larger. 00:23:38.550 --> 00:23:42.290 And so heteroatoms can be included if you 00:23:42.290 --> 00:23:44.990 use a different value of alpha. 00:23:44.990 --> 00:23:50.160 And now, alpha and beta are both negative. 00:23:50.160 --> 00:23:52.821 Now, this is a little bit fraudulent. 00:23:52.821 --> 00:23:53.320 Yeah. 00:23:53.320 --> 00:23:55.540 AUDIENCE: Do you also have to pick a new beta? 00:23:55.540 --> 00:23:56.360 Or-- 00:23:56.360 --> 00:23:57.190 ROBERT FIELD: Yes. 00:23:57.190 --> 00:23:58.900 But the main thing is alpha. 00:23:58.900 --> 00:24:00.660 AUDIENCE: I see. 00:24:00.660 --> 00:24:01.870 Why is that? 00:24:01.870 --> 00:24:02.980 ROBERT FIELD: Why is that? 00:24:02.980 --> 00:24:11.410 Because beta comes from overlap, even though we're neglecting 00:24:11.410 --> 00:24:12.950 overlap. 00:24:12.950 --> 00:24:19.150 And so the bond distances between normal and heteroatoms 00:24:19.150 --> 00:24:22.880 are not usually that different. 00:24:22.880 --> 00:24:26.320 But the thing is, you put what you need into the model. 00:24:26.320 --> 00:24:28.870 And the first thing you do is, you 00:24:28.870 --> 00:24:30.400 solve the most simple model. 00:24:30.400 --> 00:24:33.160 And you say, this is not quite what I wanted. 00:24:33.160 --> 00:24:36.920 And so I allow a couple of extra degrees of freedom. 00:24:36.920 --> 00:24:40.760 And it's really instructive how these things work. 00:24:40.760 --> 00:24:44.950 But the thing is, you're not calculating the matrix element 00:24:44.950 --> 00:24:46.930 of a real operator. 00:24:46.930 --> 00:24:48.920 It's all make believe. 00:24:48.920 --> 00:24:52.810 But it's really powerful, because what you're comparing 00:24:52.810 --> 00:24:55.090 is families of molecules. 00:24:55.090 --> 00:24:58.400 And the reality might be really complicated, 00:24:58.400 --> 00:25:00.940 but the complexity in each member of the family 00:25:00.940 --> 00:25:02.320 is pretty much the same. 00:25:02.320 --> 00:25:04.960 And what it's allowing you to see 00:25:04.960 --> 00:25:07.042 is, what are the differences? 00:25:07.042 --> 00:25:10.890 It allows you to see the big picture. 00:25:10.890 --> 00:25:12.420 I love this. 00:25:12.420 --> 00:25:15.900 And in fact, at an early stage of my education, 00:25:15.900 --> 00:25:18.250 I thought Huckel theory was wonderful. 00:25:18.250 --> 00:25:21.130 And it was what got me interested in quantum 00:25:21.130 --> 00:25:21.630 mechanics. 00:25:21.630 --> 00:25:24.327 Because you normally see Huckel theory before you 00:25:24.327 --> 00:25:25.910 know anything about quantum mechanics, 00:25:25.910 --> 00:25:28.190 because it's just a game. 00:25:28.190 --> 00:25:30.120 OK So heteroatoms-- 00:25:30.120 --> 00:25:33.570 You can fiddle with alpha. 00:25:33.570 --> 00:25:40.180 Now, the ionization energies for carbon and nitrogen 00:25:40.180 --> 00:25:42.371 are not that different. 00:25:42.371 --> 00:25:42.870 I'm sorry. 00:25:42.870 --> 00:25:46.410 They're very different, but the effect 00:25:46.410 --> 00:25:50.910 on the alpha value in Huckel theory is very small. 00:25:50.910 --> 00:25:53.000 Well, so it is. 00:25:53.000 --> 00:26:00.170 But the more electronegative or the higher the ionization 00:26:00.170 --> 00:26:05.660 energy, the alpha value becomes increasingly negative. 00:26:05.660 --> 00:26:07.790 Now, I was starting to say something is fraudulent, 00:26:07.790 --> 00:26:12.590 and I was distracted by a really good question. 00:26:12.590 --> 00:26:14.840 Alpha is on the diagonal. 00:26:14.840 --> 00:26:21.840 And we know we can determine the sign of a diagonal element. 00:26:21.840 --> 00:26:25.400 And we know we can't determine the sign 00:26:25.400 --> 00:26:28.070 of an off-diagonal element. 