WEBVTT 00:01.690 --> 00:05.390 Prof: Okay, let's get started. 00:05.390 --> 00:11.000 So last time we looked at methane and ammonia, 00:11.000 --> 00:14.650 and saw something interesting about molecular orbitals: 00:14.650 --> 00:18.300 we could give them the same name as atomic orbitals; 00:18.300 --> 00:22.130 that is, if we, especially if we look at a low 00:22.127 --> 00:26.717 electron density contour of the molecular orbitals, 00:26.720 --> 00:32.070 we see that what it looks like is an atom where the nucleus is 00:32.070 --> 00:33.650 split into pieces. 00:33.650 --> 00:34.440 Right? 00:34.440 --> 00:39.010 Now that splitting into pieces changes the potential energy for 00:39.005 --> 00:40.105 the electron. 00:40.110 --> 00:42.290 So you expect that to distort the orbital. 00:42.290 --> 00:45.950 The electrons will move in the directions that pieces of the 00:45.949 --> 00:47.189 nucleus have gone. 00:47.190 --> 00:51.310 But then, in addition to potential energy controlling the 00:51.313 --> 00:54.703 shape of orbitals, kinetic energy also does. 00:54.700 --> 00:57.040 And that's the same thing as it is in an atom. 00:57.040 --> 01:00.390 You have a thing with no nodes, a thing with one node -- 01:00.390 --> 01:04.060 and when you have one node you can have either a spherical 01:04.055 --> 01:05.855 node, or a planar node, 01:05.863 --> 01:08.683 and there can be three planar nodes; 01:08.680 --> 01:10.920 so a 2s and three 2p's. 01:10.920 --> 01:14.880 Exactly the same considerations apply in a molecular orbital as 01:14.879 --> 01:16.349 in an atomic orbital. 01:16.349 --> 01:19.369 You have the kinetic energy, which comes with curvature, 01:19.367 --> 01:20.737 which comes with nodes. 01:20.739 --> 01:25.239 Now, as you split the nucleus up and pull pieces in different 01:25.239 --> 01:27.219 directions, it doesn't have the same 01:27.223 --> 01:29.483 symmetry it had when it was all together in the nucleus, 01:29.480 --> 01:30.860 a spherical kind of symmetry. 01:30.860 --> 01:32.780 So the nodes get distorted. 01:32.780 --> 01:34.300 But still you can see them there. 01:34.300 --> 01:38.710 And we saw them last time and went through all the occupied 01:38.706 --> 01:42.806 and vacant valance orbitals of ammonia and methane, 01:42.810 --> 01:45.430 and saw how they looked like atomic orbitals. 01:45.430 --> 01:48.570 This, not surprisingly, because it's so fundamental, 01:48.572 --> 01:51.222 the potential energy and the kinetic energy, 01:51.224 --> 01:52.954 applies to every system. 01:52.950 --> 01:56.750 It applies to you, viewed as a single atom -- 01:56.748 --> 01:57.438 right? 01:57.440 --> 02:00.240 -- with a zillion electrons. 02:00.239 --> 02:00.899 Okay? 02:00.900 --> 02:04.310 But pieces have moved around, so the orbitals change. 02:04.310 --> 02:07.690 We'll look at two more complicated cases and then we'll 02:07.689 --> 02:10.819 get on to a different way of looking at bonding. 02:10.818 --> 02:12.768 So we'll look at ethane and methanol. 02:12.770 --> 02:16.180 And we use -- I didn't tell you last time, explicitly, 02:16.177 --> 02:18.877 where I got the molecular orbitals from. 02:18.879 --> 02:20.369 I got them from my laptop. 02:20.370 --> 02:23.270 There's a program -- the particular one is called 02:23.274 --> 02:26.054 Spartan, that I use -- and it calculates 02:26.054 --> 02:28.514 what molecular orbitals look like, 02:28.508 --> 02:31.938 using approximate molecular orbital (that is, 02:35.370 --> 02:38.770 Okay, so let's just look at the ones of ethane and methanol. 02:38.770 --> 02:43.130 Now both of these have seven pairs of valence electrons. 