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. 00:00:21.540 --> 00:00:23.350 ROBERT FIELD: So in the previous lecture, 00:00:23.350 --> 00:00:26.430 I told you about many electron atoms. 00:00:26.430 --> 00:00:30.870 And in order to describe their structure, 00:00:30.870 --> 00:00:33.930 we have to build a Slater determinant, 00:00:33.930 --> 00:00:37.410 because that builds in the antisymmetry 00:00:37.410 --> 00:00:39.910 between all pairs of orbitals. 00:00:39.910 --> 00:00:43.480 Now, Slater determinants are expanded 00:00:43.480 --> 00:00:46.980 into a very large number of additive terms. 00:00:46.980 --> 00:00:50.340 And there are algebraic methods, which 00:00:50.340 --> 00:00:52.740 I describe in a supplement, which 00:00:52.740 --> 00:00:57.450 enable you to reduce these algebraic terms, the sum 00:00:57.450 --> 00:00:59.490 over an enormous number of terms, 00:00:59.490 --> 00:01:03.720 to something almost identical to what you would get 00:01:03.720 --> 00:01:05.790 if you didn't antisymmetrize. 00:01:05.790 --> 00:01:08.160 They're just a few extra things. 00:01:08.160 --> 00:01:12.940 And so once you know the mechanics, it's not a big deal. 00:01:12.940 --> 00:01:16.830 It's a little bit more work than just calculating 00:01:16.830 --> 00:01:21.840 the individual orbital matrix elements, 00:01:21.840 --> 00:01:24.780 but it's nothing special. 00:01:24.780 --> 00:01:27.630 And Henry has produced a beautiful thing, 00:01:27.630 --> 00:01:32.080 a beautiful handout on Slater-Condon parameters, 00:01:32.080 --> 00:01:36.050 which is what you get when you evaluate these Slater 00:01:36.050 --> 00:01:39.090 determinal matrix elements. 00:01:39.090 --> 00:01:42.510 And they are the things that you use to describe 00:01:42.510 --> 00:01:44.340 electronic structure. 00:01:44.340 --> 00:01:48.570 So electronic structure is a really complicated 00:01:48.570 --> 00:01:52.470 many electron descriptive thing. 00:01:52.470 --> 00:01:58.300 But if you can reduce for every atom to a small number 00:01:58.300 --> 00:02:02.230 of parameters, most of which you understand intuitively 00:02:02.230 --> 00:02:06.460 how they depend on properties of the atom, 00:02:06.460 --> 00:02:11.230 like the ionization energy of the individual orbitals, 00:02:11.230 --> 00:02:19.220 you can say, yes, I understand all of the periodic table. 00:02:19.220 --> 00:02:21.060 That's where it comes from. 00:02:21.060 --> 00:02:26.880 The rules enable you to identify the lowest lying states 00:02:26.880 --> 00:02:29.310 and to calculate their properties 00:02:29.310 --> 00:02:34.140 and to be able to intuit their properties 00:02:34.140 --> 00:02:37.644 and to be aware of when something unexpected happens, 00:02:37.644 --> 00:02:39.060 which is what we're always looking 00:02:39.060 --> 00:02:40.590 for as physical chemists. 00:02:40.590 --> 00:02:43.510 We're always looking for a simple minded picture 00:02:43.510 --> 00:02:45.940 that breaks. 00:02:45.940 --> 00:02:48.460 Because then we can go deeper into our understanding 00:02:48.460 --> 00:02:49.600 of all the pieces. 00:02:49.600 --> 00:02:54.900 But we're always parameterizing. 00:02:54.900 --> 00:02:57.170 OK. 00:02:57.170 --> 00:03:00.630 So there's a lot of subtleties that I didn't have time 00:03:00.630 --> 00:03:04.950 to talk about in the evaluation of these Slater determinal 00:03:04.950 --> 00:03:06.900 matrix elements. 00:03:06.900 --> 00:03:10.240 And since whatever problems I give 00:03:10.240 --> 00:03:13.420 you connected with Slater determinal matrix elements 00:03:13.420 --> 00:03:16.690 or Slater determinants are going to be really simple. 00:03:16.690 --> 00:03:19.660 They're probably going to be 2 by 2 or 3 by 3. 00:03:19.660 --> 00:03:22.270 You don't really need to know and understand 00:03:22.270 --> 00:03:24.370 all the rules and all the shortcuts. 00:03:24.370 --> 00:03:29.110 But you've got the basic mechanism. 00:03:29.110 --> 00:03:34.190 So a Hamiltonian doesn't depend on spins. 00:03:34.190 --> 00:03:37.880 But because of antisymmetry, the spatial part 00:03:37.880 --> 00:03:43.190 of the wave function does depend on which spin state you've got. 00:03:43.190 --> 00:03:46.670 And so the Hamiltonian knows about spins 00:03:46.670 --> 00:03:51.020 in a peculiar, indirect way, which is beautiful. 00:03:51.020 --> 00:03:54.260 You've got to admire how clever this Hamiltonian is 00:03:54.260 --> 00:03:58.450 to be able to capture these spin effects. 00:03:58.450 --> 00:04:00.950 And of course, you have to capture them too. 00:04:00.950 --> 00:04:03.740 All right. 00:04:03.740 --> 00:04:09.320 So today we're finally getting to molecules. 00:04:09.320 --> 00:04:18.860 And the simplest molecule is the homonuclear molecule. 00:04:18.860 --> 00:04:21.740 Two hydrogens with one electron. 00:04:21.740 --> 00:04:23.570 That's the simplest problem. 00:04:23.570 --> 00:04:28.280 And because it is representative of all 00:04:28.280 --> 00:04:30.650 of the qualitative effects and most 00:04:30.650 --> 00:04:34.040 of the quantitative calculations you do, 00:04:34.040 --> 00:04:36.200 it's a really great introduction, 00:04:36.200 --> 00:04:40.660 because it captures most of the important stuff 00:04:40.660 --> 00:04:42.740 that molecules or small molecules 00:04:42.740 --> 00:04:45.960 do, that the electronic structure of small molecule. 00:04:45.960 --> 00:04:52.310 So the key to being able to describe small molecules using 00:04:52.310 --> 00:04:56.900 this LCAO, Linear Combination of Atomic Orbitals method, 00:04:56.900 --> 00:04:59.180 is the variational method. 00:04:59.180 --> 00:05:02.280 And this variational method says, 00:05:02.280 --> 00:05:07.070 we're going to throw a whole bunch of basis functions 00:05:07.070 --> 00:05:08.270 at the problem. 00:05:08.270 --> 00:05:12.150 A whole bunch of orbitals, which we choose because we like them 00:05:12.150 --> 00:05:15.870 or because the computer likes them. 00:05:15.870 --> 00:05:22.100 And we find that the linear combination 00:05:22.100 --> 00:05:24.670 of coefficients of these orbitals 00:05:24.670 --> 00:05:26.120 that minimizes the energy. 00:05:29.550 --> 00:05:34.410 And you can do this in wave mechanics or matrix mechanics. 00:05:34.410 --> 00:05:37.110 It's more elegant in matrix mechanics, 00:05:37.110 --> 00:05:40.270 and you should be comfortable going from one to the other. 00:05:40.270 --> 00:05:42.760 Now, I'm only going to be talking about a 2 by 2 problem. 