00:26:28.070 --> 00:26:32.300 But we say that alpha and beta are both less than 0. 00:26:34.890 --> 00:26:39.350 And the reason for this is that, when you saw the secular 00:26:39.350 --> 00:26:41.600 equation, especially-- 00:26:45.130 --> 00:26:48.820 You get two eigenvalues-- one where 00:26:48.820 --> 00:26:52.360 you have minus beta and one where you have plus beta. 00:26:52.360 --> 00:26:54.640 And so in a sense, beta is present, 00:26:54.640 --> 00:26:56.680 but it's just a sign choice. 00:26:56.680 --> 00:27:00.520 And since alpha and beta have to do with stability, 00:27:00.520 --> 00:27:03.200 we just say alpha is negative. 00:27:03.200 --> 00:27:04.000 We know that. 00:27:04.000 --> 00:27:08.900 And beta is chosen to be always negative. 00:27:08.900 --> 00:27:12.310 There's no, you could have had beta be positive. 00:27:12.310 --> 00:27:13.840 And you could do all the theory. 00:27:13.840 --> 00:27:18.961 It's just a lot more complicated explaining all the cases. 00:27:18.961 --> 00:27:19.460 All right. 00:27:19.460 --> 00:27:23.738 So let's continue with this. 00:27:23.738 --> 00:27:25.490 And so we can do heteroatoms. 00:27:32.840 --> 00:27:37.150 So if we have a non-planar system, 00:27:37.150 --> 00:27:42.190 we can say that beta is a function of the dihedral angle. 00:27:42.190 --> 00:27:43.160 We can put that in. 00:27:43.160 --> 00:27:45.290 We can do anything we want. 00:27:45.290 --> 00:27:47.840 We have a molecule. 00:27:47.840 --> 00:27:49.840 We're trying to describe its properties 00:27:49.840 --> 00:27:53.020 relative to the normal members of the group, which 00:27:53.020 --> 00:27:55.570 are planar and no heteroatoms. 00:27:55.570 --> 00:27:58.120 And we can do stuff that will accommodate 00:27:58.120 --> 00:28:02.020 these interesting differences, which you can impose 00:28:02.020 --> 00:28:06.310 by you putting the molecule in a constrained environment 00:28:06.310 --> 00:28:11.970 or doing stuff that distorts the geometry. 00:28:11.970 --> 00:28:13.855 So we do this. 00:28:17.700 --> 00:28:19.700 So when we do this, we get a Hamiltonian. 00:28:19.700 --> 00:28:25.960 We get the energies of the orbitals, 00:28:25.960 --> 00:28:35.370 and we get the eigenvector that corresponds to each energy. 00:28:35.370 --> 00:28:37.570 And the total energy involves the sum 00:28:37.570 --> 00:28:41.500 over the energies of each of the occupied orbitals-- the number 00:28:41.500 --> 00:28:43.240 of electrons in that. 00:28:43.240 --> 00:28:50.485 And we can also get bond order and charge. 00:28:53.608 --> 00:28:58.660 So sometimes, we want to know, what 00:28:58.660 --> 00:29:01.990 is the charge on each atom if the charge is 00:29:01.990 --> 00:29:04.880 going to be different from 0? 00:29:04.880 --> 00:29:08.570 Because that also controls chemistry. 00:29:08.570 --> 00:29:12.110 Negatively charged atoms are sought out 00:29:12.110 --> 00:29:14.300 by certain classes of reactants. 00:29:14.300 --> 00:29:18.370 Anyway, so you get all this stuff. 00:29:18.370 --> 00:29:21.510 And of course, some of it will require 00:29:21.510 --> 00:29:27.000 a little bit of patchwork, but you do this for your career. 00:29:27.000 --> 00:29:29.160 And you discover that there are certain things 00:29:29.160 --> 00:29:31.210 that I know how to handle. 00:29:31.210 --> 00:29:33.510 And I use my favorite parameters for it. 00:29:33.510 --> 00:29:36.520 And you get closer to the truth. 00:29:36.520 --> 00:29:39.400 And you always want to be surprised, 00:29:39.400 --> 00:29:46.150 because when the crude theory cannot be made consistent with 00:29:46.150 --> 00:29:49.630 observations, you know you did something special. 