WEBVTT 00:00:00.090 --> 00:00:02.490 The following content is provided under a Creative 00:00:02.490 --> 00:00:04.030 Commons license. 00:00:04.030 --> 00:00:06.330 Your support will help MIT OpenCourseWare 00:00:06.330 --> 00:00:10.690 continue to offer high-quality educational resources for free. 00:00:10.690 --> 00:00:13.320 To make a donation or view additional materials 00:00:13.320 --> 00:00:17.250 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:17.250 --> 00:00:18.200 at ocw.mit.edu. 00:00:22.072 --> 00:00:23.405 ROBERT FIELD: Let's get started. 00:00:26.210 --> 00:00:34.010 William Klemperer was my thesis advisor, and he died yesterday. 00:00:34.010 --> 00:00:37.850 It also happens that the subject of this lecture 00:00:37.850 --> 00:00:42.170 is really the core of what I got from him. 00:00:42.170 --> 00:00:46.870 He showed me how to evaluate matrix elements 00:00:46.870 --> 00:00:51.130 of many-electron operators, which is the key to being able 00:00:51.130 --> 00:00:53.530 to interpret-- 00:00:53.530 --> 00:00:57.820 not just tabulate-- electronic properties 00:00:57.820 --> 00:01:00.710 of atoms and molecules. 00:01:00.710 --> 00:01:06.980 Our goal is to be able to reduce the complexity 00:01:06.980 --> 00:01:10.280 of electronic structure, which is really complicated. 00:01:10.280 --> 00:01:13.270 The electrons interact with each other really strongly, 00:01:13.270 --> 00:01:14.660 and there are a lot of them. 00:01:14.660 --> 00:01:21.020 And it's very hard to separate the complexity 00:01:21.020 --> 00:01:24.650 of the many-body interactions into things 00:01:24.650 --> 00:01:28.290 that we can put in our head and interpret. 00:01:28.290 --> 00:01:30.930 And the whole goal of this course 00:01:30.930 --> 00:01:33.600 is to give you the tools to interpret 00:01:33.600 --> 00:01:37.090 complicated phenomena. 00:01:37.090 --> 00:01:41.290 We have the vibrational problem as a way of understanding 00:01:41.290 --> 00:01:45.040 internuclear interactions-- 00:01:45.040 --> 00:01:47.230 nuclear motions. 00:01:47.230 --> 00:01:50.770 We have electronic structure and the hydrogen atom 00:01:50.770 --> 00:01:54.580 as a way of understanding what electronic structure is, 00:01:54.580 --> 00:01:57.580 and to reduce it to, basically, the things 00:01:57.580 --> 00:02:01.456 we learn about hydrogen. 00:02:01.456 --> 00:02:03.080 When we go to molecular orbital theory, 00:02:03.080 --> 00:02:05.330 we take what we know about atoms, 00:02:05.330 --> 00:02:11.030 and build a minimally-complex interpretive picture, 00:02:11.030 --> 00:02:14.300 which is sort of a framework for understanding 00:02:14.300 --> 00:02:17.270 complicated molecular interactions. 00:02:17.270 --> 00:02:21.350 So one of the most important things about understanding 00:02:21.350 --> 00:02:23.870 electronic structure is, how do we 00:02:23.870 --> 00:02:28.616 deal with many-electron wave functions? 00:02:28.616 --> 00:02:34.190 And one of the terrible problems is that the electrons 00:02:34.190 --> 00:02:36.410 are indistinguishable. 00:02:36.410 --> 00:02:40.970 And so we have to ensure that the wave functions are 00:02:40.970 --> 00:02:43.310 anti-symmetric with respect to permutation 00:02:43.310 --> 00:02:47.150 of every pair of electrons, not just two. 00:02:47.150 --> 00:02:52.004 In helium we just dealt with two, and that wasn't so bad. 00:02:52.004 --> 00:02:57.010 But when we deal with n electrons, what 00:02:57.010 --> 00:03:00.910 we are going to discover is that in order 00:03:00.910 --> 00:03:03.520 to anti-symmetrize the wave function, 00:03:03.520 --> 00:03:07.630 we have to write a determinantal wave function, a determinant 00:03:07.630 --> 00:03:09.440 of orbitals. 00:03:09.440 --> 00:03:11.590 And when you expand an n by n determinant, 00:03:11.590 --> 00:03:14.080 you get n factorial times. 00:03:14.080 --> 00:03:15.790 And when you calculate matrix elements, 00:03:15.790 --> 00:03:20.090 you have n factorial squared integrals. 00:03:20.090 --> 00:03:23.900 So you're not going to be handling these one at a time, 00:03:23.900 --> 00:03:28.130 and looking at them lovingly. 00:03:28.130 --> 00:03:31.340 You're going to want to be able to take these things 00:03:31.340 --> 00:03:34.100 and extract what is the important thing 00:03:34.100 --> 00:03:36.320 about the electronic structure that you're 00:03:36.320 --> 00:03:39.150 going to need to know. 00:03:39.150 --> 00:03:44.970 And as a graduate student, I was collecting numbers. 00:03:44.970 --> 00:03:49.020 I was collecting numbers about spectroscopic perturbations, 00:03:49.020 --> 00:03:51.990 where non-degenerate perturbation 3 breaks down, 00:03:51.990 --> 00:03:53.700 and interesting things happen. 00:03:53.700 --> 00:03:55.680 But this was something that nobody in the world 00:03:55.680 --> 00:03:58.830 was interested in because it was the breaking 00:03:58.830 --> 00:04:00.990 of the usual patterns. 00:04:00.990 --> 00:04:05.400 And I was convinced that I had collected some stuff that 00:04:05.400 --> 00:04:07.970 told an interesting story. 00:04:07.970 --> 00:04:13.370 And I told Klemperer about this, and he said, well, 00:04:13.370 --> 00:04:17.620 have you thought about how to evaluate these integrals-- 00:04:17.620 --> 00:04:21.430 these numbers that you are extracting from the spectrum, 00:04:21.430 --> 00:04:25.950 by doing some tricks with the many-electron wave functions? 00:04:25.950 --> 00:04:29.580 And then he showed me, on a scrap of paper, how to do it. 00:04:29.580 --> 00:04:31.120 And I was launched. 00:04:31.120 --> 00:04:31.750 That was it. 00:04:31.750 --> 00:04:35.740 That has been the foundation of my career 00:04:35.740 --> 00:04:38.910 for the last 50 years. 00:04:38.910 --> 00:04:41.252 And I didn't think that Klemperer knew that. 00:04:41.252 --> 00:04:42.960 I didn't think anybody knew it, because I 00:04:42.960 --> 00:04:45.090 didn't think it was knowable. 00:04:45.090 --> 00:04:47.490 But he just gave it to me on a silver platter. 00:04:47.490 --> 00:04:50.970 And so I'm going to try to give you 00:04:50.970 --> 00:04:54.840 at least the rudiments of what it is you're up against, 00:04:54.840 --> 00:04:57.120 and how you reduce them to things 00:04:57.120 --> 00:05:00.570 that you care about, that you can think about. 