02:43.128 --> 02:47.188 There are also core electrons, and if we were looking at the 02:47.186 --> 02:50.826 orbitals for all the electrons, we'd include those. 02:50.830 --> 02:54.230 And exactly how we're going to count those -- 02:54.229 --> 02:56.099 you could do it one way or the other -- 02:56.098 --> 02:58.198 whether you consider the core electrons, 02:58.199 --> 03:01.089 the 1s electrons to just be part of the nucleus and then 03:01.092 --> 03:03.662 treat the rest as the electrons you're interested in; 03:03.658 --> 03:08.018 or whether you want to count the core electrons too. 03:08.020 --> 03:10.750 You can do it either way, but what you analogize to what 03:10.745 --> 03:13.615 depends on whether you count them as part of the nuclei. 03:13.620 --> 03:17.800 So anyhow, we're going to compare these molecular orbitals 03:17.796 --> 03:22.266 to the atomic orbitals of argon, which has also seven electron 03:22.266 --> 03:22.996 pairs. 03:23.000 --> 03:26.640 Okay, so there's the 2s orbital. 03:26.639 --> 03:29.489 I'm going to start with that, because I'm going to pretend 03:29.490 --> 03:32.340 that that red part is the core -- that there's a 1s 03:32.340 --> 03:33.740 which is core electrons. 03:33.740 --> 03:37.460 But in this case -- here's my pedantic note on this 03:37.464 --> 03:39.364 subject which I just added. 03:39.360 --> 03:42.910 So if you have -- before we had just one heavy atom, 03:42.913 --> 03:45.563 carbon in methane, nitrogen in ammonia. 03:45.562 --> 03:46.262 Right? 03:46.258 --> 03:48.878 So there was one 1s orbital. 03:48.882 --> 03:49.502 Right? 03:49.500 --> 03:54.110 Now we have two heavy atoms, carbon and carbon in methane, 03:54.110 --> 03:56.700 carbon and oxygen in methanol. 03:56.699 --> 04:01.459 So there are two heavy atoms and therefore two boring core 04:01.461 --> 04:02.381 orbitals. 04:02.378 --> 04:05.118 So for purposes of making analogies, 04:05.120 --> 04:08.080 we'll use the atomic 1s orbital, 04:08.080 --> 04:11.260 of the atom that we're analogizing things to, 04:11.258 --> 04:14.258 to stand for all the molecular core orbitals. 04:14.258 --> 04:15.488 You can do it any way you want to. 04:15.490 --> 04:16.870 We're not really interested in that. 04:16.870 --> 04:18.150 We're interested in the valence orbitals. 04:18.149 --> 04:20.889 So whether I start with the 1s or the 2s 04:20.887 --> 04:23.417 depends on how I'm handling the core electrons; 04:23.420 --> 04:24.630 it's not a big deal. 04:24.629 --> 04:28.849 Anyhow, let's pretend it's the 2s of argon here, 04:28.845 --> 04:33.445 and we're going to compare it with this lowest valence level 04:33.451 --> 04:35.951 molecular orbital of ethane. 04:35.949 --> 04:40.969 Now we can -- as I do this, on the left side of the 04:40.971 --> 04:44.891 pictures I'm going to show one view, 04:44.889 --> 04:49.469 and then, because it's more complicated than methane and 04:49.468 --> 04:52.048 ammonia were, I'm going to also show a 04:52.052 --> 04:53.882 picture rotated by ninety degrees. 04:53.879 --> 04:56.809 So I'm going to rotate around that axis, and on the right I'll 04:56.807 --> 04:58.917 show a different view of the same orbital. 04:58.920 --> 05:02.300 Okay, so that's the lowest valence level molecular orbital 05:02.298 --> 05:03.008 of ethane. 05:03.009 --> 05:07.769 And it's not very exciting; it's just a distorted sphere. 05:07.769 --> 05:09.799 And you can see the way in which it's distorted. 05:09.800 --> 05:12.140 It's distorted vertically, because the two carbons are 05:12.137 --> 05:14.837 pulled apart; so it's got sort of a narrow 05:14.841 --> 05:15.791 waist to it. 05:15.790 --> 05:20.360 And then it's pulled out, where each of the protons left 05:20.357 --> 05:24.257 the middle atom to come out and be hydrogens. 