00:29:49.630 --> 00:29:54.040 You have a molecule, which has a property which is unexpected, 00:29:54.040 --> 00:29:54.880 and we like that. 00:29:58.360 --> 00:29:59.930 OK. 00:29:59.930 --> 00:30:05.820 So when you're doing quantum chemistry, 00:30:05.820 --> 00:30:06.900 there are five steps-- 00:30:06.900 --> 00:30:11.060 or when you're doing molecular orbital theory. 00:30:11.060 --> 00:30:15.200 And this is one of Troy's rules. 00:30:15.200 --> 00:30:17.210 We have a five-step procedure. 00:30:17.210 --> 00:30:25.580 And we define the atomic orbital basis set. 00:30:28.940 --> 00:30:31.550 And the basis set is one p z orbital per atom. 00:30:35.160 --> 00:30:45.280 And so we can have a molecular orbital, which is the sum i 00:30:45.280 --> 00:30:52.260 equals 1 to n c i u p z i. 00:30:52.260 --> 00:30:56.160 So this is the p z orbital on the i-th atom. 00:30:56.160 --> 00:31:00.116 This is the coefficient for this particular linear combination. 00:31:04.141 --> 00:31:04.640 OK. 00:31:04.640 --> 00:31:09.190 So we have a bunch of molecular orbitals. 00:31:09.190 --> 00:31:16.140 Next step-- compute h and s. 00:31:16.140 --> 00:31:21.860 But s is 0 because we made this ridiculous assumption. 00:31:21.860 --> 00:31:24.510 It's convenient. 00:31:24.510 --> 00:31:28.090 This is called the complete neglect of overlap. 00:31:28.090 --> 00:31:30.430 And we can do that. 00:31:30.430 --> 00:31:32.730 It's wrong. 00:31:32.730 --> 00:31:36.900 But when you include overlap, it leads 00:31:36.900 --> 00:31:39.300 to greater complexity of the calculation, 00:31:39.300 --> 00:31:42.550 and not much improvement in the results. 00:31:42.550 --> 00:31:47.160 And so we normally don't worry about the overlap. 00:31:47.160 --> 00:31:49.770 So all we care about is this. 00:31:49.770 --> 00:31:56.040 And as I said, h has this structure 00:31:56.040 --> 00:32:01.360 of alpha, beta, beta-- 00:32:01.360 --> 00:32:04.301 tri-diagonal structure and maybe something up here. 00:32:04.301 --> 00:32:04.800 That's it. 00:32:08.700 --> 00:32:15.360 Then we diagonalize the Hamiltonian. 00:32:18.420 --> 00:32:24.680 And so this gives us the eigenvectors or eigenvalues-- 00:32:24.680 --> 00:32:28.010 the energies and eigenvectors associated 00:32:28.010 --> 00:32:31.070 with each orbital energy. 00:32:31.070 --> 00:32:34.870 So we fill electrons into orbitals. 00:32:34.870 --> 00:32:40.710 Now, for benzene, what you would get-- 00:32:40.710 --> 00:32:43.150 Now, this is important. 00:32:43.150 --> 00:32:47.340 How many carbon atoms in benzene? 00:32:47.340 --> 00:32:48.310 Right. 00:32:48.310 --> 00:32:50.890 And how many orbitals will you get 00:32:50.890 --> 00:32:54.240 from six primitive orbitals? 00:32:54.240 --> 00:32:55.440 Right. 00:32:55.440 --> 00:32:58.800 And so there are six energy levels. 00:32:58.800 --> 00:33:04.530 And it happens that when you solve the secular equation, 00:33:04.530 --> 00:33:05.940 you get this pattern. 00:33:16.450 --> 00:33:19.090 This is a problem where the number of nodal planes 00:33:19.090 --> 00:33:21.160 determines the order of energy. 00:33:21.160 --> 00:33:24.070 And so if you have benzene, you can 00:33:24.070 --> 00:33:29.740 imagine that the orbitals that you can have will be no nodes-- 00:33:29.740 --> 00:33:32.830 nodal planes-- one nodal plane. 00:33:32.830 --> 00:33:36.130 And you could have the nodal planes between the atoms 00:33:36.130 --> 00:33:38.410 or through atoms-- 00:33:38.410 --> 00:33:42.860 and two nodal planes or three. 00:33:42.860 --> 00:33:46.680 You can't have any more with six atoms. 