00:05:00.570 --> 00:05:02.940 And you can understand the hydrogen atom 00:05:02.940 --> 00:05:05.160 in rather complete detail. 00:05:05.160 --> 00:05:08.400 Or at least you can understand how one observable relates 00:05:08.400 --> 00:05:10.690 to another. 00:05:10.690 --> 00:05:16.480 And so the relationship between the effective quantum 00:05:16.480 --> 00:05:20.080 number and the ionization energy of a state 00:05:20.080 --> 00:05:24.010 then provides a hydrogen-atom-based structural 00:05:24.010 --> 00:05:26.950 model for everything you can observe. 00:05:26.950 --> 00:05:36.410 Now spectroscopists have the unfortunate habit 00:05:36.410 --> 00:05:38.570 of saying we're interested in structure. 00:05:38.570 --> 00:05:41.030 Structure is static. 00:05:41.030 --> 00:05:46.070 Dynamics is magical, and special, and hard. 00:05:46.070 --> 00:05:52.040 But if you understand structure in a way which is not 00:05:52.040 --> 00:05:55.880 the exact eigenstates, not the exact wave functions, 00:05:55.880 --> 00:06:00.200 but something that the molecule was trying to do and sort of 00:06:00.200 --> 00:06:00.920 missed. 00:06:00.920 --> 00:06:03.920 And the dynamics is just what happens 00:06:03.920 --> 00:06:07.200 when this preparation isn't in eigenstate, 00:06:07.200 --> 00:06:09.440 which would be boring. 00:06:09.440 --> 00:06:12.110 And you get dynamics, which you can understand, 00:06:12.110 --> 00:06:14.180 as opposed to just saying, I'm going 00:06:14.180 --> 00:06:17.370 to tabulate the dynamics too. 00:06:17.370 --> 00:06:21.230 You don't know anything unless you have a reductionist picture 00:06:21.230 --> 00:06:23.040 of what's going on. 00:06:23.040 --> 00:06:28.520 And since the hardest part of dealing with molecules 00:06:28.520 --> 00:06:32.370 is the fact that they have a lot of electrons, 00:06:32.370 --> 00:06:37.520 this is really the core of being able to do important stuff. 00:06:37.520 --> 00:06:44.770 Now it's a horrendously complicated problem, 00:06:44.770 --> 00:06:47.340 and notationally awkward, too. 00:06:47.340 --> 00:06:51.500 And let me just try to explain it. 00:06:51.500 --> 00:06:54.130 And I'm going to try to do this without too much reliance 00:06:54.130 --> 00:06:57.085 on my notes, because they're terrible. 00:07:01.750 --> 00:07:04.150 We talked about helium. 00:07:04.150 --> 00:07:07.390 And helium has two electrons. 00:07:07.390 --> 00:07:13.090 And there's this 1 over r12 interaction between electrons, 00:07:13.090 --> 00:07:14.530 which looks innocent enough. 00:07:14.530 --> 00:07:15.480 You can write it down. 00:07:15.480 --> 00:07:18.680 You know, it's just a few symbols. 00:07:18.680 --> 00:07:26.680 And we can call it the first order perturbation. 00:07:26.680 --> 00:07:29.380 But that's really a lie, because it's as 00:07:29.380 --> 00:07:32.870 big as almost everything else. 00:07:32.870 --> 00:07:37.790 And so, yeah, we can, in fact, do a series of approximations. 00:07:37.790 --> 00:07:42.370 One is, ignore it, the non-interacting electron 00:07:42.370 --> 00:07:43.730 approximation. 00:07:43.730 --> 00:07:46.210 And that's basically repackaging hydrogen, 00:07:46.210 --> 00:07:48.280 and it's not quite enough. 00:07:48.280 --> 00:07:53.650 And then we can say, OK, let's calculate the first order 00:07:53.650 --> 00:08:01.030 energy by calculating expectation values of h12. 00:08:01.030 --> 00:08:03.350 So that's E1. 00:08:03.350 --> 00:08:08.690 And that's almost enough to give us a sense of what is going on. 00:08:18.130 --> 00:08:29.890 1 over r12, commutate with any electron, that's not 0. 00:08:32.720 --> 00:08:38.620 1 over r12, commutated with any orbital angular momentum-- 00:08:38.620 --> 00:08:42.039 any momentum is not 0. 00:08:42.039 --> 00:08:53.572 So that means that l and n are not good quantum numbers. 00:08:53.572 --> 00:08:54.780 What's a good quantum number? 00:08:54.780 --> 00:08:58.022 What's the definition of a good quantum number? 00:08:58.022 --> 00:09:00.600 Come on, this is an important question. 00:09:00.600 --> 00:09:01.182 Yes. 00:09:01.182 --> 00:09:02.890 AUDIENCE: [INAUDIBLE] count, and then you 00:09:02.890 --> 00:09:04.420 put them into some formula, and then 00:09:04.420 --> 00:09:08.240 you can read off eigenvalues. 00:09:08.240 --> 00:09:12.810 ROBERT FIELD: That's maybe 70% of what I want. 00:09:12.810 --> 00:09:14.160 You can put it into a formula. 00:09:14.160 --> 00:09:17.440 That means it's a rigorously good thing. 00:09:17.440 --> 00:09:19.720 It means it commutes with the Hamiltonian. 00:09:19.720 --> 00:09:22.710 A rigorously good quantum number corresponds 00:09:22.710 --> 00:09:24.990 to a eigenvalue of an operator that 00:09:24.990 --> 00:09:27.000 commutes with the Hamiltonian. 00:09:27.000 --> 00:09:33.050 So hydrogen, we rely on n and l to get almost everything. 00:09:33.050 --> 00:09:36.530 But here we find that, in addition to this 00:09:36.530 --> 00:09:43.630 being that small, it destroys the foundation of our picture. 00:09:43.630 --> 00:09:48.610 And so how do we think that we can 00:09:48.610 --> 00:09:57.930 make any sense of many-electron atoms and molecules? 00:09:57.930 --> 00:10:03.700 Well it turns out we can hide most of the complexity. 00:10:03.700 --> 00:10:09.430 And most of the complexity is just working out the rules 00:10:09.430 --> 00:10:12.800 for calculating these matrix elements. 00:10:12.800 --> 00:10:15.770 The matrix elements of operators that we care about, 00:10:15.770 --> 00:10:22.080 like transition moments, spin orbit, Zeeman effect, things 00:10:22.080 --> 00:10:25.520 that correspond to how we observe 00:10:25.520 --> 00:10:28.490 atomic and molecular structure. 00:10:28.490 --> 00:10:32.930 And so the main obstacle to being 00:10:32.930 --> 00:10:35.660 able to evaluate these matrix elements 00:10:35.660 --> 00:10:38.577 is the permutation requirement. 00:10:44.650 --> 00:10:52.250 And it turns out that there is a really simple way of dealing 00:10:52.250 --> 00:10:57.080 with the requirements for electron permutation, 00:10:57.080 --> 00:11:01.400 and that is to write the wave function as a determinant 00:11:01.400 --> 00:11:02.465 of one-electron orbitals. 