05:24.259 --> 05:24.879 Okay? 05:24.879 --> 05:29.249 Now if we look at methanol, it'll be a little different. 05:29.250 --> 05:32.480 Can you anticipate how it might be different if, 05:32.480 --> 05:35.990 on the bottom we'll have CH_3 again, but on the top, 05:35.985 --> 05:39.485 instead of having CH_3, we're going to have OH. 05:39.490 --> 05:42.610 How do you think it might be different, in the way it looks? 05:42.610 --> 05:46.140 Student: It might be skewed more towards the OH 05:46.144 --> 05:48.864 because oxygen is -- Prof: Okay, 05:48.855 --> 05:51.465 so oxygen has a bigger nuclear charge. 05:51.470 --> 05:56.120 So that's going to pull the lowest energy orbital toward the 05:56.120 --> 05:56.830 oxygen. 05:56.829 --> 05:57.669 That'll be one thing. 05:57.670 --> 06:00.330 We expect it to be bigger, top-heavy. 06:00.329 --> 06:00.939 Okay? 06:00.939 --> 06:04.199 What else, about how the top will look? 06:04.199 --> 06:07.349 How will it be different from if it were a CH_3? 06:07.350 --> 06:08.680 Yes, Alex? 06:08.680 --> 06:10.580 I can't hear very well. 06:10.579 --> 06:11.329 Student: Oxygen has lone pairs. 06:11.329 --> 06:12.949 Prof: Oxygen has lone pairs. 06:12.949 --> 06:15.179 Now, how is that going to change things? 06:15.180 --> 06:18.440 Or another way of saying the same thing is it has only one 06:18.437 --> 06:19.577 hydrogen up there. 06:19.579 --> 06:21.989 How's that going to change it, do you think? 06:21.990 --> 06:23.160 You have to speak up. 06:23.160 --> 06:24.320 Student: It's not going to be symmetrical. 06:24.319 --> 06:25.499 Prof: It's not going to be symmetrical. 06:25.500 --> 06:30.690 Which way is it going to be distorted to be unsymmetrical? 06:30.689 --> 06:31.829 You have to speak up. 06:31.829 --> 06:33.439 Student: Towards the electron pair. 06:33.440 --> 06:35.690 Prof: Toward the electron pair did you say? 06:35.690 --> 06:36.210 I couldn't hear. 06:36.209 --> 06:37.269 Student: Away from hydrogen. 06:37.269 --> 06:39.629 Prof: Away from hydrogen? 06:39.629 --> 06:41.639 Well how about -- the proton goes out; 06:41.639 --> 06:45.489 does it bring electrons with it, or does it repel electrons, 06:45.490 --> 06:46.340 the proton? 06:46.339 --> 06:46.969 Speak up. 06:46.970 --> 06:47.810 Student: It brings them. 06:47.810 --> 06:50.100 Prof: It pulls, so it should distort toward the 06:50.095 --> 06:50.565 hydrogen. 06:50.569 --> 06:52.289 So here's what it looks like. 06:52.293 --> 06:52.713 Right? 06:52.709 --> 06:54.749 It's top-heavy, as Angela said, 06:54.745 --> 06:57.795 and it's distorted out toward the hydrogen; 06:57.800 --> 07:00.710 there are no protons pulling it to the top left. 07:00.709 --> 07:03.269 And you see the same thing end-on there, 07:03.266 --> 07:04.246 on the right. 07:04.250 --> 07:07.060 Okay, this is the next orbital. 07:07.060 --> 07:09.610 What does that look like? 07:09.610 --> 07:12.060 Obviously you can peek in the middle and see. 07:12.060 --> 07:16.020 It's obviously a 2p_z orbital, with the node across 07:16.016 --> 07:16.916 the middle. 07:16.920 --> 07:17.310 Okay? 07:17.310 --> 07:18.380 In both cases. 07:18.379 --> 07:22.089 Now the ethane case is symmetrical. 07:22.089 --> 07:23.829 The other case is unsymmetrical. 07:23.829 --> 07:25.449 Why is it unsymmetrical? 07:25.449 --> 07:29.459 Because the first orbital pulled electrons to the top. 07:29.464 --> 07:30.074 Right? 