00:33:46.680 --> 00:33:52.490 And so you almost don't have to solve this secular equation 00:33:52.490 --> 00:33:53.210 at all. 00:33:53.210 --> 00:33:56.480 You can anticipate what the structure 00:33:56.480 --> 00:34:00.630 is going to be just by saying, no nodes, one node, two 00:34:00.630 --> 00:34:02.240 node, three node. 00:34:02.240 --> 00:34:07.140 And you can even anticipate double degeneracies, 00:34:07.140 --> 00:34:09.989 because if you have two nodal planes, 00:34:09.989 --> 00:34:17.489 you can put them through opposite bonds 00:34:17.489 --> 00:34:19.540 or through opposite atoms. 00:34:19.540 --> 00:34:25.060 And those are examples of the forms that you would deal with. 00:34:25.060 --> 00:34:28.949 The only thing you can't do is to know what the order 00:34:28.949 --> 00:34:31.389 the energies are. 00:34:31.389 --> 00:34:32.860 But there are tricks for that, too. 00:34:35.949 --> 00:34:38.380 The tricks for that include-- 00:34:38.380 --> 00:34:42.244 The sum of the eigenvalues is equal to sum 00:34:42.244 --> 00:34:43.160 of the diagonal nodes. 00:34:46.210 --> 00:34:50.080 And so we know that the six orbital energies 00:34:50.080 --> 00:34:52.530 will sum to 0-- 00:34:52.530 --> 00:34:55.557 will sum to 6 alpha. 00:34:55.557 --> 00:34:58.100 The betas go away. 00:34:58.100 --> 00:35:00.620 And there are other tricks that you can do, 00:35:00.620 --> 00:35:02.780 but generally, you solve the equation. 00:35:02.780 --> 00:35:09.110 You don't want to push your requirements for symmetry 00:35:09.110 --> 00:35:10.010 too far. 00:35:10.010 --> 00:35:13.040 So we end up doing that. 00:35:13.040 --> 00:35:18.550 And then stick diagrams-- 00:35:22.700 --> 00:35:27.500 We fill electrons into orbitals in energy order. 00:35:27.500 --> 00:35:29.970 And for benzene, there are six of them. 00:35:29.970 --> 00:35:35.675 And so this is, then, the lowest energy state of benzene. 00:35:38.740 --> 00:35:43.570 This is called the independent electron approximation. 00:35:43.570 --> 00:35:46.010 They don't know about each other. 00:35:46.010 --> 00:35:48.860 This is illegal. 00:35:48.860 --> 00:35:50.360 But it's legal in a sense, if you 00:35:50.360 --> 00:35:52.730 have two electrons in every orbital, 00:35:52.730 --> 00:35:54.950 you have nothing but singlet states. 00:35:54.950 --> 00:35:58.040 The ground state is always going to be a singlet state 00:35:58.040 --> 00:36:01.130 unless you have something really weird going on-- 00:36:01.130 --> 00:36:03.450 like in O2-- 00:36:03.450 --> 00:36:05.690 But that's not Huckel theory. 00:36:05.690 --> 00:36:09.190 But it's examinable. 00:36:09.190 --> 00:36:11.180 OK. 00:36:11.180 --> 00:36:15.660 Why does oxygen have a triplet ground state? 00:36:15.660 --> 00:36:18.465 That's something that every textbook says. 00:36:18.465 --> 00:36:19.340 You got to know that. 00:36:19.340 --> 00:36:21.830 And of course, they don't really tell you anything more 00:36:21.830 --> 00:36:24.301 except something to memorize. 00:36:24.301 --> 00:36:24.800 OK. 00:36:24.800 --> 00:36:27.950 So generally, you have two electrons 00:36:27.950 --> 00:36:30.940 in each orbital in singlets. 00:36:30.940 --> 00:36:33.190 Now, if you're going to do spectroscopy 00:36:33.190 --> 00:36:36.850 you would perhaps promote one of these guys to a higher state. 00:36:39.400 --> 00:36:43.270 And so you'll have singlets and triplets. 00:36:43.270 --> 00:36:45.400 But of course the only transition you would see 00:36:45.400 --> 00:36:47.590 would be a singlet-to-singlet transition. 00:36:47.590 --> 00:36:50.040 So you might as well forget about triplets. 00:36:50.040 --> 00:36:52.570 And you might as well forget about having 00:36:52.570 --> 00:36:53.650 to add [INAUDIBLE]. 00:36:56.270 --> 00:37:00.750 You can get away with murder, especially with Huckel theory. 00:37:00.750 --> 00:37:03.230 So we have a stick diagram. 00:37:03.230 --> 00:37:05.750 We fill electrons into the thing. 