00:05:42.760 --> 00:05:44.820 So you can go easily from one to the other 00:05:44.820 --> 00:05:49.740 without even taking a breath. 00:05:49.740 --> 00:05:51.730 But anyway, OK. 00:05:51.730 --> 00:05:54.570 In approaching this, you're going 00:05:54.570 --> 00:05:59.160 to see what's called the general eigenvalue problem where 00:05:59.160 --> 00:06:02.610 instead of diagonalizing a simple matrix, 00:06:02.610 --> 00:06:06.540 you have to deal with the Hamiltonian matrix 00:06:06.540 --> 00:06:09.750 and an overlap matrix, which makes 00:06:09.750 --> 00:06:13.650 the solution of the secular equation, 00:06:13.650 --> 00:06:18.600 finding the eigenvalues and eigenstates a little bit more 00:06:18.600 --> 00:06:20.020 complicated. 00:06:20.020 --> 00:06:22.650 So you're going to see it first appear here. 00:06:22.650 --> 00:06:26.100 And it can be solved by a simple matrix 00:06:26.100 --> 00:06:29.430 equation, which is in a supplement to a later set 00:06:29.430 --> 00:06:30.670 of lecture notes. 00:06:30.670 --> 00:06:36.480 But this is the thing that consumes the vast majority 00:06:36.480 --> 00:06:40.140 of computer time in maybe in the world, 00:06:40.140 --> 00:06:43.330 but certainly among chemists. 00:06:43.330 --> 00:06:48.820 So the crucial thing, and I wrote it all in caps, 00:06:48.820 --> 00:06:56.830 in dealing with this complicated many body problem, 00:06:56.830 --> 00:06:59.050 we make a crucial assumption. 00:06:59.050 --> 00:07:02.760 And that's called the Born-Oppenheimer approximation. 00:07:02.760 --> 00:07:08.120 It's based on the fact that electrons move really fast 00:07:08.120 --> 00:07:12.930 and nuclei move really slowly relative to electrons. 00:07:12.930 --> 00:07:16.620 And so what we do is we solve the problem 00:07:16.620 --> 00:07:19.640 with the nuclei clamped. 00:07:19.640 --> 00:07:22.430 Because that's a good approximation. 00:07:22.430 --> 00:07:25.870 And then after we solve the electronic problem, 00:07:25.870 --> 00:07:29.950 we have what we call a potential energy surface, which 00:07:29.950 --> 00:07:35.070 is the electronic energy as a function of nuclear coordinate. 00:07:35.070 --> 00:07:37.260 And it's parametric. 00:07:37.260 --> 00:07:40.620 But then once we have the potential energy surface, 00:07:40.620 --> 00:07:43.530 we can calculate the rotation and vibration. 00:07:43.530 --> 00:07:46.050 So basically, what we're doing in making 00:07:46.050 --> 00:07:48.120 the Born-Oppenheimer approximation is 00:07:48.120 --> 00:07:51.630 we're setting up to treat vibration and rotation 00:07:51.630 --> 00:07:54.220 by perturbation theory. 00:07:54.220 --> 00:07:56.950 Now, you never see it mentioned in that way, 00:07:56.950 --> 00:07:59.410 but that is, in fact, what we're doing. 00:07:59.410 --> 00:08:01.270 And the reason we can get away with it 00:08:01.270 --> 00:08:04.915 is the difference in velocity of electrons versus nuclei. 00:08:07.450 --> 00:08:11.050 But it's a fundamental philosophical point of view 00:08:11.050 --> 00:08:13.690 that we take about molecular structure 00:08:13.690 --> 00:08:18.400 that we can clamp the nuclei, get most of the insight, 00:08:18.400 --> 00:08:22.120 and then allow the nuclei to move and the electrons 00:08:22.120 --> 00:08:25.460 with them and calculate everything else. 00:08:25.460 --> 00:08:29.110 So this is central to chemistry, the Born-Oppenheimer 00:08:29.110 --> 00:08:29.950 approximation. 00:08:29.950 --> 00:08:32.335 Without that, we would have a bag of atoms. 00:08:35.200 --> 00:08:38.650 And that would be a very bad starting point, 00:08:38.650 --> 00:08:41.409 because we believe in bonds. 00:08:41.409 --> 00:08:43.730 And a bag of atoms is saying, well, yeah, 00:08:43.730 --> 00:08:44.900 we could be statistical. 00:08:44.900 --> 00:08:48.710 We could say every atom is capable of being anywhere 00:08:48.710 --> 00:08:51.470 and they don't interact in any rational way, 00:08:51.470 --> 00:08:54.040 and we would have no insight. 00:08:54.040 --> 00:08:58.210 So this is where most of the ideas that you learn 00:08:58.210 --> 00:09:03.880 to use as physical chemists and probably any kind of chemist 00:09:03.880 --> 00:09:06.670 comes from the Born-Oppenheimer approximation. 00:09:06.670 --> 00:09:11.005 And it's widely misunderstood and misrepresented. 00:09:11.005 --> 00:09:13.250 But we won't go into that. 00:09:13.250 --> 00:09:21.040 OK, we're going to derive two orbitals for hydrogen. H2 plus. 00:09:21.040 --> 00:09:25.570 The sigma bonding orbital and the sigma star 00:09:25.570 --> 00:09:29.030 anti-bonding orbital. 00:09:29.030 --> 00:09:32.770 This is enough to basically understand 00:09:32.770 --> 00:09:36.070 how you get bonding and anti-bonding in any molecule. 00:09:39.000 --> 00:09:42.930 And I like to say that we understand 00:09:42.930 --> 00:09:46.840 bonding in a very simple way. 00:09:46.840 --> 00:09:51.850 If you have wave functions for two adjacent atoms 00:09:51.850 --> 00:09:57.040 and they overlap in the region between the two nuclei, 00:09:57.040 --> 00:09:59.460 you get constructive interference, 00:09:59.460 --> 00:10:03.810 you get twice as much amplitude in the region 00:10:03.810 --> 00:10:09.570 where the electrons interact with both nuclei, which means 00:10:09.570 --> 00:10:11.860 four times the probability. 00:10:11.860 --> 00:10:14.730 So a little bit of overlap leads to a big energy 00:10:14.730 --> 00:10:19.990 effect because of this constructive interference. 00:10:19.990 --> 00:10:22.340 And that's really, really important. 00:10:22.340 --> 00:10:25.190 And all sorts of hand waving arguments 00:10:25.190 --> 00:10:28.370 about what makes a chemical bond have a lot of currency, 00:10:28.370 --> 00:10:31.640 even here at MIT, but that's what a bond is. 00:10:31.640 --> 00:10:33.416 It's constructive interference. 00:10:39.740 --> 00:10:43.580 All right, so let's get started. 00:10:43.580 --> 00:10:48.350 And the lecture notes are extremely long and extremely 00:10:48.350 --> 00:10:49.730 beautiful and clear. 00:10:49.730 --> 00:10:51.800 They're written by Troy. 00:10:51.800 --> 00:10:56.340 I have reduced them to a minimal picture, 00:10:56.340 --> 00:11:02.090 but I don't want to detract from them. 00:11:02.090 --> 00:11:05.270 The insight is very rich. 00:11:05.270 --> 00:11:08.030 OK, so what we have to do first is 00:11:08.030 --> 00:11:10.430 to look at a coordinate system. 00:11:10.430 --> 00:11:13.880 So we have one hydrogen nucleus. 