07:30.069 --> 07:33.549 So the next- in the next orbital, electrons aren't going 07:33.545 --> 07:35.815 to want to be at the top so much, 07:35.819 --> 07:37.959 because they're going to be repelled by the other electrons; 07:37.959 --> 07:40.929 so there'll be more toward the bottom, because the first ones 07:40.928 --> 07:41.818 went to the top. 07:41.819 --> 07:44.409 Okay? 07:44.410 --> 07:47.610 Okay, then here's the next orbital. 07:47.610 --> 07:50.420 You can see there are energies marching up here. 07:50.420 --> 07:53.770 The lowest one was s, then 2p_z, 07:53.769 --> 07:56.969 now 2p_x, if we define the horizontal 07:56.971 --> 07:59.951 axis here on the left as the x axis. 07:59.949 --> 08:03.079 So again, it's what you expect, and it's pulled out, 08:03.084 --> 08:06.844 stretched vertically from being a dumbbell by where the nuclei 08:06.836 --> 08:07.386 went. 08:07.389 --> 08:11.719 And here's 2p_y, which we can see more clearly 08:11.718 --> 08:14.468 on the right, perpendicular to the 08:14.466 --> 08:15.796 2p_x. 08:15.800 --> 08:16.800 Okay? 08:16.800 --> 08:22.160 And notice that it's obviously so, that the methanol orbital 08:22.161 --> 08:25.431 will be less symmetric, in all cases, 08:25.434 --> 08:27.984 than the one for ethane. 08:27.980 --> 08:31.010 But still we can recognize the nodes, because they must go that 08:31.007 --> 08:31.297 way. 08:31.300 --> 08:35.340 It must be no nodes, one node of three different 08:35.344 --> 08:36.984 kinds, and so on. 08:36.980 --> 08:40.160 Okay, now this is 3s. 08:40.158 --> 08:43.238 You notice it has -- the node that we don't see, 08:43.240 --> 08:47.120 the one that's down near the nuclei, 08:47.120 --> 08:49.090 in fact two nodes down near the nuclei, 08:49.090 --> 08:51.670 one for each of the heavy atoms. 08:51.668 --> 08:54.858 But then this now has another spherical node, 08:54.861 --> 08:57.331 or approximately spherical node. 08:57.330 --> 09:01.050 So we have that extra red lump in the middle on the top, 09:01.048 --> 09:02.738 or blue, in the middle. 09:02.740 --> 09:05.800 Just to review, what's the difference between 09:05.804 --> 09:06.854 red and blue? 09:06.850 --> 09:10.170 On the top, the computer decided to draw it with red in 09:10.169 --> 09:10.969 the middle. 09:10.970 --> 09:13.580 On the bottom it decided to draw it with blue in the middle. 09:13.580 --> 09:17.350 What do those colors mean? 09:17.350 --> 09:18.520 Yes, Cathy? 09:18.519 --> 09:19.419 Student: Positive and negative signs. 09:19.418 --> 09:20.618 Prof: The mean positive and negative. 09:20.620 --> 09:21.760 So which one is right? 09:21.759 --> 09:24.919 Should it be positive in the middle or negative in the 09:24.922 --> 09:25.462 middle? 09:25.460 --> 09:29.090 Student: Positive. 09:29.090 --> 09:29.940 Prof: Positive is a guess. 09:29.940 --> 09:30.780 Wilson, what do you say? 09:30.778 --> 09:31.458 Student: It doesn't matter. 09:31.460 --> 09:32.180 Prof: Why not? 09:32.178 --> 09:33.738 Student: Because it's not really positive or negative; 09:33.740 --> 09:36.380 it's just kind of a phase of it. 09:36.379 --> 09:37.939 Prof: Yes, it's the sign of the wave 09:37.941 --> 09:38.351 function. 09:38.350 --> 09:40.990 But you can multiply the wave function by minus one, 09:40.990 --> 09:43.270 any constant, and it's just as good as it was 09:43.269 --> 09:43.839 before. 09:43.840 --> 09:46.610 So it's arbitrary, and the computer was arbitrary 09:46.614 --> 09:48.064 in choosing the colors. 09:48.058 --> 09:51.028 I think there's actually a function that I could've changed 09:51.027 --> 09:53.737 it, if I'd thought to do so, so they'd be the same. 09:53.740 --> 09:57.520 But actually it tells a story if I leave them different. 