00:37:05.750 --> 00:37:17.440 And then we compute the energy of the many electron problem. 00:37:17.440 --> 00:37:20.990 And so let's just do this. 00:37:20.990 --> 00:37:22.490 And you've all seen this. 00:37:35.560 --> 00:37:38.980 So we number the atoms. 00:37:38.980 --> 00:37:42.580 We have our symmetric structure. 00:37:42.580 --> 00:37:52.375 And psi mu is going to be sum from i equals 1 to 6. 00:37:52.375 --> 00:37:59.106 C i mu p z i. 00:37:59.106 --> 00:38:12.140 And c mu is c 1 mu c 2 mu to c 6 mu. 00:38:12.140 --> 00:38:13.690 These are the mixing coefficients. 00:38:16.520 --> 00:38:18.430 Well, for benzene, it's pretty simple 00:38:18.430 --> 00:38:24.370 because you know that if you have no nodes and symmetry, 00:38:24.370 --> 00:38:27.470 all of these are going to be the same. 00:38:27.470 --> 00:38:30.540 And so if you put 1s here, you put a 1 over square root of 6 00:38:30.540 --> 00:38:34.410 out in front for normalization. 00:38:34.410 --> 00:38:38.160 And you can figure out the eigenvectors for benzene-- 00:38:38.160 --> 00:38:41.490 all of them-- just by counting the number of nodes. 00:38:41.490 --> 00:38:45.660 And that's useful, because if you know the eigenvectors, then 00:38:45.660 --> 00:38:48.480 you can show what the eigenvalue is 00:38:48.480 --> 00:38:51.993 by multiplying the original Hamiltonian by an eigenvalue. 00:38:55.470 --> 00:39:01.380 So anyway, we have this. 00:39:01.380 --> 00:39:05.580 And then the Hamiltonian-- 00:39:05.580 --> 00:39:12.700 we have alpha, beta, and then 0s. 00:39:12.700 --> 00:39:17.610 And beta, alpha, beta, and then 0s, et cetera. 00:39:17.610 --> 00:39:22.710 And so we have this tri-diagonal structure. 00:39:22.710 --> 00:39:29.420 And because it's a ring, we have a beta here and here. 00:39:32.885 --> 00:39:35.360 OK. 00:39:35.360 --> 00:39:40.270 So then, we ask our computer to diagonalize this, 00:39:40.270 --> 00:39:45.320 or we use clever tricks from linear algebra 00:39:45.320 --> 00:39:49.340 to find the eigenvalues, but I don't recommend it. 00:39:49.340 --> 00:39:52.700 I mean, the general problem is going 00:39:52.700 --> 00:39:56.070 to be something where you have to use a computer. 00:39:56.070 --> 00:39:58.730 So don't develop tricks unless you 00:39:58.730 --> 00:40:01.040 want to check to see whether you programmed 00:40:01.040 --> 00:40:02.602 the computer correctly. 00:40:05.320 --> 00:40:13.380 And so when you do this, you get-- 00:40:13.380 --> 00:40:19.090 E 1, the lowest energy, is alpha plus 2 beta. 00:40:19.090 --> 00:40:23.575 E 2 is alpha plus beta. 00:40:23.575 --> 00:40:27.280 And E 3 is alpha minus beta. 00:40:32.940 --> 00:40:34.110 I'm sorry. 00:40:34.110 --> 00:40:37.080 E 2 and e 3 are both this. 00:40:37.080 --> 00:40:40.820 And E 4 and e 5 are this. 00:40:40.820 --> 00:40:46.482 And e 6 is alpha minus 2 beta. 00:40:46.482 --> 00:40:48.440 Now, we don't really care about these orbitals, 00:40:48.440 --> 00:40:50.148 because they don't put electrons in them. 00:40:52.810 --> 00:40:56.200 And so when you put the electrons in the orbitals, 00:40:56.200 --> 00:40:59.661 you get 2 for this, 2 for this, 2 for that. 00:40:59.661 --> 00:41:00.160 OK? 00:41:00.160 --> 00:41:03.580 And so we just calculate the sum of the energies. 00:41:03.580 --> 00:41:12.520 And we end up getting the energy levels for benzene. 00:41:12.520 --> 00:41:21.790 The ground state is going to be 2 alpha, 00:41:21.790 --> 00:41:36.580 plus 2 beta, plus 2 alpha, plus beta, plus 2 alpha, plus beta. 00:41:36.580 --> 00:41:38.370 Right? 00:41:38.370 --> 00:41:44.330 So this gives you 6 alpha plus 8 beta. 00:41:54.290 --> 00:41:58.620 Now, this-- You might have guessed it, 00:41:58.620 --> 00:41:59.620 but you didn't guess it. 00:41:59.620 --> 00:42:02.236 Your computer told you that. 00:42:02.236 --> 00:42:07.590 And the computer actually likes numbers rather than symbols. 