00:11:05.620 --> 00:11:13.150 Because a determinant has three really important properties. 00:11:13.150 --> 00:11:19.270 One, it changes its sign when you permute any two columns. 00:11:19.270 --> 00:11:24.340 Two, it changes its sign when you permute any two rows. 00:11:24.340 --> 00:11:27.190 And three, if you have two identical columns or rows, 00:11:27.190 --> 00:11:29.560 it's 0. 00:11:29.560 --> 00:11:33.720 That's really fantastic. 00:11:33.720 --> 00:11:37.460 And that is the Pauli exclusion principle-- not what 00:11:37.460 --> 00:11:40.300 you learned in high school. 00:11:40.300 --> 00:11:44.260 What you learned is a small consequence of that. 00:11:47.080 --> 00:11:51.510 So if we can build anti-symmetric wave functions, 00:11:51.510 --> 00:11:55.890 we have aufbau, we can only put one electron in an orbital. 00:11:55.890 --> 00:12:00.810 We have all sorts of stuff, but it's too complicated to tell 00:12:00.810 --> 00:12:07.170 a student in high school that you can't-- 00:12:07.170 --> 00:12:10.650 just the question of indistinguishable electrons 00:12:10.650 --> 00:12:14.350 is such a subtle thing that you can't say, 00:12:14.350 --> 00:12:18.010 well, they have to be anti-symmetric. 00:12:18.010 --> 00:12:21.200 But it's easy to say, you can't put more than one electron 00:12:21.200 --> 00:12:23.670 in a spin orbital. 00:12:23.670 --> 00:12:25.510 But we don't talk about spin orbitals. 00:12:25.510 --> 00:12:30.350 We say, we can't put more than two electrons in an orbital, 00:12:30.350 --> 00:12:35.580 because we're protecting you from unnecessary knowledge. 00:12:35.580 --> 00:12:37.606 OK, well, I'm not going to protect you. 00:12:37.606 --> 00:12:39.390 [LAUGHTER] 00:12:39.390 --> 00:12:50.430 OK, so we know that the Hamiltonian 00:12:50.430 --> 00:12:53.400 has to commute with-- 00:12:57.880 --> 00:13:01.940 these capital letters mean, many electrons' angular momenta. 00:13:01.940 --> 00:13:05.500 And this is the spin, this is the projection of the spin. 00:13:05.500 --> 00:13:08.470 We know this is true because the Hamiltonian doesn't 00:13:08.470 --> 00:13:11.930 operate on spin. 00:13:11.930 --> 00:13:22.770 It's a trivial result, but it's a very important result. 00:13:22.770 --> 00:13:32.980 OK, so we have to worry about spin, and spin eigenstates, 00:13:32.980 --> 00:13:34.930 and other things like that. 00:13:34.930 --> 00:13:41.005 OK, so Slater determinants. 00:13:52.650 --> 00:13:56.490 J.C. Slater was an MIT professor in physics. 00:13:59.070 --> 00:14:03.480 He invented these things in 1929. 00:14:03.480 --> 00:14:05.170 I have a reference. 00:14:05.170 --> 00:14:07.620 I don't know if I've ever read this paper, 00:14:07.620 --> 00:14:12.000 but it's probably beautiful. 00:14:21.900 --> 00:14:25.770 So basically what Slater did is showed, 00:14:25.770 --> 00:14:34.190 yeah, you can do the necessary algebra to deal with any atom, 00:14:34.190 --> 00:14:38.980 and to be able to reduce an atom to a small number of integrals 00:14:38.980 --> 00:14:42.000 that you really care about. 00:14:42.000 --> 00:14:44.150 And there are two ways of doing this. 00:14:44.150 --> 00:14:47.140 One is the truth, and one is the fit model. 00:14:50.000 --> 00:14:54.860 Now the truth is really boring, because you 00:14:54.860 --> 00:14:58.130 lose all the insights, and the fit model 00:14:58.130 --> 00:15:01.670 gives you the things you have to think about and understand. 00:15:01.670 --> 00:15:05.990 And a fit model also tells you what are the import the actors. 00:15:05.990 --> 00:15:08.480 And maybe they're in costume, maybe they're not, 00:15:08.480 --> 00:15:12.520 but we can deal with them. 00:15:12.520 --> 00:15:16.420 But the truth is really very complicated. 00:15:16.420 --> 00:15:26.870 And as I said many times, when you go from hydrogen to helium, 00:15:26.870 --> 00:15:30.360 you can't solve the Schrodinger equation exactly. 00:15:30.360 --> 00:15:34.390 This was perhaps a little bit of a surprise, 00:15:34.390 --> 00:15:36.480 but I think it was only a surprise in newspapers. 00:15:36.480 --> 00:15:38.550 I think physicists knew immediately, 00:15:38.550 --> 00:15:40.320 when you go for a two-bodied problem 00:15:40.320 --> 00:15:42.600 to a three-bodied problem, there is no way 00:15:42.600 --> 00:15:45.830 you can have an exact solution. 00:15:45.830 --> 00:15:47.550 And that's the truth. 00:15:47.550 --> 00:15:52.340 You can't solve helium or any more-than-one-electron problem 00:15:52.340 --> 00:15:54.250 exactly. 00:15:54.250 --> 00:15:58.660 But you can do it really well, and it just 00:15:58.660 --> 00:16:00.820 costs computer time. 00:16:00.820 --> 00:16:05.470 And if the computer is doing the work, you don't really care. 00:16:05.470 --> 00:16:08.530 Because once you've told the computer the rules, 00:16:08.530 --> 00:16:09.940 then it's off to the races. 00:16:09.940 --> 00:16:14.380 You can go have lives or you can go have a life, and come back, 00:16:14.380 --> 00:16:18.910 and the computer will tell you whether you made a mistake 00:16:18.910 --> 00:16:21.070 and you're getting a nonsense result, 00:16:21.070 --> 00:16:26.690 or that you have the correct result. 00:16:26.690 --> 00:16:32.690 So what we know is this permutation operator, 00:16:32.690 --> 00:16:40.220 operating on any two-electron function, has to make-- 00:16:49.880 --> 00:16:54.510 OK, I'm skipping steps, and my notes are really kind of stupid 00:16:54.510 --> 00:16:55.620 sometimes. 00:16:55.620 --> 00:16:59.625 P 2, 1, which has to be equal to minus. 00:17:04.050 --> 00:17:07.410 And now if you apply the permutation operator twice, 00:17:07.410 --> 00:17:08.849 you get back to the same thing. 00:17:08.849 --> 00:17:11.490 So there's only two possible eigenvalues. 00:17:11.490 --> 00:17:15.960 You can have minus 1 or plus 1. 00:17:15.960 --> 00:17:18.690 And the minus 1 corresponds to fermions, 00:17:18.690 --> 00:17:22.950 things that have half-integer spin, like electrons. 00:17:22.950 --> 00:17:25.890 And the plus 1 corresponds to things that have integer spin, 00:17:25.890 --> 00:17:31.300 like photons, and vibrons, and other things. 00:17:31.300 --> 00:17:36.810 And actually, it's harder to construct a symmetric function 00:17:36.810 --> 00:17:39.060 than an anti-symmetric function. 