00:11:16.460 --> 00:11:17.180 Another one. 00:11:22.190 --> 00:11:23.900 There's a center of mass. 00:11:23.900 --> 00:11:30.590 And so we say we have RA, this vector from the center mass 00:11:30.590 --> 00:11:35.630 to this RB like that. 00:11:35.630 --> 00:11:39.350 And up here is the electron. 00:11:39.350 --> 00:11:47.520 And it's at position R. This is at position RB. 00:11:50.550 --> 00:11:52.410 Position RA. 00:11:55.130 --> 00:12:00.310 Now, the center of mass can be anywhere in space, 00:12:00.310 --> 00:12:02.930 and we don't really care. 00:12:02.930 --> 00:12:06.290 So we tend to not talk about where the center of mass is. 00:12:06.290 --> 00:12:09.200 But there is another set of coordinates for it. 00:12:09.200 --> 00:12:09.800 OK. 00:12:09.800 --> 00:12:15.420 And now we have this, which is called R sub A. 00:12:15.420 --> 00:12:16.490 And it's a vector. 00:12:16.490 --> 00:12:23.000 And this is R sub B. So we have uppercase symbols 00:12:23.000 --> 00:12:26.520 for nuclear positions and lowercase for electrons, 00:12:26.520 --> 00:12:30.340 except they're between electrons and nuclei. 00:12:30.340 --> 00:12:32.250 OK. 00:12:32.250 --> 00:12:34.580 And once we have this picture, we 00:12:34.580 --> 00:12:38.570 can write what the Hamiltonian is, because we're basically 00:12:38.570 --> 00:12:42.990 looking at the interactions between these three particles. 00:12:42.990 --> 00:12:46.520 And so we have kinetic energy and we have potential energy. 00:12:46.520 --> 00:12:54.350 And the Hamiltonian-- excuse me. 00:12:54.350 --> 00:13:04.810 --in atomic units is this del squared, 00:13:04.810 --> 00:13:10.140 which is the second derivatives with respect to whatever we 00:13:10.140 --> 00:13:12.270 have here, which is little r. 00:13:12.270 --> 00:13:14.590 And we have 1/2. 00:13:14.590 --> 00:13:17.940 It's 1/2 because of atomic units and because the mass 00:13:17.940 --> 00:13:21.980 of the electron is treated as one. 00:13:21.980 --> 00:13:30.340 And then we have a similar sort of term for del for nucleus A. 00:13:30.340 --> 00:13:33.500 And we have a two, because 1/2 in atomic units. 00:13:33.500 --> 00:13:39.067 But we have the mass of A. Now, the minus signs, 00:13:39.067 --> 00:13:40.400 why do we have minus signs here? 00:13:45.830 --> 00:13:48.450 You know this. 00:13:48.450 --> 00:13:50.520 AUDIENCE: This is the Schrodinger. 00:13:50.520 --> 00:13:52.130 ROBERT FIELD: Yeah, right. 00:13:52.130 --> 00:13:54.920 Or we say it's p squared over 2m. 00:13:54.920 --> 00:13:57.850 And p is-- right. 00:13:57.850 --> 00:14:01.640 OK, but what we're trying to understand stuff, 00:14:01.640 --> 00:14:02.640 this is negative. 00:14:02.640 --> 00:14:06.860 That means it sort of leads to a stabilization. 00:14:06.860 --> 00:14:12.881 And many people talk about bonding in terms of velocity 00:14:12.881 --> 00:14:16.100 or momentum, but I don't. 00:14:16.100 --> 00:14:21.860 And then we have del squared B over 2 00:14:21.860 --> 00:14:25.190 and B and then a repulsion term. 00:14:28.250 --> 00:14:30.240 A whole bunch of terms that we have. 00:14:30.240 --> 00:14:32.860 So these are the kinetic energy terms. 00:14:32.860 --> 00:14:35.960 And then we have the potential energy terms. 00:14:35.960 --> 00:14:48.920 1 over RA vector minus little r vector minus 1 over RB 00:14:48.920 --> 00:14:51.800 minus little r. 00:14:51.800 --> 00:14:57.000 And then the repulsion between RA minus RB. 00:15:01.430 --> 00:15:03.110 OK, that's it. 00:15:03.110 --> 00:15:05.330 Now, that's a lot of stuff. 00:15:05.330 --> 00:15:08.780 And we'd like to make a lot of it go away. 00:15:08.780 --> 00:15:10.910 But again, we have potential. 00:15:10.910 --> 00:15:13.240 We have kinetic energy and we have potential energy. 00:15:18.480 --> 00:15:21.200 So as I've written it, the Hamiltonian, 00:15:21.200 --> 00:15:24.200 which is our big problem, is a function 00:15:24.200 --> 00:15:33.920 of ra rb r big RA big RB. 00:15:37.690 --> 00:15:41.860 That's just ridiculous. 00:15:41.860 --> 00:15:44.770 We don't want to look at a function of all 00:15:44.770 --> 00:15:46.210 these variables. 00:15:46.210 --> 00:15:48.070 We would like to be able to simplify it 00:15:48.070 --> 00:15:52.900 so that we can draw pictures on a two dimensional surface 00:15:52.900 --> 00:15:55.060 or just to reduce it now. 00:15:57.930 --> 00:16:02.220 We like to think in terms of orders of magnitude. 00:16:02.220 --> 00:16:06.320 And so it is a very simple matter to say, 00:16:06.320 --> 00:16:08.690 well, let's simplify this Hamiltonian 00:16:08.690 --> 00:16:11.120 by clamping the nuclei. 00:16:11.120 --> 00:16:12.530 They don't move. 00:16:12.530 --> 00:16:17.950 If they don't move, then the big Rs are no longer operators. 00:16:17.950 --> 00:16:20.060 They're just parameters. 00:16:20.060 --> 00:16:28.820 And the contributions to the kinetic energy 00:16:28.820 --> 00:16:32.360 of the heavy particles gone. 00:16:32.360 --> 00:16:35.240 So there's no rotation and no vibration. 00:16:35.240 --> 00:16:36.360 Now, it doesn't go away. 00:16:36.360 --> 00:16:37.830 We have to come back with it. 00:16:37.830 --> 00:16:40.220 But we come back with it when we have 00:16:40.220 --> 00:16:44.330 a good representation of the electronic structure 00:16:44.330 --> 00:16:46.340 for the nuclei clamped. 00:16:46.340 --> 00:16:50.540 And if we can do it for one internuclear separation, 00:16:50.540 --> 00:16:53.550 we can do it for all internuclear separations. 00:16:53.550 --> 00:16:59.150 And so if you have a grid of say 1,000 points, 00:16:59.150 --> 00:17:03.250 you do the same calculation 1,000 times, except you don't. 00:17:03.250 --> 00:17:06.970 Because you know how to extrapolate from one 00:17:06.970 --> 00:17:08.200 point to another. 00:17:08.200 --> 00:17:10.510 But basically, you are thinking about this 00:17:10.510 --> 00:17:14.500 as the problem of finding the electronic wave 00:17:14.500 --> 00:17:18.220 function at a grid of points. 00:17:18.220 --> 00:17:24.849 And so that turns all of the big R's into parameters, not 00:17:24.849 --> 00:17:32.110 operators, and simplifies the problem enormously. 00:17:32.110 --> 00:17:37.010 And so the reduced Hamiltonian becomes-- 00:17:37.010 --> 00:17:39.270 we'll call it the electronic Hamiltonian. 00:17:39.270 --> 00:17:44.280 R semicolon ra rb. 00:17:44.280 --> 00:17:46.860 So r is the variable, and these are just parameters 00:17:46.860 --> 00:17:48.680 that we sometimes include. 00:17:51.960 --> 00:17:58.560 And then the Hamiltonian is just del squared little r over 2 00:17:58.560 --> 00:18:00.480 plus V effective. 