09:57.519 --> 09:59.739 Okay, so here's the next one. 09:59.740 --> 10:02.780 Now this is -- if you look on the left now, 10:02.782 --> 10:07.202 it's more clear where the nodes are, that that's a d_xz 10:07.200 --> 10:08.070 orbital. 10:08.070 --> 10:14.040 The name 'xz' means that the product of x and z appears 10:14.042 --> 10:16.102 in the wave function. 10:16.097 --> 10:16.977 Right? 10:16.980 --> 10:21.560 So when both x and z are positive, then the product is 10:21.558 --> 10:25.308 positive; on the top right, red. Right? 10:25.308 --> 10:28.158 When x is negative, and z is positive, 10:28.155 --> 10:30.225 the top left, it's negative, 10:30.232 --> 10:32.082 the product of them. 10:32.080 --> 10:35.690 So that's why it has the name xz. 10:35.690 --> 10:39.400 Okay, and there's d_yz, which you see on the right, 10:39.398 --> 10:40.958 turned ninety degrees. 10:40.960 --> 10:44.560 And here's the d_z^2, which is that thing that has a 10:44.562 --> 10:46.802 doughnut that goes around the middle. 10:46.798 --> 10:47.418 Right? 10:47.419 --> 10:49.489 It's hard to see the doughnut. 10:49.490 --> 10:50.050 Can you see? 10:50.049 --> 10:51.339 It's blue on top. 10:51.340 --> 10:54.370 You can easily see the red on the top left, 10:54.370 --> 10:57.040 which is what's blue in the middle; 10:57.039 --> 10:58.909 the sign has changed. Right? 10:58.908 --> 11:01.508 But the doughnut is highly distorted, 11:01.509 --> 11:04.619 because as you go around, first a proton on the top pulls 11:04.620 --> 11:06.340 it up, then a proton on the bottom 11:06.344 --> 11:07.634 pulls it down, then up, down. 11:07.629 --> 11:11.309 So it's like a crown, the doughnut has been made into 11:11.306 --> 11:15.046 a crown around the end, by the protons pulling in that 11:15.053 --> 11:15.623 way. 11:15.620 --> 11:16.360 Okay. 11:16.360 --> 11:19.900 Then here's the 3p_z. 11:19.899 --> 11:24.069 So it has the horizontal nodal plane, but also it has a 11:24.068 --> 11:27.308 spherical node, which you can see in either 11:27.312 --> 11:28.782 picture really. 11:28.779 --> 11:29.969 Okay? 11:29.970 --> 11:32.170 Then here's the 3p _y orbital, 11:32.168 --> 11:34.078 which again has that spherical node, 11:34.080 --> 11:37.020 but now the planar node, or approximately planar node, 11:37.019 --> 11:39.509 is vertical instead of horizontal. 11:39.509 --> 11:43.969 So say on the top right there you have red on the right and 11:43.969 --> 11:45.429 blue on the left. 11:45.428 --> 11:49.348 There's a vertical node over which it changes sign, 11:49.346 --> 11:51.066 going right to left. 11:51.070 --> 11:54.560 Or here's the 3_px. 11:54.559 --> 11:59.979 Or here's the 3d_xy; which you don't see so well 11:59.976 --> 12:02.456 here; well, but if you turned it down 12:02.460 --> 12:03.230 you would. 12:03.230 --> 12:06.880 And here's the 3d_x^2-y^2; 12:06.879 --> 12:08.999 which again, to see it well, 12:08.996 --> 12:10.876 you'd have to turn it. 12:10.879 --> 12:14.819 And here's the 4f orbital. 12:14.820 --> 12:18.500 So you can see, especially say in the top ones, 12:18.500 --> 12:21.570 compared with the atomic orbital, that it's exactly the 12:21.566 --> 12:26.156 same general pattern of nodes, slightly distorted. 12:26.158 --> 12:29.778 And incidentally, remember all the n 12:29.775 --> 12:32.355 equals whatever, n=3; 12:32.360 --> 12:35.630 all the orbitals had the same energy before. 12:35.629 --> 12:37.759 Now they don't have all the same energy. 12:37.759 --> 12:39.709 Notice that they all have different energies. 12:39.710 --> 12:41.190 None of them are degenerate. 