00:17:39.060 --> 00:17:42.580 But the thing is, you've got lots of electrons, 00:17:42.580 --> 00:17:49.270 and you have very few quanta of vibrations in a single mode, 00:17:49.270 --> 00:17:51.760 and you have very few photons interacting 00:17:51.760 --> 00:17:53.840 with a molecule at once. 00:17:53.840 --> 00:17:59.920 And so the boson symmetry is less 00:17:59.920 --> 00:18:04.130 important in most applications. 00:18:04.130 --> 00:18:07.430 And so we just have to kill this one. 00:18:07.430 --> 00:18:22.300 OK, so suppose we want to talk about something like this, 00:18:22.300 --> 00:18:25.290 the 1s, 2s configuration. 00:18:25.290 --> 00:18:29.640 A configuration is a list of the occupied orbitals-- 00:18:29.640 --> 00:18:32.430 not the occupied spin orbitals, which 00:18:32.430 --> 00:18:36.540 is a spin associated with an orbital. 00:18:36.540 --> 00:18:39.240 The world of spin orbitals is where I live, 00:18:39.240 --> 00:18:42.450 but we do that for a reason. 00:18:42.450 --> 00:18:52.990 And so this two-electron thing can be expressed as a space 00:18:52.990 --> 00:18:53.490 part-- 00:19:08.660 --> 00:19:10.510 there are various conventions that-- 00:19:16.320 --> 00:19:17.450 times the spin part. 00:19:21.960 --> 00:19:34.890 And alpha 1, beta 2, and then we have minus or plus beta 1, 00:19:34.890 --> 00:19:36.280 alpha 2. 00:19:36.280 --> 00:19:39.910 I'm looking at my notes because some people always 00:19:39.910 --> 00:19:43.030 keep the electron in the first position, 00:19:43.030 --> 00:19:47.710 and some people keep always the orbital in the first position. 00:19:47.710 --> 00:19:50.080 And it doesn't matter, because you 00:19:50.080 --> 00:19:52.180 can permute rows or columns. 00:19:52.180 --> 00:19:55.500 But I just want to write what is in my notes. 00:19:55.500 --> 00:20:00.970 OK, so this thing, this two-electron function, 00:20:00.970 --> 00:20:05.780 has two anti-symmetrized possibilities. 00:20:05.780 --> 00:20:10.530 And one is a singlet, and one is a triplet. 00:20:10.530 --> 00:20:15.140 So s equals zero, s equals 1. 00:20:15.140 --> 00:20:19.850 We recognize this alpha beta minus beta alpha 00:20:19.850 --> 00:20:26.570 as the singlet spin state, and alpha beta plus beta alpha 00:20:26.570 --> 00:20:29.610 as the triplet spin state. 00:20:29.610 --> 00:20:34.610 So we have alpha beta plus beta alpha, and alpha alpha 00:20:34.610 --> 00:20:40.800 and beta beta, and we have alpha beta minus beta alpha. 00:20:40.800 --> 00:20:45.810 So we call s = 0 a singlet, and this a triplet, 00:20:45.810 --> 00:20:47.280 because of the number of states. 00:20:49.930 --> 00:20:55.900 And this wave function has the necessary spin symmetry 00:20:55.900 --> 00:20:57.775 and the necessary permutation symmetry. 00:21:01.180 --> 00:21:10.270 OK, so if, instead of two electrons, we have 1, 00:21:10.270 --> 00:21:22.220 2 dot dot dot N, then Mr. Slater says we do this-- 00:21:22.220 --> 00:21:22.720 whoops. 00:21:32.030 --> 00:21:46.960 OK, N, 1, and then K1, and KN N. So that's 00:21:46.960 --> 00:21:49.780 a determinant-- an N by N determinant. 00:21:49.780 --> 00:21:54.220 And the rows correspond to the electrons, 00:21:54.220 --> 00:21:59.010 and the columns corresponding to the orbitals. 00:21:59.010 --> 00:22:04.080 Now this, because of the properties of the determinant, 00:22:04.080 --> 00:22:08.130 is anti-symmetric with respect to permutation of any two 00:22:08.130 --> 00:22:12.974 electrons or any two orbitals. 00:22:12.974 --> 00:22:14.890 But we don't really care about the permutation 00:22:14.890 --> 00:22:16.760 of the orbitals, because it's really the same thing 00:22:16.760 --> 00:22:18.140 is permuting the electrons. 00:22:18.140 --> 00:22:24.340 And so this N factorial is a consequence of normalization, 00:22:24.340 --> 00:22:27.490 because when you expand an N by N determinant, 00:22:27.490 --> 00:22:33.126 you get N factorial, additive products of N functions. 00:22:33.126 --> 00:22:34.250 It looks horrible. 00:22:37.540 --> 00:22:40.630 And because we're normalizing, we 00:22:40.630 --> 00:22:44.950 need this 1 over the square root of N factorial 00:22:44.950 --> 00:22:46.570 in order to have this thing come out 00:22:46.570 --> 00:22:49.940 to be 1 when you calculate the normalization integral. 00:22:54.450 --> 00:22:58.540 Now this notation is horrible because you've 00:22:58.540 --> 00:23:00.740 got too many symbols. 00:23:00.740 --> 00:23:04.180 And so depending on what you're trying to convey, 00:23:04.180 --> 00:23:08.620 you reduce the symbols, and you can reduce it simply 00:23:08.620 --> 00:23:15.630 by, instead of writing psi every time, just writing the state. 00:23:15.630 --> 00:23:18.360 Or you can-- since you don't need psi, 00:23:18.360 --> 00:23:20.940 you don't need the state letter, you can just 00:23:20.940 --> 00:23:22.110 have the state number. 00:23:25.000 --> 00:23:28.360 But the best way to do this is simply to say-- 00:23:40.200 --> 00:23:44.350 this is just the main diagonal of the determinant. 00:23:44.350 --> 00:23:48.090 It conveys everything you need. 00:23:48.090 --> 00:23:52.820 Again, if you permute any two of these guys, any adjacent pair, 00:23:52.820 --> 00:23:54.720 the sign changes. 00:23:54.720 --> 00:23:56.470 And it contains everything you need, 00:23:56.470 --> 00:23:59.910 and it doesn't require you to look at stuff 00:23:59.910 --> 00:24:02.650 you're not going to use. 00:24:02.650 --> 00:24:05.740 And your goal is going to be to take these things, 00:24:05.740 --> 00:24:08.410 and calculate matrix of them. 00:24:08.410 --> 00:24:11.680 And so you'll be dealing with the orbitals one or two 00:24:11.680 --> 00:24:13.240 at a time. 00:24:13.240 --> 00:24:15.880 And this is very convenient. 00:24:15.880 --> 00:24:20.170 And soon, you start to take this for granted. 00:24:20.170 --> 00:24:22.210 And it's a very simple thing, but it 00:24:22.210 --> 00:24:27.090 isn't, because you're doing a huge number of tricks. 00:24:27.090 --> 00:24:29.950 OK, I'm going to skip over what's in my notes. 00:24:29.950 --> 00:24:33.490 Demonstrating that for a two by two, that what I asserted 00:24:33.490 --> 00:24:36.115 is correct, you can do that very easily. 