00:18:03.240 --> 00:18:10.230 And this would be as a function of RA and RB. 00:18:10.230 --> 00:18:12.750 But it's because of R approximation, 00:18:12.750 --> 00:18:14.850 it's just a function of R. So these we pick. 00:18:14.850 --> 00:18:19.130 And this is the thing that we're worrying about. 00:18:19.130 --> 00:18:25.970 And then there's the 1 over RA minus RB, 00:18:25.970 --> 00:18:27.170 which is just a constant. 00:18:27.170 --> 00:18:31.420 This is the repulsion between two charged particles. 00:18:31.420 --> 00:18:33.150 So this is a lot nicer. 00:18:33.150 --> 00:18:35.520 We can also sort of forget about this. 00:18:35.520 --> 00:18:37.800 I mean, it's there, but it's a trivial thing 00:18:37.800 --> 00:18:39.000 that we can add in. 00:18:39.000 --> 00:18:41.940 So we have basically the kind of problem we like. 00:18:41.940 --> 00:18:45.050 A kinetic energy problem and a potential energy problem. 00:18:50.200 --> 00:18:52.964 So we're going to take this approach 00:18:52.964 --> 00:18:54.630 and we're going to learn how to apply it 00:18:54.630 --> 00:18:56.340 to the simplest system. 00:18:56.340 --> 00:18:59.010 And the simplest system is homonuclear. 00:19:03.000 --> 00:19:04.710 That's better than heteronuclear, 00:19:04.710 --> 00:19:08.370 because if we had two different atoms, 00:19:08.370 --> 00:19:13.500 we'd have to somehow represent their difference in properties. 00:19:13.500 --> 00:19:16.050 But because it's homonuclear, we just basically 00:19:16.050 --> 00:19:22.150 have an atom, the same atom, repeated, 00:19:22.150 --> 00:19:24.820 and there's a lot of symmetry that results. 00:19:24.820 --> 00:19:29.596 And the other is only two electrons. 00:19:29.596 --> 00:19:30.970 I mean, sorry, only one electron. 00:19:35.560 --> 00:19:39.620 Remember, we did hydrogen first, and that was wonderful. 00:19:39.620 --> 00:19:42.100 We got everything that was-- 00:19:42.100 --> 00:19:45.970 basically everything is related to anything else. 00:19:45.970 --> 00:19:49.120 Any property is related to any other property of the hydrogen 00:19:49.120 --> 00:19:52.280 atom through this orbital approximation. 00:19:52.280 --> 00:19:56.920 And so one electron makes us feel really comfortable, 00:19:56.920 --> 00:20:01.570 because we are going to expect that there 00:20:01.570 --> 00:20:05.590 is some structure, which means not just a random number 00:20:05.590 --> 00:20:08.950 or random collection of descriptive observations 00:20:08.950 --> 00:20:11.230 but some relationships. 00:20:11.230 --> 00:20:13.600 And we don't have to even antisymmetrize. 00:20:13.600 --> 00:20:16.510 So it's really something we can hit out of a park 00:20:16.510 --> 00:20:20.340 without getting too stressed. 00:20:20.340 --> 00:20:24.030 But there is a problem, and that is, 00:20:24.030 --> 00:20:29.970 how do we find the wave functions if we 00:20:29.970 --> 00:20:34.710 take a simple basis set? 00:20:34.710 --> 00:20:39.480 1s a plus 1s b. 00:20:39.480 --> 00:20:43.330 So these are hydrogenic wave functions. 00:20:43.330 --> 00:20:45.570 We know them. 00:20:45.570 --> 00:20:48.057 Now, they may not be the most convenient thing 00:20:48.057 --> 00:20:49.890 when you're doing a large scale calculation. 00:20:49.890 --> 00:20:52.590 In fact, nobody uses hydrogenic functions, 00:20:52.590 --> 00:20:56.140 even though they make a lot of sense, 00:20:56.140 --> 00:20:57.790 because they're not computationally as 00:20:57.790 --> 00:20:59.440 easy as Gaussian functions. 00:21:02.160 --> 00:21:09.080 So if you need to describe a problem with, say, 10 00:21:09.080 --> 00:21:15.290 hydrogenic functions and 1,000 Gaussians, 00:21:15.290 --> 00:21:17.730 you're way ahead of the game when you use 1,000 Gaussians, 00:21:17.730 --> 00:21:20.660 because the integrals are trivial. 00:21:20.660 --> 00:21:22.790 But in terms of understanding, it's 00:21:22.790 --> 00:21:27.050 really nice to take things we know and do everything 00:21:27.050 --> 00:21:27.880 in terms of them. 00:21:30.530 --> 00:21:32.140 OK. 00:21:32.140 --> 00:21:34.436 And so this is where we get the variational. 00:21:39.300 --> 00:21:42.020 So what we do is we say we're going 00:21:42.020 --> 00:21:46.250 to express the wave function as a linear combination 00:21:46.250 --> 00:21:48.800 of atomic orbitals. 00:21:48.800 --> 00:21:52.280 And that means we can adjust the coefficient 00:21:52.280 --> 00:21:57.510 of each atomic orbital to minimize the energy. 00:21:57.510 --> 00:22:00.370 And the variational principle or the variational theorem 00:22:00.370 --> 00:22:06.050 says, no matter what you do, you can't 00:22:06.050 --> 00:22:08.900 calculate an energy lower than the lowest 00:22:08.900 --> 00:22:13.290 energy of the system. 00:22:13.290 --> 00:22:15.290 Even if you don't know what it is, 00:22:15.290 --> 00:22:17.510 you know that anything you calculate 00:22:17.510 --> 00:22:19.110 is a little bit higher. 00:22:19.110 --> 00:22:20.690 And what you want to do, then, is 00:22:20.690 --> 00:22:25.460 to make the energy as small as possible by doing a min max 00:22:25.460 --> 00:22:29.010 problem on the coefficients. 00:22:29.010 --> 00:22:31.190 And so the variational method is basically that. 00:22:36.190 --> 00:22:38.220 OK. 00:22:38.220 --> 00:22:42.545 So basis set. 00:22:46.150 --> 00:22:49.590 I'm probably not going to finish my notes, even though these 00:22:49.590 --> 00:22:54.386 are a massively reduced version of what Troy had written, 00:22:54.386 --> 00:22:56.010 because I have some other things to say 00:22:56.010 --> 00:22:59.930 that aren't in the notes, and they're really important. 00:22:59.930 --> 00:23:04.000 When we do perturbation theory, we implicitly 00:23:04.000 --> 00:23:08.170 are dealing with an infinite basis set and an infinite set 00:23:08.170 --> 00:23:12.820 of known matrix elements. 00:23:12.820 --> 00:23:21.440 But we can't ask a computer to diagonalize an infinite matrix. 00:23:21.440 --> 00:23:24.220 So we have a tricky way of doing this 00:23:24.220 --> 00:23:27.850 so that we can use perturbation theory 00:23:27.850 --> 00:23:33.750 to get a good approximation of almost anything you want. 00:23:33.750 --> 00:23:35.940 But if your basis set is really terrible, 00:23:35.940 --> 00:23:39.500 then the perturbation theory is going to be terrible too. 00:23:39.500 --> 00:23:41.340 I mean, it's going to give correct answers. 00:23:41.340 --> 00:23:43.810 You're just going to have to do more calculations. 