12:41.190 --> 12:42.460 Why? 12:42.460 --> 12:46.710 Because if you broke the nucleus apart, 12:46.710 --> 12:49.510 how stable a thing that has that general shape, 12:49.509 --> 12:52.389 like a dumbbell or a cloverleaf or something, 12:52.389 --> 12:57.779 how stable the electrons are in those lumps depends on whether a 12:57.779 --> 13:00.689 proton got pulled into the lump. 13:00.690 --> 13:03.810 If there happens to be a proton in the lump, where kinetic 13:03.812 --> 13:06.002 energy, the node pattern, wants it to be, 13:06.003 --> 13:08.033 then that'll be unusually stable. 13:08.028 --> 13:12.018 If the protons have been pulled someplace where there is a node, 13:12.024 --> 13:15.644 because of kinetic energy, then it won't be stabilized. 13:15.639 --> 13:19.979 So that breaks the degeneracy of these different patterns that 13:19.975 --> 13:22.245 have the same number of nodes. 13:22.250 --> 13:25.650 It depends on where the -- how the potential energy changed. 13:25.649 --> 13:27.919 Okay, now just finally and very quickly, 13:27.918 --> 13:30.208 I want to look at 1-flouoroethanol, 13:30.210 --> 13:31.460 which is a molecule that would have, 13:31.460 --> 13:34.380 I think, no stability at all, but you can put it into the -- 13:34.379 --> 13:36.919 as a practical matter, you're never going to put it in 13:36.923 --> 13:38.843 a bottle, but you can easily put it into 13:38.841 --> 13:41.481 the computer and calculate what its molecular orbitals would 13:41.477 --> 13:42.057 look like. 13:42.058 --> 13:45.278 And I did it to show you something that's very, 13:45.279 --> 13:48.779 very unsymmetrical, and has atoms of very different 13:48.779 --> 13:50.109 nuclear charge. 13:50.110 --> 13:50.940 Okay? 13:50.940 --> 13:55.200 So what will the very, very, very lowest orbital look 13:55.198 --> 13:58.228 like, do you think, for this thing? 13:58.230 --> 14:02.250 It has a fluorine, an oxygen, two carbons and five 14:02.254 --> 14:03.244 hydrogens. 14:03.240 --> 14:04.800 So what do you think? 14:04.798 --> 14:08.258 If you were an electron, and you had that set of nuclei, 14:08.264 --> 14:11.294 arranged this way, where would you want to go? 14:11.289 --> 14:11.949 Elizabeth? 14:11.950 --> 14:14.850 Student: Very biased to the left, especially around the 14:14.846 --> 14:15.356 fluorine. 14:15.360 --> 14:15.960 Prof: Yes. 14:15.956 --> 14:17.636 Now what would the very, very, very lowest one? 14:17.639 --> 14:21.439 It should be more toward the left, and really close to the 14:21.437 --> 14:22.167 fluorine. 14:22.168 --> 14:25.248 It would be a 1s orbital, and mostly on fluorine, 14:25.254 --> 14:27.724 because that's where most of the protons are, 14:27.721 --> 14:29.631 the most concentrated protons. 14:29.629 --> 14:30.479 So let's look. 14:30.480 --> 14:34.320 There's a smaller scale ball model, 14:34.320 --> 14:36.900 so that we can see really small orbitals, 14:36.899 --> 14:39.099 and there that's, you're absolutely right, 14:39.100 --> 14:42.490 it's essentially the 1s orbital of fluorine, 14:42.490 --> 14:44.660 is the very lowest orbital. 14:44.659 --> 14:47.769 What would be next? 14:47.769 --> 14:50.239 Suppose this one, you came up and this seat was 14:50.240 --> 14:52.230 already taken, now where would you go, 14:52.227 --> 14:53.407 the next electron? 14:53.409 --> 14:53.519 Zack? 14:53.519 --> 14:54.179 Student: Oxygen. 14:54.178 --> 14:57.558 Prof: To the oxygen, because that's the next highest 14:57.557 --> 14:59.127 concentration of protons. 14:59.129 --> 15:02.699 And next? 15:02.700 --> 15:03.890 Where will you go next? 15:03.889 --> 15:05.669 Steve, what do you say? 15:05.668 --> 15:08.088 The 1s of fluorine is taken. 15:08.090 --> 15:10.000 The 1s of oxygen is taken. 