00:24:47.480 --> 00:24:53.780 OK, so we can count, and we have an atom, 00:24:53.780 --> 00:24:57.020 and we know how many electrons it has. 00:24:57.020 --> 00:25:00.550 And so we immediately know what our job is going to be. 00:25:00.550 --> 00:25:04.300 We're going to be having to write some Slater determinant 00:25:04.300 --> 00:25:07.000 of those number of electrons. 00:25:07.000 --> 00:25:11.980 And the goal is to be able to do the algebra in a way 00:25:11.980 --> 00:25:14.050 that maybe you can't describe to your friends 00:25:14.050 --> 00:25:16.010 because it's too complicated. 00:25:16.010 --> 00:25:19.330 I'm faced with the problem of trying to explain 00:25:19.330 --> 00:25:21.190 how to do this algebra. 00:25:21.190 --> 00:25:25.210 But it is something that you can learn, 00:25:25.210 --> 00:25:28.000 and you can ask a computer to do it, 00:25:28.000 --> 00:25:30.850 and there are all sorts of intuitive shortcuts 00:25:30.850 --> 00:25:32.980 where you can look at a problem, and you could say, 00:25:32.980 --> 00:25:35.540 I understand. 00:25:35.540 --> 00:25:40.240 OK, so you're used to orbitals. 00:25:43.540 --> 00:25:46.950 And that's perfectly reasonable, because for hydrogen, we 00:25:46.950 --> 00:25:50.260 have orbitals, and there's only one orbital, 00:25:50.260 --> 00:25:51.510 and it could have either spin. 00:25:51.510 --> 00:25:52.920 We don't mess with that. 00:25:52.920 --> 00:25:55.215 But now we're going to talk about spin orbitals. 00:25:58.940 --> 00:26:01.950 And that's just the combination of the name 00:26:01.950 --> 00:26:05.750 of the orbital with whether the spin is up or down. 00:26:05.750 --> 00:26:10.870 And the reason for this is it's easier to do the algebra. 00:26:10.870 --> 00:26:15.070 And the reason the algebra is-- 00:26:15.070 --> 00:26:18.580 it's initially harder to do the algebra, 00:26:18.580 --> 00:26:21.631 because there are certain selection rules, and stuff 00:26:21.631 --> 00:26:22.130 like that. 00:26:22.130 --> 00:26:25.640 But once you know how to do it you do the algebra. 00:26:25.640 --> 00:26:27.730 And then all of a sudden, everything 00:26:27.730 --> 00:26:29.640 pops out in a very useful form. 00:26:36.630 --> 00:26:38.910 So the stick diagrams are very important. 00:26:38.910 --> 00:26:42.420 But now I'm specifying the stick diagrams 00:26:42.420 --> 00:26:45.790 as spin orbitals rather than orbitals. 00:26:45.790 --> 00:26:53.100 Now another point, there are rules for how-- 00:26:58.720 --> 00:27:01.360 the number of spin orbitals is different 00:27:01.360 --> 00:27:04.270 between the left-hand side and the right-hand side of a matrix 00:27:04.270 --> 00:27:05.020 element. 00:27:05.020 --> 00:27:07.540 There are rules that are easily described-- and so 00:27:07.540 --> 00:27:14.800 for every kind of orbital, an orbital that 00:27:14.800 --> 00:27:20.170 is a scalar, that doesn't depend on quantum numbers, that 00:27:20.170 --> 00:27:23.750 has a selection rule delta SO of 0; 00:27:23.750 --> 00:27:30.010 and for something like a one-electron operator that 00:27:30.010 --> 00:27:35.290 has a selection rule delta spin orbital of 1 and 0; 00:27:35.290 --> 00:27:37.780 and then we have our friend 1 over rij, 00:27:37.780 --> 00:27:43.320 that has a selection rule delta spin orbital of 2, 1, and 0. 00:27:43.320 --> 00:27:46.270 And the algebra for each is something you work out, 00:27:46.270 --> 00:27:47.970 and then you know how to do it. 00:27:47.970 --> 00:27:50.260 And I'm going to try to give you just a little bit 00:27:50.260 --> 00:27:51.130 of a taste of this. 00:27:53.800 --> 00:28:02.980 So we already looked at something like this. 00:28:02.980 --> 00:28:05.930 But we use a slightly different notation. 00:28:05.930 --> 00:28:07.210 So I'm going to go back. 00:28:07.210 --> 00:28:16.720 And we have 1s alpha, 1s beta, 2s alpha, 2s beta. 00:28:16.720 --> 00:28:20.290 And so for the ground state of helium 1s 00:28:20.290 --> 00:28:23.080 squared, and we would do this. 00:28:27.330 --> 00:28:30.540 And the stick diagrams are great, 00:28:30.540 --> 00:28:36.750 because it's easier to see on a picture, who are the actors, 00:28:36.750 --> 00:28:41.880 and have I included all of them, or have I left something out? 00:28:41.880 --> 00:28:45.570 And so now we're interested in the stick diagram 00:28:45.570 --> 00:28:52.020 for the 1s 2s configuration. 00:28:52.020 --> 00:28:58.090 And there are several kinds of 1s/2s configurations, 00:28:58.090 --> 00:29:00.300 depending on what the alpha and beta are. 00:29:00.300 --> 00:29:05.020 So we have 1s alpha, 2s alpha. 00:29:05.020 --> 00:29:11.645 And we have 1s alpha, 2s beta. 00:29:19.890 --> 00:29:29.830 1s beta, 2s alpha, 2s alpha, 1s beta, 2s beta. 00:29:29.830 --> 00:29:33.370 So there's four guys, and we can put our arrows on these things, 00:29:33.370 --> 00:29:40.010 and we know everything we need to know about these guys. 00:29:40.010 --> 00:29:41.050 It tells us what to do. 00:29:46.670 --> 00:29:57.420 Well, when we do this, the diagonal matrix elements 00:29:57.420 --> 00:30:02.990 of the 1 over rij Hamiltonian can be expressed. 00:30:16.000 --> 00:30:24.620 And we use this notation, J tilde minus K tilde. 00:30:24.620 --> 00:30:27.980 So for every two-electron thing, we're going to get this kind 00:30:27.980 --> 00:30:28.730 of-- 00:30:28.730 --> 00:30:33.470 now these are simple integrals, and some of them are 0. 00:30:33.470 --> 00:30:37.805 Because this doesn't operate on spins. 00:30:37.805 --> 00:30:58.150 And so if you had a 1s alpha, 2s beta, 1 over rij, 1s beta, 00:30:58.150 --> 00:31:09.050 2s alpha, then the 1s alpha with the 1s beta is 0. 00:31:09.050 --> 00:31:14.600 The 1s alpha with the 2s alpha is not 0, et cetera. 00:31:14.600 --> 00:31:16.500 There are all sorts of stuff. 00:31:16.500 --> 00:31:20.025 But this tilde notation says, well, this 00:31:20.025 --> 00:31:22.430 is what we start with, and we have to convert it 00:31:22.430 --> 00:31:25.950 into things that really matter. 00:31:25.950 --> 00:31:28.850 So the operation of removing the tilde 00:31:28.850 --> 00:31:32.990 requires a little bit of work, a little bit of thought. 