00:23:43.810 --> 00:23:46.590 The variational method says, OK, let's 00:23:46.590 --> 00:23:51.720 choose not an infinite basis set but a physically appropriate 00:23:51.720 --> 00:23:56.340 basis set or a computationally minimal basis set. 00:23:56.340 --> 00:24:03.280 And so we are making decisions about form and size. 00:24:03.280 --> 00:24:06.450 So we can use hydrogenic orbitals, 00:24:06.450 --> 00:24:07.770 because we know what they are. 00:24:07.770 --> 00:24:10.560 We know how to evaluate lots of integrals with them 00:24:10.560 --> 00:24:13.890 without too much trouble. 00:24:13.890 --> 00:24:15.700 And how many? 00:24:15.700 --> 00:24:20.260 Well, we've got two atoms. 00:24:20.260 --> 00:24:27.000 And so you sort of need one orbital on one atom 00:24:27.000 --> 00:24:29.880 and one orbital on another to even begin 00:24:29.880 --> 00:24:33.430 to capture the molecularness of the problem. 00:24:33.430 --> 00:24:38.550 So we've chosen the simplest possible form but not 00:24:38.550 --> 00:24:40.290 necessarily the most convenient form. 00:24:43.010 --> 00:24:46.950 And the size is two. 00:24:46.950 --> 00:24:48.250 Now, we could use four. 00:24:48.250 --> 00:24:49.840 We could use 100. 00:24:49.840 --> 00:24:52.000 And you could use a million. 00:24:52.000 --> 00:24:55.330 Most quantum chemical calculations, 00:24:55.330 --> 00:24:57.310 which you're going to be doing soon, 00:24:57.310 --> 00:25:01.720 involve not just millions but often 00:25:01.720 --> 00:25:04.240 billions of basis functions. 00:25:04.240 --> 00:25:05.710 And the computer does all the work 00:25:05.710 --> 00:25:07.720 and doesn't complain to you. 00:25:07.720 --> 00:25:12.260 It doesn't require a trip to the beach or anything like that. 00:25:12.260 --> 00:25:15.230 It just says, OK, here is what you wanted me to do. 00:25:15.230 --> 00:25:17.770 I don't know whether you-- it's a stupid question 00:25:17.770 --> 00:25:23.770 or a smart question, but I'm answering not what you thought 00:25:23.770 --> 00:25:27.640 you asked but what you did ask. 00:25:27.640 --> 00:25:28.390 OK. 00:25:28.390 --> 00:25:31.990 But we're going to show how it works with the simplest 00:25:31.990 --> 00:25:35.420 possible basis set. 00:25:35.420 --> 00:25:39.350 So our electronic wave function will 00:25:39.350 --> 00:25:43.190 be a function of electron position 00:25:43.190 --> 00:25:46.650 and parametrically nuclear position. 00:25:46.650 --> 00:25:56.290 C1, 1S A R R. I'm going to stop including 00:25:56.290 --> 00:26:02.310 this parametrically dependent on R. But eventually I'll do that. 00:26:02.310 --> 00:26:03.010 But for now. 00:26:08.810 --> 00:26:12.120 OK, so that's the wave function. 00:26:12.120 --> 00:26:13.820 The variational function. 00:26:13.820 --> 00:26:17.910 And we need to be able to calculate the average energy, 00:26:17.910 --> 00:26:21.710 because we know the average energy is going to be larger 00:26:21.710 --> 00:26:23.850 than the true energy. 00:26:23.850 --> 00:26:27.530 But if we do a good calculation, the energy 00:26:27.530 --> 00:26:30.530 difference between the truth and what you get 00:26:30.530 --> 00:26:32.970 is small as possible. 00:26:32.970 --> 00:26:35.660 And so you can say for a particular basis that we have, 00:26:35.660 --> 00:26:38.690 the best possible energies, if you want to do better, 00:26:38.690 --> 00:26:41.570 you have to use a bigger or smarter basis set. 00:26:45.720 --> 00:26:48.180 And so how do we calculate the average energy? 00:26:48.180 --> 00:26:50.600 Well, we do the usual thing where 00:26:50.600 --> 00:26:59.280 you say psi H psi d tau over psi star psi d tau. 00:26:59.280 --> 00:27:03.670 Now, I'm including this because we need it 00:27:03.670 --> 00:27:07.470 when we don't have orthonormal basis functions. 00:27:07.470 --> 00:27:11.730 And for this basis set, they're not orthonormal. 00:27:11.730 --> 00:27:13.930 They can't be. 00:27:13.930 --> 00:27:20.070 The wave functions for one atom are all mutually orthogonal. 00:27:20.070 --> 00:27:22.080 But because you have two atoms and there's 00:27:22.080 --> 00:27:25.560 overlap between them, they are not orthogonal, 00:27:25.560 --> 00:27:28.200 and they're not normalized. 00:27:28.200 --> 00:27:30.270 And so we have to do this. 00:27:34.030 --> 00:27:43.240 And so the problem is we want to minimize 00:27:43.240 --> 00:27:55.960 the energy with respect to C1 or C2 or C1 star or C2 star. 00:27:58.900 --> 00:28:01.870 Now, this is a subtle point. 00:28:01.870 --> 00:28:08.840 The complex conjugate of C is linearly independent of C. 00:28:08.840 --> 00:28:11.650 But they're related to each other. 00:28:11.650 --> 00:28:18.240 And so you can say that these derivatives are 0, 00:28:18.240 --> 00:28:22.640 and you can choose C1 and C2 or C1 star and C2 star, 00:28:22.640 --> 00:28:27.880 and you get the same result. And so it's 00:28:27.880 --> 00:28:30.940 important to understand this. 00:28:30.940 --> 00:28:33.430 So this is all by means of introduction. 00:28:33.430 --> 00:28:36.167 I'm going to start repeating myself. 00:28:36.167 --> 00:28:37.375 But let's draw some pictures. 00:28:42.930 --> 00:28:50.270 So the complicated thing is the potential. 00:28:50.270 --> 00:28:53.610 And so to illustrate the R dependence, 00:28:53.610 --> 00:28:57.900 I'm going to try to draw things. 00:28:57.900 --> 00:29:00.990 So we have r small and R large. 00:29:00.990 --> 00:29:03.390 This is the positions of the nuclei. 00:29:03.390 --> 00:29:06.525 And the potential will look like this. 00:29:21.280 --> 00:29:25.370 So the potential goes to minus infinity 00:29:25.370 --> 00:29:28.790 at the position of the nucleus, because the electron 00:29:28.790 --> 00:29:31.830 is attracted to the nucleus. 00:29:31.830 --> 00:29:34.420 And when the distance is 0, we have an-- 00:29:34.420 --> 00:29:38.410 it's a forgivable infinity because it-- 00:29:38.410 --> 00:29:39.730 anyway. 00:29:39.730 --> 00:29:44.600 So what I should really have done is exaggerated here. 00:29:44.600 --> 00:29:47.210 So when the particles are close together, 00:29:47.210 --> 00:29:52.340 the potential in the region between the nuclei 00:29:52.340 --> 00:29:55.430 doesn't go as close to 0 as it does when we do this. 00:30:00.300 --> 00:30:02.820 And so that's an important fact. 00:30:05.930 --> 00:30:10.640 So as the particles are brought together, 00:30:10.640 --> 00:30:15.450 something happens in this intermediate region. 