00:42:07.590 --> 00:42:14.120 And so you actually obtain this simple structure 00:42:14.120 --> 00:42:18.000 from the computer-- requires a little bit of manipulation. 00:42:18.000 --> 00:42:21.960 But you still get 6 alpha plus 8 beta. 00:42:21.960 --> 00:42:29.750 And you also get the eigenvectors. 00:42:29.750 --> 00:42:32.530 And that's the stuff that you know. 00:42:32.530 --> 00:42:38.330 So c 1 is 1 over square root of 6. 00:42:38.330 --> 00:42:38.830 1. 00:42:38.830 --> 00:42:39.330 1. 00:42:39.330 --> 00:42:39.920 All of those. 00:42:39.920 --> 00:42:48.020 And c 2 is going to be something a little bit more complicated. 00:42:48.020 --> 00:42:53.050 We can have the nodal plane going through atoms. 00:42:53.050 --> 00:42:55.780 And so if it goes through atom one, 00:42:55.780 --> 00:42:58.030 it's also going to go through atom four, 00:42:58.030 --> 00:43:03.370 and so we have 0 and 0, 1, 1, 1, 1. 00:43:03.370 --> 00:43:09.900 And now, to figure out how to normalize that, we just-- 00:43:09.900 --> 00:43:13.040 1 over square root of 4, and so on. 00:43:13.040 --> 00:43:14.970 We can figure these things out. 00:43:14.970 --> 00:43:20.020 C 3 is going to be an eigenvector. 00:43:20.020 --> 00:43:24.880 Now, instead of having the nodal plane going through atoms, 00:43:24.880 --> 00:43:26.710 it's going between atoms. 00:43:26.710 --> 00:43:29.770 And so instead of having any 0s, you're 00:43:29.770 --> 00:43:33.760 going to have something more complicated. 00:43:33.760 --> 00:43:41.200 And now I can see that there is something in my notes 00:43:41.200 --> 00:43:51.170 which is subject to how you'd actually impose the symmetry. 00:43:51.170 --> 00:43:53.630 But suppose you have something like this-- 00:43:53.630 --> 00:44:02.830 2, 1, minus 1, minus 2, minus 1, 1. 00:44:02.830 --> 00:44:05.380 So why did I use these numbers? 00:44:05.380 --> 00:44:14.080 Well, I had to have a nodal plane here between atoms two 00:44:14.080 --> 00:44:15.310 and three. 00:44:15.310 --> 00:44:19.467 And the corresponding guy will be-- 00:44:19.467 --> 00:44:21.300 Well, that should have been 2 at the bottom. 00:44:24.830 --> 00:44:27.481 Let me just make sure I'm doing this right. 00:44:27.481 --> 00:44:27.980 No. 00:44:27.980 --> 00:44:28.621 I had a 2. 00:44:28.621 --> 00:44:29.120 OK. 00:44:29.120 --> 00:44:30.280 It's a 1. 00:44:30.280 --> 00:44:30.860 OK. 00:44:30.860 --> 00:44:37.790 And so the last 1 is here. 00:44:37.790 --> 00:44:42.530 So the sign change occurs twice. 00:44:42.530 --> 00:44:46.160 And those correspond to opposite bonds. 00:44:49.760 --> 00:44:51.590 Why were there 2s? 00:44:51.590 --> 00:44:59.180 Well, the 2s are between two atoms that have 1s, 00:44:59.180 --> 00:45:02.070 and so it's going to have a larger eigenvector. 00:45:02.070 --> 00:45:03.830 And you can figure it out lots of ways. 00:45:03.830 --> 00:45:07.610 You can also say, well, every atom has to be used up. 00:45:07.610 --> 00:45:08.840 Yes. 00:45:08.840 --> 00:45:10.870 AUDIENCE: So should there also be a sign change? 00:45:10.870 --> 00:45:11.828 ROBERT FIELD: I can't-- 00:45:11.828 --> 00:45:14.276 AUDIENCE: Should there also be a sign change in c 2? 00:45:14.276 --> 00:45:15.942 AUDIENCE: Yeah. 00:45:15.942 --> 00:45:22.182 In c 2 eigen c 2 6 should be [INAUDIBLE].. 00:45:22.182 --> 00:45:23.390 ROBERT FIELD: Did I screw up? 00:45:23.390 --> 00:45:24.015 AUDIENCE: Yeah. 00:45:27.770 --> 00:45:29.110 No, not-- in c 2. 00:45:29.110 --> 00:45:30.030 Not c 3. 00:45:30.030 --> 00:45:30.600 ROBERT FIELD: I'm sorry? 