00:31:32.990 --> 00:31:36.680 And that's why my notes are crap, 00:31:36.680 --> 00:31:43.370 because I can't explain it well enough to really teach this. 00:31:45.980 --> 00:32:02.790 So when we do the 1s squared, the J 1s 00:32:02.790 --> 00:32:10.240 squared tilde, is equal to the J 1s squared, 00:32:10.240 --> 00:32:14.250 because the spins take care of themselves. 00:32:14.250 --> 00:32:22.840 But k tilde 1s squared is equal to 0. 00:32:22.840 --> 00:32:27.700 Because when we do 1s squared, we 00:32:27.700 --> 00:32:31.120 have an alpha with an alpha for the J 00:32:31.120 --> 00:32:35.320 term, and an alpha with the beta for the K term. 00:32:35.320 --> 00:32:40.890 And alpha with beta is 0, because the operator cannot 00:32:40.890 --> 00:32:42.790 change the alpha into beta. 00:32:50.570 --> 00:32:53.150 So this tilde notation is a convenient thing, 00:32:53.150 --> 00:32:56.960 because you can use any Slater determinant, 00:32:56.960 --> 00:33:00.020 and you can express it in terms of J's and K's. 00:33:00.020 --> 00:33:04.310 And the sign comes from switching 00:33:04.310 --> 00:33:06.410 the order of the orbitals. 00:33:06.410 --> 00:33:09.140 That's how the determinants work. 00:33:09.140 --> 00:33:12.740 And so you're going to see a whole bunch of stuff. 00:33:12.740 --> 00:33:17.210 But removing the tildes is the tricky business. 00:33:17.210 --> 00:33:22.400 OK, now when you get a problem where 00:33:22.400 --> 00:33:34.590 you have a configuration where a single Slater is not 00:33:34.590 --> 00:33:35.668 sufficient-- 00:33:40.250 --> 00:33:45.230 in other words, in order to make an eigenstate of s squared 00:33:45.230 --> 00:33:51.230 or sz, you sometimes need two or more Slaters, 00:33:51.230 --> 00:33:54.260 and you have to use a particular linear combination of them 00:33:54.260 --> 00:33:59.750 to get the right value of s and sz. 00:33:59.750 --> 00:34:02.030 And then what happens is you're looking 00:34:02.030 --> 00:34:10.179 at matrix elements of the 1 over rij operator, between Slaters. 00:34:10.179 --> 00:34:12.170 Now this is a headache. 00:34:12.170 --> 00:34:15.820 And I could talk until I'm blue in the face, 00:34:15.820 --> 00:34:20.679 and I cannot make it clear how to do this. 00:34:20.679 --> 00:34:24.090 Because it's just awful. 00:34:24.090 --> 00:34:28.420 But some things in life are worth suffering for. 00:34:28.420 --> 00:34:36.929 And so anyway, in the 1s, 2s situation, 00:34:36.929 --> 00:34:42.150 when you do everything right, you get-- 00:34:42.150 --> 00:34:45.179 this is just a general notation for a two-electron wave 00:34:45.179 --> 00:34:46.199 function-- 00:34:46.199 --> 00:34:52.469 1 over r12, psi 2. 00:34:52.469 --> 00:34:56.460 So these guys are eigenfunctions of s squared and sz. 00:34:56.460 --> 00:35:05.500 And when you do that, you get 1/2 times 2 J, 00:35:05.500 --> 00:35:13.960 1s, 2s, minus or plus 2K, 1s, 2s. 00:35:13.960 --> 00:35:19.840 Remember, when you have mismatched alpha and beta, 00:35:19.840 --> 00:35:20.920 the K's are 0. 00:35:20.920 --> 00:35:25.780 But when you have K, the 1 over rij matrix element between two 00:35:25.780 --> 00:35:28.620 Slaters, you can fix that. 00:35:28.620 --> 00:35:32.550 And so this is why it's so hard to explain, because-- 00:35:32.550 --> 00:35:36.460 yes, I'm not even going to apologize anymore. 00:35:36.460 --> 00:35:39.780 OK, so this is what you do. 00:35:39.780 --> 00:35:42.480 And the notes are pretty clear about how to do them, 00:35:42.480 --> 00:35:43.950 and what the problems are. 00:35:43.950 --> 00:35:46.170 But lecturing on it would be a little bit hard. 00:35:46.170 --> 00:35:48.970 OK, so now what are we going to do with it? 00:35:48.970 --> 00:35:57.840 Well, we'd like qualitative stuff and interpretive stuff. 00:36:06.490 --> 00:36:09.835 Qualitative is Hund's rules. 00:36:12.520 --> 00:36:19.352 Now if you looked at 100 textbooks, I think 95% of them 00:36:19.352 --> 00:36:20.560 will have Hund's rules wrong. 00:36:24.440 --> 00:36:26.860 You're never going to make that mistake. 00:36:26.860 --> 00:36:30.200 And interpretive-- well, we want to know the trends of things, 00:36:30.200 --> 00:36:33.560 and we want to be able to do something 00:36:33.560 --> 00:36:36.831 like what you did in freshman chemistry on shielding. 00:36:39.720 --> 00:36:43.600 Now you probably memorized some rules about what shields what. 00:36:43.600 --> 00:36:45.410 But I'm going to give you a little bit more 00:36:45.410 --> 00:36:46.250 insight into that. 00:36:46.250 --> 00:36:47.625 So we're going to talk about this 00:36:47.625 --> 00:36:48.980 for the rest of the lecture. 00:36:52.160 --> 00:37:01.340 OK, so you specify a configuration. 00:37:04.910 --> 00:37:08.180 And this configuration might be two electrons, two spin 00:37:08.180 --> 00:37:11.645 orbitals, two orbitals times e, or three, or 10. 00:37:14.560 --> 00:37:20.950 And often, when you specify the occupied orbitals, 00:37:20.950 --> 00:37:26.120 you neglect the field ones, which 00:37:26.120 --> 00:37:30.170 is nice, because you have fewer things to worry about. 00:37:30.170 --> 00:37:34.750 Because field orbitals have spin 0. 00:37:34.750 --> 00:37:37.140 And you don't have to do anything. 00:37:37.140 --> 00:37:39.220 They're automatically asymmetrized, 00:37:39.220 --> 00:37:43.510 and they basically act as a charged distribution 00:37:43.510 --> 00:37:49.630 in the core that is sampled by the electrons outside. 00:37:49.630 --> 00:37:52.870 And so you need some sort of a set of rules 00:37:52.870 --> 00:37:55.320 for how does that work. 00:37:55.320 --> 00:37:57.810 And that's shielding. 00:37:57.810 --> 00:38:01.720 So first, we specify a configuration. 00:38:01.720 --> 00:38:05.470 And you also learned-- in high school, probably-- 00:38:05.470 --> 00:38:10.750 how to determine the L, S, J terms that 00:38:10.750 --> 00:38:14.590 result from this configuration by some magical crossing-out 00:38:14.590 --> 00:38:16.170 of boxes. 00:38:16.170 --> 00:38:18.670 And if you didn't, I'm glad. 00:38:18.670 --> 00:38:21.870 Because it would have just clouded your mind, 00:38:21.870 --> 00:38:26.410 and caused earlier insanity than MIT causes. 00:38:26.410 --> 00:38:31.520 So anyway, so we have orbital angular momentum. 00:38:31.520 --> 00:38:33.310 And we can add the orbital angular 00:38:33.310 --> 00:38:37.300 momenta of the electrons following certain rules. 