00:30:15.450 --> 00:30:17.750 And when they're completely separated, 00:30:17.750 --> 00:30:19.520 it's as if they are completely separated. 00:30:19.520 --> 00:30:20.936 They're not talking to each other. 00:30:23.770 --> 00:30:27.360 And the wave functions look like this. 00:30:43.810 --> 00:30:46.012 I missed. 00:30:46.012 --> 00:30:47.470 And this is what I was telling you. 00:30:47.470 --> 00:30:48.326 This is a bond. 00:30:48.326 --> 00:30:49.450 This is why we have a bond. 00:30:52.430 --> 00:30:54.920 Because we have constructive interference 00:30:54.920 --> 00:30:57.890 in the spatial region where the electron can 00:30:57.890 --> 00:31:00.020 be attracted to both nuclei. 00:31:00.020 --> 00:31:05.240 Whereas here, there is no amplitude in between them 00:31:05.240 --> 00:31:08.060 or negligible amplitude, and so there 00:31:08.060 --> 00:31:11.030 isn't much of a contribution from the attraction 00:31:11.030 --> 00:31:12.780 to both nuclei. 00:31:12.780 --> 00:31:15.920 And so this is the separated atom or approaching separated 00:31:15.920 --> 00:31:16.920 atom limit. 00:31:16.920 --> 00:31:18.530 And this is the molecular limit. 00:31:18.530 --> 00:31:19.970 And now it's possible that you can 00:31:19.970 --> 00:31:24.610 squeeze the atoms too close together 00:31:24.610 --> 00:31:27.640 and bad things start to happen. 00:31:27.640 --> 00:31:29.710 Because then what you're doing is you're 00:31:29.710 --> 00:31:33.260 putting two electrons into the same spin orbital. 00:31:33.260 --> 00:31:37.030 And we know that that's overlap repulsion 00:31:37.030 --> 00:31:39.490 or it violates the exclusion principle 00:31:39.490 --> 00:31:42.520 and there is an energy penalty that you 00:31:42.520 --> 00:31:45.190 pay in order to get too close. 00:31:45.190 --> 00:31:48.840 But basically, this is what we're going to be explaining. 00:31:48.840 --> 00:31:51.220 And we're going to recover this from a real calculation. 00:31:59.450 --> 00:32:01.370 OK. 00:32:01.370 --> 00:32:07.940 So again, we want to minimize the average energy. 00:32:07.940 --> 00:32:11.810 And the average energy, as I've written somewhere, 00:32:11.810 --> 00:32:25.240 is given by psi star H psi d tau psi star psi d tau. 00:32:25.240 --> 00:32:27.510 And if we are writing these things 00:32:27.510 --> 00:32:34.440 as C1 1S A times C2 1S B, then we 00:32:34.440 --> 00:32:39.600 end up getting stuff that we know how to calculate now. 00:32:44.720 --> 00:33:03.320 1S A. This orbital is not orthogonal to 1S B. 00:33:03.320 --> 00:33:04.100 It can't be. 00:33:07.210 --> 00:33:09.180 And so this is something that we had 00:33:09.180 --> 00:33:12.510 been able to avoid in all of our one dimensional problems. 00:33:12.510 --> 00:33:17.250 We could always think about a orthonormalized basis set. 00:33:17.250 --> 00:33:20.870 But the effect that we're really interested in 00:33:20.870 --> 00:33:25.320 is based on the fact that these guys aren't orthogonal. 00:33:25.320 --> 00:33:31.650 You can't just say we're going to write an orthonormal basis 00:33:31.650 --> 00:33:32.490 set. 00:33:32.490 --> 00:33:34.170 It's not possible. 00:33:34.170 --> 00:33:38.250 If you want to construct an orthonormal basis set, 00:33:38.250 --> 00:33:41.010 there is a transformation you can do. 00:33:41.010 --> 00:33:43.770 And it's a simple transformation, 00:33:43.770 --> 00:33:45.820 and it's done all the time. 00:33:45.820 --> 00:33:50.280 And you'll see it, but not today. 00:33:50.280 --> 00:33:52.230 But you still do the calculations 00:33:52.230 --> 00:33:56.610 on the non-orthogonal basis functions. 00:33:56.610 --> 00:33:59.850 And then you do a transformation that takes into account 00:33:59.850 --> 00:34:03.150 the lack of orthonormality. 00:34:03.150 --> 00:34:06.310 For now just accept it. 00:34:06.310 --> 00:34:10.380 And so we're going to have to worry about an integral 00:34:10.380 --> 00:34:11.864 like this one. 00:34:11.864 --> 00:34:15.310 SA 1 SB. 00:34:18.300 --> 00:34:20.340 And we call this the overlap integral. 00:34:27.760 --> 00:34:32.280 And this is something that we have to not ignore. 00:34:35.179 --> 00:34:40.010 So now we can approach this problem using wave function 00:34:40.010 --> 00:34:41.239 notation. 00:34:41.239 --> 00:34:45.170 And it's most familiar. 00:34:45.170 --> 00:34:49.580 But we can also use these things saying, OK, this 00:34:49.580 --> 00:34:54.320 is the complete set of the variational coefficients. 00:34:54.320 --> 00:34:58.770 And this is a step towards the matrix picture. 00:34:58.770 --> 00:35:07.410 Because it's basically a linear array of mixing coefficients. 00:35:07.410 --> 00:35:11.240 And you could think of that as a vector. 00:35:11.240 --> 00:35:14.210 And in fact, that's how you do the vector picture, 00:35:14.210 --> 00:35:18.880 but we'll use sort of a hybrid notation for the time being. 00:35:18.880 --> 00:35:28.970 But we can also say, OK, this means C1 C2 Cn. 00:35:28.970 --> 00:35:38.420 And this means C1 star Cn star. 00:35:38.420 --> 00:35:42.380 And here we are in the matrix picture. 00:35:42.380 --> 00:35:45.100 OK, but I will use sort of a hybrid picture 00:35:45.100 --> 00:35:48.470 until you're ready to take the complete jump. 00:35:48.470 --> 00:35:53.050 And so the average energy expressed 00:35:53.050 --> 00:35:55.900 as a function of this set of coefficients 00:35:55.900 --> 00:35:57.740 is going to be integral. 00:35:57.740 --> 00:36:26.180 Psi C star H psi C d tau over integral psi C star psi. 00:36:30.120 --> 00:36:31.830 The reason I'm being hesitant here 00:36:31.830 --> 00:36:35.400 is because I work in the matrix picture, 00:36:35.400 --> 00:36:41.720 and this hybrid picture is away from both comfort zones. 00:36:41.720 --> 00:36:44.850 But it is leading you towards what you need to see, 00:36:44.850 --> 00:36:46.910 and so we're going to do it. 00:36:46.910 --> 00:36:47.520 OK. 00:36:47.520 --> 00:36:50.400 So in the wave function picture, this thing 00:36:50.400 --> 00:37:02.635 becomes just C1 1S A plus C2 1S B, 00:37:02.635 --> 00:37:15.250 the whole thing conjugated, H C1 1S A C2 psi 1S B d tau. 00:37:19.200 --> 00:37:24.780 And the same sort of thing down here, except no Hamiltonian. 00:37:24.780 --> 00:37:29.140 And so we can now slavishly write down 00:37:29.140 --> 00:37:33.190 all of the terms that result. And what 00:37:33.190 --> 00:37:40.100 we get is from the Hamiltonian matrix element. 00:37:40.100 --> 00:37:52.900 So for H, we get C1 star C1 H11 plus C1 star C2 00:37:52.900 --> 00:38:09.130 H1 H12 plus C2 star C1 H21 plus C2 star C2 H22. 