00:45:30.600 --> 00:45:31.725 AUDIENCE: The previous one. 00:45:33.992 --> 00:45:34.950 ROBERT FIELD: Oh, yeah. 00:45:34.950 --> 00:45:35.449 Yeah. 00:45:38.040 --> 00:45:41.540 So opposite sides. 00:45:41.540 --> 00:45:42.040 OK? 00:45:45.440 --> 00:45:45.940 All right. 00:45:45.940 --> 00:45:46.990 It doesn't matter. 00:45:46.990 --> 00:45:48.280 The computer tells you. 00:45:48.280 --> 00:45:52.360 But sometimes you can approach the problem very quickly, 00:45:52.360 --> 00:45:55.042 and just say, I know what the eigenvalues are going to be, 00:45:55.042 --> 00:45:56.250 and I have to normalize them. 00:46:01.580 --> 00:46:03.270 But one of the things that you often 00:46:03.270 --> 00:46:08.970 do if you're trying to be smart and skip steps is to say, OK. 00:46:08.970 --> 00:46:10.950 We have six eigenvectors. 00:46:10.950 --> 00:46:17.070 And each of the atomic orbitals gets used up completely 00:46:17.070 --> 00:46:19.560 among the six eigenvectors. 00:46:19.560 --> 00:46:22.170 And I do that all the time, because it's 00:46:22.170 --> 00:46:26.410 a very useful way of making sure I haven't screwed up. 00:46:26.410 --> 00:46:27.870 OK. 00:46:27.870 --> 00:46:37.510 So we get this, and that's perfectly OK. 00:46:37.510 --> 00:46:39.120 We don't know how significant it is, 00:46:39.120 --> 00:46:42.405 but say we had three of these-- 00:46:45.300 --> 00:46:48.900 three ethylenes-- I guess an organic chemist 00:46:48.900 --> 00:46:50.060 would just do that, right? 00:46:53.070 --> 00:46:56.580 But anyway, if we had three of these, when you do this 00:46:56.580 --> 00:47:03.464 you get 2 alpha plus 2 theta. 00:47:03.464 --> 00:47:04.350 OK? 00:47:04.350 --> 00:47:08.330 So we get 6 alpha plus 6 beta. 00:47:08.330 --> 00:47:10.670 And alpha and beta are both negative. 00:47:10.670 --> 00:47:14.660 And so the energy for three ethylenes-- 00:47:14.660 --> 00:47:15.890 did I screw up again? 00:47:15.890 --> 00:47:17.265 AUDIENCE: How are those ethylenes 00:47:17.265 --> 00:47:18.950 arranged with one another? 00:47:18.950 --> 00:47:20.306 ROBERT FIELD: They're separate. 00:47:25.584 --> 00:47:26.500 We could draw benzene. 00:47:29.130 --> 00:47:31.720 And so we could say, we have three isolated bonds. 00:47:31.720 --> 00:47:34.210 And since we don't care about the sigma structure, 00:47:34.210 --> 00:47:39.880 and no bonds are adjacent, you know that they're additive. 00:47:39.880 --> 00:47:42.730 And so we represent benzene. 00:47:42.730 --> 00:47:46.300 The primitive structure for benzene is three ethylenes. 00:47:46.300 --> 00:47:49.280 And benzene is 2 beta better. 00:47:49.280 --> 00:47:50.270 AUDIENCE: Yeah. 00:47:50.270 --> 00:47:52.270 ROBERT FIELD: That's the resonant stabilization. 00:47:52.270 --> 00:47:54.910 That's a great thing. 00:47:54.910 --> 00:47:57.110 OK. 00:47:57.110 --> 00:48:02.760 So we get a resonant stabilization. 00:48:02.760 --> 00:48:09.250 And now, the more subtle and wonderful stuff 00:48:09.250 --> 00:48:16.550 is, we have these eigenvectors. 00:48:16.550 --> 00:48:18.120 And we have the energies. 00:48:18.120 --> 00:48:19.370 And we can do stuff with them. 00:48:19.370 --> 00:48:20.930 And one thing is bond order. 00:48:27.550 --> 00:48:29.410 And so we have a formula. 00:48:29.410 --> 00:48:37.925 The bond order between atoms i and j is equal to the sum 00:48:37.925 --> 00:48:50.160 for mu equals 1, including only the occupied orbitals, of c i 00:48:50.160 --> 00:48:52.650 mu c j mu. 00:48:52.650 --> 00:48:56.320 So we have the new molecular orbitals, 00:48:56.320 --> 00:48:59.040 and we have the adjacent atoms. 00:48:59.040 --> 00:49:01.900 And this gives you the bond order. 00:49:01.900 --> 00:49:04.180 And so we can calculate the bond order. 