00:38:37.300 --> 00:38:39.650 And we have spin angular momentum. 00:38:39.650 --> 00:38:46.934 And J is equal to the vector sum of L and S. 00:38:46.934 --> 00:38:51.470 And we say we have an LS term-- 00:38:51.470 --> 00:38:55.910 like triplet P. And it can have J 00:38:55.910 --> 00:39:06.860 is equal to L plus S, L plus S minus 1, down to L minus S 00:39:06.860 --> 00:39:09.050 absolute value. 00:39:09.050 --> 00:39:11.000 These are the possible J's. 00:39:11.000 --> 00:39:13.520 And so Hund's rules is all about, 00:39:13.520 --> 00:39:18.650 of all of the states that belong to a particular configuration, 00:39:18.650 --> 00:39:20.840 which one is the lowest? 00:39:20.840 --> 00:39:23.910 One-- which one, not the second lowest. 00:39:23.910 --> 00:39:25.040 Which one is the lowest? 00:39:25.040 --> 00:39:26.330 And why do we care? 00:39:26.330 --> 00:39:29.510 Because in statistical mechanics everything 00:39:29.510 --> 00:39:33.380 is dominated by the lowest energy state. 00:39:33.380 --> 00:39:36.340 And so if you can figure out what is the lowest energy 00:39:36.340 --> 00:39:39.790 state, you've basically got as much as most people are 00:39:39.790 --> 00:39:42.430 going to want. 00:39:42.430 --> 00:39:48.280 So you want to know what are L, S, and J for the lowest energy 00:39:48.280 --> 00:39:51.040 state of a configuration. 00:39:51.040 --> 00:39:54.830 Configurations are typically far apart in energy. 00:39:54.830 --> 00:39:58.210 So if you know what the lowest energy configuration is, 00:39:58.210 --> 00:40:01.210 and the lowest energy state of it, as far as your friends-- 00:40:01.210 --> 00:40:03.730 the statistical machinations, you 00:40:03.730 --> 00:40:07.590 can tell them how to write their partition functions. 00:40:07.590 --> 00:40:09.490 And the rest is details. 00:40:09.490 --> 00:40:11.710 And mostly, you don't want details. 00:40:11.710 --> 00:40:13.779 If your friends tell you they want details, 00:40:13.779 --> 00:40:15.820 well, you tell them, this is what you have to do, 00:40:15.820 --> 00:40:19.420 but it's no simple three Hund's rules. 00:40:19.420 --> 00:40:22.910 OK, so Hund's rules-- 00:40:22.910 --> 00:40:25.840 you look at all of the L, S, J states that 00:40:25.840 --> 00:40:29.860 are possible for a particular configuration. 00:40:29.860 --> 00:40:34.530 And you can use the crossing out of ML/MS boxes if you want. 00:40:34.530 --> 00:40:36.790 And I could tell you why you would do that. 00:40:36.790 --> 00:40:41.170 But I don't want to cause insanity at this stage, either. 00:40:41.170 --> 00:40:43.180 But I'm an expert at that. 00:40:43.180 --> 00:40:47.370 And you can also use lowering operators 00:40:47.370 --> 00:40:50.370 to generate all the states, once you know stuff. 00:40:50.370 --> 00:40:57.820 OK, so once you know all the states, 00:40:57.820 --> 00:41:04.410 Hund says, which one of these has the largest S? 00:41:04.410 --> 00:41:05.666 which one? 00:41:05.666 --> 00:41:07.860 And that's easy to know. 00:41:07.860 --> 00:41:14.610 And for example, if you had 2p squared, 00:41:14.610 --> 00:41:26.730 you're going to get singlet D, triplet P, and singlet S. 00:41:26.730 --> 00:41:30.210 And well, here's the triplet. 00:41:30.210 --> 00:41:33.440 That has the largest S. So the triplet P 00:41:33.440 --> 00:41:37.250 is the lowest energy state. 00:41:37.250 --> 00:41:43.850 Now if there were multiple triplets, as there would be, 00:41:43.850 --> 00:41:56.340 say, for 2p3d, then you'd have to decide 00:41:56.340 --> 00:42:01.030 which of those triplets is the lowest. 00:42:01.030 --> 00:42:03.610 And all you care about, all you're allowed to say 00:42:03.610 --> 00:42:06.380 is which one is the lowest. 00:42:06.380 --> 00:42:10.900 And it's the one with the maximum L. 00:42:10.900 --> 00:42:15.610 And then the last step is, what is the lowest 00:42:15.610 --> 00:42:18.940 J for that LS state? 00:42:18.940 --> 00:42:20.740 And that's kind of cute. 00:42:20.740 --> 00:42:25.580 Because you have the P shell, there's-- 00:42:36.900 --> 00:42:46.320 for a P shell, you can have six P orbitals to fill the shells. 00:42:46.320 --> 00:42:51.450 1 alpha, 1 beta, 0 alpha, 0 beta, et cetera. 00:42:51.450 --> 00:42:54.515 So the degeneracy of a P orbital is 6. 00:42:57.650 --> 00:43:05.690 If you have p to N, where N is less than 3, 00:43:05.690 --> 00:43:08.630 you have a less-than-half-filled shell. 00:43:08.630 --> 00:43:19.430 And then lowest is J equal L minus S, absolute value. 00:43:19.430 --> 00:43:24.680 And if N is greater than 3, you have the lowest being-- 00:43:24.680 --> 00:43:29.060 the highest possible value of J is equal to L plus S-- 00:43:29.060 --> 00:43:33.310 so for L, N greater than 3. 00:43:33.310 --> 00:43:35.410 And now when you have a half-filled shell, 00:43:35.410 --> 00:43:48.114 the lowest state is usually an S state with maximum spin. 00:43:48.114 --> 00:43:49.030 But it doesn't matter. 00:43:52.510 --> 00:43:55.000 When you have less-than or more-than-half-filled shell, 00:43:55.000 --> 00:43:58.300 you have generally a state with orbital angular momentum not 00:43:58.300 --> 00:43:59.890 equal to 0. 00:43:59.890 --> 00:44:02.260 And you have spin orbit splitting of that. 00:44:02.260 --> 00:44:06.160 And so you do want to care what is the lowest J. 00:44:06.160 --> 00:44:09.580 But when N is equal to 3, the lowest state 00:44:09.580 --> 00:44:10.870 is usually an S state. 00:44:10.870 --> 00:44:12.520 It doesn't have a spin orbit splitting, 00:44:12.520 --> 00:44:20.440 and it just has one value of J, which is whatever the spin is. 00:44:20.440 --> 00:44:27.190 So Hund's rules tell you how to identify, 00:44:27.190 --> 00:44:32.030 without knowing beans, what is the lowest energy state, 00:44:32.030 --> 00:44:34.300 and it's never wrong. 00:44:34.300 --> 00:44:36.280 Well, maybe sometimes wrong, but that's 00:44:36.280 --> 00:44:38.410 because of one of my things where 00:44:38.410 --> 00:44:40.460 you have a perturbation between states belonging 00:44:40.460 --> 00:44:41.980 to two configurations. 00:44:41.980 --> 00:44:44.500 But people get really excited when they discover 00:44:44.500 --> 00:44:47.140 a violation of Hund's rules. 