00:38:09.130 --> 00:38:11.470 OK, we think we know some stuff. 00:38:11.470 --> 00:38:15.590 We think this is going to be the energy of the hydrogen atom. 00:38:15.590 --> 00:38:17.140 But it isn't. 00:38:17.140 --> 00:38:23.990 Because the orbital one sees both nuclei. 00:38:23.990 --> 00:38:28.150 And so this is something which we just call this guy. 00:38:28.150 --> 00:38:30.100 We're calling this epsilon. 00:38:30.100 --> 00:38:32.635 And that is a function of internuclear distance. 00:38:35.400 --> 00:38:41.275 But this guy is the same, because it's homonuclear. 00:38:47.220 --> 00:38:52.200 And this, well, we can call it V12 or V. 00:38:52.200 --> 00:38:55.570 And these two guys are the same thing. 00:38:55.570 --> 00:39:02.900 So we've reduced the problem to a simple equation. 00:39:02.900 --> 00:39:07.260 But I'm not going to go all the way there. 00:39:07.260 --> 00:39:10.330 But you know how to do this. 00:39:10.330 --> 00:39:12.390 You know how to do the overlap integral. 00:39:12.390 --> 00:39:14.610 You're gonna have a similar sort of thing. 00:39:14.610 --> 00:39:19.950 S11 is the overlap of the orbital 1 with itself. 00:39:19.950 --> 00:39:21.330 And that one is 1. 00:39:25.390 --> 00:39:27.460 But S12 is S. 00:39:27.460 --> 00:39:37.240 So anyway, what we end up doing is a lot of algebra. 00:39:37.240 --> 00:39:41.250 And what we end up getting when we impose the condition 00:39:41.250 --> 00:39:47.350 that the partial of everything with respect to C1 star 00:39:47.350 --> 00:39:50.230 is equal to 0, we get an equation 00:39:50.230 --> 00:39:51.820 for the average energy. 00:39:51.820 --> 00:39:56.170 And the average energy is-- 00:39:56.170 --> 00:39:57.530 oh, I'm sorry. 00:39:57.530 --> 00:39:58.590 C1. 00:39:58.590 --> 00:40:02.390 When we take the derivative with respect to C1, what we get 00:40:02.390 --> 00:40:04.040 is this equation. 00:40:04.040 --> 00:40:21.793 C1 star H11 plus C2 star H22 over C1 star S11 plus C2 star 00:40:21.793 --> 00:40:22.293 S21. 00:40:25.920 --> 00:40:29.380 And so we could also do the same thing, 00:40:29.380 --> 00:40:33.150 taking the partial derivative with respect to C2 00:40:33.150 --> 00:40:35.144 and setting that equal to 0. 00:40:35.144 --> 00:40:36.310 But we get another equation. 00:40:39.890 --> 00:40:52.350 And so that other equation would be E average is equal to-- 00:40:59.380 --> 00:41:01.010 no, I'm not even going to write it, 00:41:01.010 --> 00:41:02.593 because I didn't write it in my notes, 00:41:02.593 --> 00:41:06.860 and I'm operating on batteries. 00:41:06.860 --> 00:41:08.390 You can write that. 00:41:08.390 --> 00:41:11.900 And so we have two equations for the average energy. 00:41:11.900 --> 00:41:16.580 And well, that's kind of good, because we have two unknowns. 00:41:16.580 --> 00:41:19.940 So we're going to be able to solve for them. 00:41:19.940 --> 00:41:21.130 OK. 00:41:21.130 --> 00:41:31.480 Now, we could also approach this problem using matrix methods. 00:41:31.480 --> 00:41:33.910 And since that's the way computers think 00:41:33.910 --> 00:41:37.600 about the problem and that's the way most people think 00:41:37.600 --> 00:41:40.900 about the problem, you really want to understand the matrix 00:41:40.900 --> 00:41:41.890 picture. 00:41:41.890 --> 00:41:45.040 And at risk of repeating myself, we 00:41:45.040 --> 00:41:53.750 know for this two level problem, we have H11 H12 H21 H22. 00:41:53.750 --> 00:42:01.840 And we know that the S matrix S11 S12 S21 S22. 00:42:01.840 --> 00:42:10.180 So we have an equation C1 star C2 star 00:42:10.180 --> 00:42:26.620 times the H matrix is equal to E average times the C1 star C2 00:42:26.620 --> 00:42:31.420 star times the S matrix. 00:42:31.420 --> 00:42:34.860 So this is the H matrix and this is the S matrix. 00:42:34.860 --> 00:42:37.410 And you can write this down. 00:42:37.410 --> 00:42:43.650 And when you take the derivatives with respect to C, 00:42:43.650 --> 00:42:57.670 you get C dagger H is equal to E average times C dagger S. 00:42:57.670 --> 00:43:01.780 Now, if instead of taking derivatives with respect 00:43:01.780 --> 00:43:04.780 to the Cs we took the derivatives with respect 00:43:04.780 --> 00:43:10.130 to the C stars, we'd get a different equation. 00:43:10.130 --> 00:43:22.870 And that equation would be HC is equal to E average times S then 00:43:22.870 --> 00:43:26.970 C. This guy looks familiar. 00:43:29.510 --> 00:43:33.610 It looks like Hamiltonian times something 00:43:33.610 --> 00:43:38.444 is equal to the energy times-- 00:43:38.444 --> 00:43:40.040 uh oh. 00:43:40.040 --> 00:43:43.730 We have something else here. 00:43:43.730 --> 00:43:45.042 It's not just the energy. 00:43:45.042 --> 00:43:46.250 We have the overlap integral. 00:43:49.040 --> 00:43:54.050 So these two things are examples of the general eigenvalue 00:43:54.050 --> 00:43:56.210 equation. 00:43:56.210 --> 00:44:01.060 And so in almost all calculations, 00:44:01.060 --> 00:44:07.480 the fact that S is present leads to problems of algebra. 00:44:11.490 --> 00:44:18.810 But in this particular problem, we know S12 is equal to S21. 00:44:18.810 --> 00:44:26.330 And this epsilon parameter can be-- 00:44:26.330 --> 00:44:32.060 so we have H11 is equal to H22. 00:44:32.060 --> 00:44:35.490 They're both parametrically dependent on internuclear 00:44:35.490 --> 00:44:39.950 distance, and they're equal to epsilon of R. 00:44:39.950 --> 00:44:43.760 So what we end up with is an equation that 00:44:43.760 --> 00:44:44.865 looks really familiar. 00:44:50.780 --> 00:44:53.450 We have a 2 by 2 matrix, which we can diagonalize. 00:44:53.450 --> 00:44:56.060 We can find the energies. 00:44:56.060 --> 00:44:59.420 And so what you end up getting when you do that. 00:44:59.420 --> 00:45:03.470 And I want to spare you my attempting 00:45:03.470 --> 00:45:06.226 to do a derivation in real time. 00:45:06.226 --> 00:45:07.600 You can look at the notes and you 00:45:07.600 --> 00:45:11.950 can see that the results are what you care about 00:45:11.950 --> 00:45:13.910 and what we're going to interpret. 00:45:13.910 --> 00:45:14.870 So here we go. 00:45:18.940 --> 00:45:20.760 So what you find. 00:45:20.760 --> 00:45:25.740 The average energy for one of the orbitals that we get. 00:45:25.740 --> 00:45:27.000 Remember, we got a 2 by 2. 00:45:27.000 --> 00:45:31.740 We're going to have two eigenvalues and eigenstates. 00:45:31.740 --> 00:45:41.880 And so for one of those, we get epsilon plus V12 1 plus S1. 