00:49:04.180 --> 00:49:09.340 And we find that every bond is a pi bond 00:49:09.340 --> 00:49:20.510 order of 2/3, which is neat, because the non-resonant 00:49:20.510 --> 00:49:24.050 structure says, three of the bonds 00:49:24.050 --> 00:49:31.900 have a pi bond order of 0, and 3 have a bond order of 1. 00:49:31.900 --> 00:49:36.340 And so the average is 1/2. 00:49:36.340 --> 00:49:40.477 And this is bigger than 1/2. 00:49:40.477 --> 00:49:41.185 And it's uniform. 00:49:46.470 --> 00:49:48.900 So you can do this, and you can say, OK. 00:49:48.900 --> 00:49:50.250 The 1, 2 bond order-- 00:49:50.250 --> 00:49:53.580 the 2, 3 bond order-- you do the laborious calculation-- 00:49:53.580 --> 00:49:54.360 always get 2/3. 00:50:02.010 --> 00:50:07.890 Now, you could also calculate the charge on atom i. 00:50:07.890 --> 00:50:13.800 And that is, again, mu for the occupied orbitals. 00:50:13.800 --> 00:50:20.580 And that would be c i mu c i mu. 00:50:20.580 --> 00:50:23.610 And so for benzene, you would expect 00:50:23.610 --> 00:50:24.780 this to come out to be 0. 00:50:24.780 --> 00:50:27.680 And it does. 00:50:27.680 --> 00:50:29.095 So there is no charge. 00:50:37.160 --> 00:50:38.740 Let me just make sure that that-- 00:50:43.921 --> 00:50:47.640 Well, there is an equal charge on each atom, 00:50:47.640 --> 00:50:53.080 whether it's 0 or 1/6. 00:50:53.080 --> 00:50:54.600 That I haven't done. 00:50:54.600 --> 00:50:57.225 And I am a little uncertain about what this 00:50:57.225 --> 00:50:58.350 is going to come out to be. 00:50:58.350 --> 00:51:03.250 But they're going to be equal on every atom in benzene. 00:51:03.250 --> 00:51:10.770 And so now, suppose, instead of benzene, you have amylin. 00:51:13.820 --> 00:51:15.630 Well, you can do stuff. 00:51:15.630 --> 00:51:17.840 And you could say, well, the amylin out here 00:51:17.840 --> 00:51:21.290 is going to affect the alpha value here a lot, 00:51:21.290 --> 00:51:23.840 and here less and less. 00:51:23.840 --> 00:51:25.260 And then you do the calculation. 00:51:25.260 --> 00:51:27.880 And you find when you do this calculation, 00:51:27.880 --> 00:51:30.710 you get unequal charges. 00:51:30.710 --> 00:51:36.950 And you get the normal rules for ortho versus meta. 00:51:36.950 --> 00:51:39.170 And everything is great. 00:51:39.170 --> 00:51:41.530 And you also, when you do this, you 00:51:41.530 --> 00:51:44.240 can actually write resonance structures 00:51:44.240 --> 00:51:46.250 and you could calculate, well, what 00:51:46.250 --> 00:51:48.260 is the energy of that resonance structure-- 00:51:48.260 --> 00:51:50.010 in a Huckel-like theory. 00:51:50.010 --> 00:51:58.640 So there are lots of things you can do in order to say, OK. 00:51:58.640 --> 00:52:01.760 We do Huckel theory to get this 0 in our picture. 00:52:01.760 --> 00:52:05.120 And then-- I'm way over time. 00:52:05.120 --> 00:52:07.460 And then we can add special effects, 00:52:07.460 --> 00:52:10.600 and we know how to parameterize them. 00:52:10.600 --> 00:52:13.750 And if we're close to being OK, we'll 00:52:13.750 --> 00:52:17.140 get results that correspond to experiment. 00:52:17.140 --> 00:52:21.300 And this is so much better than you deserve, 00:52:21.300 --> 00:52:23.430 because it's all garbage. 00:52:23.430 --> 00:52:26.510 But it's empirically-calibrated garbage. 00:52:26.510 --> 00:52:30.950 And it it's calibrated over an enormous number of molecules 00:52:30.950 --> 00:52:33.850 and a huge amount of experience. 00:52:33.850 --> 00:52:35.360 And that's good. 00:52:35.360 --> 00:52:36.860 That's what we do. 00:52:36.860 --> 00:52:37.730 OK. 00:52:37.730 --> 00:52:42.230 So I'll see you on Friday with some perturbation theory. 00:52:42.230 --> 00:52:44.570 One of my favorite problems, too. 00:52:44.570 --> 00:52:46.120 OK.