00:44:47.140 --> 00:44:49.730 And it's just trivial. 00:44:49.730 --> 00:44:53.080 So there is this. 00:44:53.080 --> 00:44:54.190 What time is it? 00:44:54.190 --> 00:44:58.390 I have a few minutes to talk about shielding, and I will. 00:44:58.390 --> 00:45:02.740 OK, so we have a nucleus. 00:45:02.740 --> 00:45:08.460 And it has a charge of z. 00:45:08.460 --> 00:45:10.740 Bare nucleus-- there's no electrons, 00:45:10.740 --> 00:45:14.150 so the atomic number is the charge. 00:45:14.150 --> 00:45:20.430 And then we have a filled shell around it. 00:45:20.430 --> 00:45:21.840 It's spherically symmetric. 00:45:24.960 --> 00:45:30.570 And so if we penetrate inside of it, what we see-- 00:45:30.570 --> 00:45:34.500 suppose we penetrate inside, to this point, what we see 00:45:34.500 --> 00:45:40.650 is only the plus z, minus the number of electrons 00:45:40.650 --> 00:45:42.510 inside this sphere. 00:45:42.510 --> 00:45:48.360 Now if you took Electricity and Magnetism, you can prove this. 00:45:48.360 --> 00:45:50.830 If you didn't, you can accept it. 00:45:50.830 --> 00:45:53.760 And so outside the nucleus, the charge 00:45:53.760 --> 00:45:57.970 is plus one, because you have a neutral atom. 00:45:57.970 --> 00:46:01.470 And then when you penetrate inside this region 00:46:01.470 --> 00:46:06.990 of dense charge, and all of the spins are generally paired, 00:46:06.990 --> 00:46:09.840 this is spherical. 00:46:09.840 --> 00:46:16.710 So what you end up seeing is z effective, as 00:46:16.710 --> 00:46:19.000 opposed to z true. 00:46:19.000 --> 00:46:21.000 And let's say here here's r0-- 00:46:21.000 --> 00:46:22.460 or this is r0. 00:46:25.020 --> 00:46:33.610 And so beyond r0, the charge that you see is plus 1. 00:46:33.610 --> 00:46:37.830 Add 0, you see a charge of z. 00:46:37.830 --> 00:46:42.460 And so what ends up happening is you get z effective, 00:46:42.460 --> 00:46:47.440 which is dependent on distance from the nucleus. 00:46:47.440 --> 00:46:49.750 And it goes from the integer value 00:46:49.750 --> 00:46:54.580 that you know, from the atomic number, down to 1, 00:46:54.580 --> 00:46:59.020 because you've taken one electron away 00:46:59.020 --> 00:47:02.650 from a neutral atom, and taken it outside. 00:47:02.650 --> 00:47:06.550 And now we have this wonderful thing 00:47:06.550 --> 00:47:08.695 called the centrifugal barrier. 00:47:13.690 --> 00:47:17.800 So if we have a state that has a non-zero of l-- 00:47:17.800 --> 00:47:19.800 well, if we have a state with a zero value of l, 00:47:19.800 --> 00:47:24.710 it can penetrate all the way into the core, to the nucleus. 00:47:24.710 --> 00:47:27.380 And so that means that the shielding is less 00:47:27.380 --> 00:47:30.070 for s orbitals. 00:47:30.070 --> 00:47:34.270 And now if we have a non-zero l, it can't get in so far. 00:47:34.270 --> 00:47:40.330 And the larger the l is, the less 00:47:40.330 --> 00:47:43.060 it can see this extra charge. 00:47:43.060 --> 00:47:45.590 So high l's are very well shielded. 00:47:45.590 --> 00:47:47.770 Low l's are not so well shielded. 00:47:47.770 --> 00:47:54.720 And the shielding goes s least shielded, p, less, so on. 00:47:54.720 --> 00:47:58.090 Now there's some other interesting things. 00:47:58.090 --> 00:48:03.200 Which, you know, I hate to say this, 00:48:03.200 --> 00:48:12.888 but comparing 5.111 or 5.112 to 3.091, 00:48:12.888 --> 00:48:16.800 there is this business of what happens when you start to-- 00:48:27.030 --> 00:48:32.280 you start with potassium, and so you put an electron in the 4s, 00:48:32.280 --> 00:48:34.560 not in the 3d orbital. 00:48:34.560 --> 00:48:35.170 Right? 00:48:35.170 --> 00:48:35.670 Why is that? 00:48:38.630 --> 00:48:45.210 Well, the 4s sees the larger charge is less shielded. 00:48:45.210 --> 00:48:47.080 So it goes in. 00:48:47.080 --> 00:48:49.770 Then when you go from potassium to calcium, 00:48:49.770 --> 00:48:54.370 you put another electron in this. 00:48:54.370 --> 00:48:55.850 And that's true. 00:48:55.850 --> 00:48:59.190 So for calcium, you have a 4s squared. 00:48:59.190 --> 00:49:03.500 And 4s 3d is a higher-lying state. 00:49:03.500 --> 00:49:05.570 Now you take an electron-- 00:49:11.640 --> 00:49:13.670 I'm cooking my own goose. 00:49:21.550 --> 00:49:26.510 If you take one of these electrons away-- 00:49:26.510 --> 00:49:28.760 this is not the way I wanted it to come out-- 00:49:28.760 --> 00:49:36.170 you find yourself in a 3d state. 00:49:36.170 --> 00:49:43.450 Because 3D penetrates a little bit under 4s. 00:49:47.580 --> 00:49:50.850 I can't explain it in a way that's going to make sense. 00:49:50.850 --> 00:49:54.960 I really wanted to, because I care so much 00:49:54.960 --> 00:49:56.970 about these simple arguments. 00:49:56.970 --> 00:50:01.390 But I will just be wasting your time. 00:50:01.390 --> 00:50:03.570 So the order in which you feel orbitals 00:50:03.570 --> 00:50:06.960 comes out, naturally, different from the order in which you 00:50:06.960 --> 00:50:09.180 remove electrons from orbitals. 00:50:09.180 --> 00:50:14.140 And the shielding arguments are capable of explaining that. 00:50:14.140 --> 00:50:18.670 OK, so this is the end of atoms. 00:50:18.670 --> 00:50:24.790 And I've asked you to observe some complicated algebra which 00:50:24.790 --> 00:50:26.500 you're never going to do, or at least 00:50:26.500 --> 00:50:27.640 never going to do much of. 00:50:30.830 --> 00:50:33.740 Everything you need to know about atoms, 00:50:33.740 --> 00:50:37.580 you can tell a computer, and it can do it. 00:50:37.580 --> 00:50:41.885 Now molecules are much more complicated. 00:50:41.885 --> 00:50:43.760 And that's we're going to start on next time. 00:50:43.760 --> 00:50:46.550 We're going to start with molecular orbital theory. 00:50:46.550 --> 00:50:50.510 And I'm not going to be presenting the normal textbook 00:50:50.510 --> 00:50:51.620 approach. 00:50:51.620 --> 00:50:54.500 I'm going to present an interpretive approach, where 00:50:54.500 --> 00:50:56.660 you understand why things happen, 00:50:56.660 --> 00:51:01.640 as opposed to memorize just symmetries, and filling orders, 00:51:01.640 --> 00:51:02.630 and so on. 00:51:02.630 --> 00:51:05.370 OK, I'll see you on Wednesday.