00:45:46.730 --> 00:45:49.430 So these are symbols that we've seen. 00:45:49.430 --> 00:45:53.430 And this tells us now this, this, 00:45:53.430 --> 00:45:56.350 and this are functions of R. So we're 00:45:56.350 --> 00:45:58.960 going to have an energy for the sigma orbital, which 00:45:58.960 --> 00:46:01.150 is a function of R. 00:46:01.150 --> 00:46:07.120 And we have another solution, which we're calling sigma star. 00:46:09.910 --> 00:46:19.650 And that one is epsilon minus V12 over 1 minus S12. 00:46:19.650 --> 00:46:24.840 So we'd like to be able to know which of these is 00:46:24.840 --> 00:46:27.210 the absolute-- one of these is-- 00:46:30.620 --> 00:46:38.850 so we have sigma and sigma star. 00:46:38.850 --> 00:46:46.100 Both as a function of R. And so if we 00:46:46.100 --> 00:46:49.430 draw a molecular orbital diagram and we say, OK, well here we 00:46:49.430 --> 00:46:59.770 have the energy of 1SA, which is epsilon as a function of R. 00:46:59.770 --> 00:47:09.650 And this is epsilon energy 1SB, which is equal to epsilon of R. 00:47:09.650 --> 00:47:13.310 Normally we choose R as equal to infinity 00:47:13.310 --> 00:47:15.770 for the separated atom. 00:47:15.770 --> 00:47:18.410 So these are actually just the energies 00:47:18.410 --> 00:47:20.810 of the individual orbitals. 00:47:20.810 --> 00:47:24.320 And what we end up getting is a molecular orbital diagram 00:47:24.320 --> 00:47:30.790 which, in an exaggerated form, is like this. 00:47:30.790 --> 00:47:33.130 We get bonding, because it's more 00:47:33.130 --> 00:47:36.140 stable than the separate atoms, and anti-bonding, 00:47:36.140 --> 00:47:37.630 which is less stable. 00:47:37.630 --> 00:47:40.600 And so one question we'd like to ask is, 00:47:40.600 --> 00:47:43.630 well, one question is why is it bonding? 00:47:43.630 --> 00:47:44.680 Why is this more stable? 00:47:44.680 --> 00:47:47.710 And the answer is constructive interference 00:47:47.710 --> 00:47:51.070 in the important energy region. 00:47:51.070 --> 00:47:53.980 But another question is, is this bigger? 00:47:53.980 --> 00:47:59.480 Is this a larger destabilization than this stabilization? 00:47:59.480 --> 00:48:02.500 And it's kind of hard to scope that out, 00:48:02.500 --> 00:48:07.790 because here we have two parameters, which are added, 00:48:07.790 --> 00:48:09.820 and here we have a difference of two parameters. 00:48:12.780 --> 00:48:14.620 And here we have a denominator, which 00:48:14.620 --> 00:48:19.210 is going to be larger for the sigma orbital 00:48:19.210 --> 00:48:21.580 than the sigma star. 00:48:21.580 --> 00:48:26.160 And so we don't know whether the plus versus minus V 00:48:26.160 --> 00:48:28.030 is dominant. 00:48:28.030 --> 00:48:30.310 But the energy denominator is saying, 00:48:30.310 --> 00:48:33.330 yeah, this one, the anti-bonding orbital 00:48:33.330 --> 00:48:36.300 is more anti-bonding than the bonding one. 00:48:36.300 --> 00:48:38.520 Now, you can actually do some algebra 00:48:38.520 --> 00:48:41.700 to calculate what is the difference in energy 00:48:41.700 --> 00:48:49.380 between the sigma star as a function of R. 00:48:49.380 --> 00:48:54.250 But evaluated at R sub E, which is the minimum-- 00:48:54.250 --> 00:49:00.000 the energy where the bonding orbital has its minimum energy. 00:49:00.000 --> 00:49:04.830 Then if you evaluate at that particular value of R 00:49:04.830 --> 00:49:13.960 and you make that comparison B 1S evaluated at R sub B, 00:49:13.960 --> 00:49:25.360 and you get epsilon times S12 minus-- 00:49:30.170 --> 00:49:32.300 so I'm doing sigma star, so that's 00:49:32.300 --> 00:49:44.120 this one minus B12 over 1 plus 1 minus S12. 00:49:44.120 --> 00:49:48.590 And for the sigma orbital, I have a similar equation, 00:49:48.590 --> 00:49:59.830 but I get epsilon S12 minus B12 over 1 minus S12. 00:49:59.830 --> 00:50:03.520 So we have the same numerator for these two cases 00:50:03.520 --> 00:50:05.330 but a different denominator. 00:50:05.330 --> 00:50:07.990 And so this is saying this simple variational calculation 00:50:07.990 --> 00:50:13.110 says you're going to get a pair of orbitals, one bonding, 00:50:13.110 --> 00:50:14.650 one anti-bonding. 00:50:14.650 --> 00:50:17.460 And the anti-bonding one is more destabilized 00:50:17.460 --> 00:50:19.540 relative to the separate atoms then 00:50:19.540 --> 00:50:22.050 the bonding one is stabilized. 00:50:22.050 --> 00:50:25.930 And this is a kind of general expectation 00:50:25.930 --> 00:50:27.370 that the anti-bonding orbitals are 00:50:27.370 --> 00:50:31.090 more destabilized than the bonding ones. 00:50:31.090 --> 00:50:33.970 But it's not a proof, because its just a proof 00:50:33.970 --> 00:50:36.490 for the simplest possible case. 00:50:36.490 --> 00:50:38.570 So I'm done. 00:50:38.570 --> 00:50:43.000 And in the next lecture, we're going 00:50:43.000 --> 00:50:46.840 to use molecular orbital diagrams like this 00:50:46.840 --> 00:50:50.405 to describe a series of examples. 00:50:53.210 --> 00:51:01.241 Homonuclear diatomic molecules, heteronuclear AH, and AB. 00:51:01.241 --> 00:51:04.300 H is special, and this is more general. 00:51:04.300 --> 00:51:06.120 And what we'd like to be able to do 00:51:06.120 --> 00:51:09.180 is without a great deal of thought 00:51:09.180 --> 00:51:11.590 construct the molecular orbital diagram 00:51:11.590 --> 00:51:14.995 for basically any diatomic molecule using some insight. 00:51:18.790 --> 00:51:23.190 Now, this also transfers to polyatomic molecules. 00:51:23.190 --> 00:51:25.020 And it's especially beautiful the way 00:51:25.020 --> 00:51:29.130 it transfers, because electronic structure is usually 00:51:29.130 --> 00:51:32.850 centered at a particular part of the molecule. 00:51:32.850 --> 00:51:35.670 And so, basically, you're reducing polyatomic molecules 00:51:35.670 --> 00:51:38.580 to a diatomic that you understanding the bonding 00:51:38.580 --> 00:51:40.050 and anti-bonding. 00:51:40.050 --> 00:51:44.910 OK, so I'll continue on Monday. 00:51:44.910 --> 00:51:48.660 And your problem set for this week is due on Monday. 00:51:48.660 --> 00:51:50.490 And the following set for the next week 00:51:50.490 --> 00:51:54.410 will also be due on Monday. 00:51:54.410 --> 00:51:55.260 Not that Monday. 00:51:55.260 --> 00:51:56.135 The following Monday. 00:51:59.547 --> 00:52:01.380 And that's the next to the last problem set. 00:52:01.380 --> 00:52:06.040 So you're inside of the home base. 00:52:06.040 --> 00:52:09.430 OK, have a good weekend.