15:10.000 --> 15:12.680 Where do we want to go next? 15:12.679 --> 15:14.309 Pardon me? 15:14.308 --> 15:16.218 Student: One of the carbons. 15:16.220 --> 15:18.450 Prof: Ah, which carbon? 15:18.450 --> 15:19.830 Student: The left carbon. 15:19.830 --> 15:20.940 Prof: Why? 15:20.940 --> 15:23.700 Student: Because it's closer to the fluorine than the 15:23.696 --> 15:24.106 oxygen. 15:24.110 --> 15:28.090 Prof: And what does the proximity have to do with it? 15:28.090 --> 15:33.660 The proximity means that the electrons that are on that atom 15:33.658 --> 15:38.008 will have been drawn toward -- and we'll talk about this more 15:38.009 --> 15:40.269 later -- will have been drawn toward the 15:40.269 --> 15:41.769 fluorine and the oxygen. 15:41.769 --> 15:44.979 Therefore that atom will have fewer electrons, 15:44.976 --> 15:46.826 or lower electron density. 15:46.827 --> 15:47.467 Right? 15:47.470 --> 15:51.440 Therefore, it's a better place for other electrons to go. 15:51.440 --> 15:51.950 Okay? 15:51.950 --> 15:55.240 So you'd expect the next one to be the middle carbon. 15:55.243 --> 15:55.753 Right! 15:55.750 --> 15:58.160 And finally, of course, it's going to be the 15:58.155 --> 16:00.615 second carbon, the one that's remote from the 16:00.619 --> 16:03.249 electronegative, high nuclear charge atoms. 16:03.250 --> 16:06.480 Okay, now we've done all the 1s's. 16:06.480 --> 16:08.570 So we can look at what's interesting, the valence 16:08.573 --> 16:11.283 orbitals, the ones that are going to be involved in bonding; 16:11.278 --> 16:13.858 these don't have anything to do with bonding. 16:13.860 --> 16:15.880 Okay, so there's the first one. 16:15.879 --> 16:19.209 Russell, tell me something about the shape of this? 16:19.210 --> 16:20.510 Does it surprise you? 16:20.509 --> 16:21.439 Student: No. 16:21.437 --> 16:23.477 It's highly distorted towards the fluorine. 16:23.480 --> 16:26.350 Prof: Yes, it has no nodes; 16:26.350 --> 16:29.340 except it actually has little nodes down near the nucleus, 16:29.340 --> 16:32.120 because it's actually more like a 2s orbital in that 16:32.121 --> 16:34.311 respect, because of the core electrons. 16:34.308 --> 16:37.538 So it has tiny nodes that we don't see around the nuclei. 16:37.538 --> 16:41.218 But it has no nodes within the valence; 16:41.220 --> 16:42.760 big orbitals that we're looking at. 16:42.759 --> 16:44.799 So it's like an s orbital. 16:44.798 --> 16:45.308 Right? 16:45.308 --> 16:48.948 And it's big where the nuclear charge is big, 16:48.952 --> 16:50.032 as you say. 16:50.029 --> 16:51.329 Okay? 16:51.330 --> 16:56.710 What's the next one going to look like? 16:56.710 --> 17:01.890 Any ideas? 17:01.889 --> 17:02.889 Pardon me? 17:02.889 --> 17:03.369 Student: Towards the oxygen. 17:03.370 --> 17:06.930 Prof: Toward the oxygen. 17:06.926 --> 17:10.136 Will it also have no nodes? 17:10.140 --> 17:10.330 No. 17:10.333 --> 17:13.243 The next highest orbital has to have a node. 17:13.240 --> 17:14.540 Where do you think the node will be? 17:14.538 --> 17:17.578 What orbital will it look like, sort of? 17:17.579 --> 17:19.329 It'll be highly distorted. 17:19.328 --> 17:21.958 But what -- if this is the s orbital, right? 17:21.960 --> 17:26.180 What's going to be next? 17:26.180 --> 17:27.450 Elizabeth? 17:27.450 --> 17:30.250 Student: It's as if it's switched but with a planar 17:30.250 --> 17:31.120 node in between. 17:31.119 --> 17:32.839 Prof: Oh let's see. 17:32.839 --> 17:34.339 Right! 17:34.338 --> 17:37.318 So the first one was -- you were right Russell that this one 17:37.317 --> 17:38.627 should be big on oxygen. 17:38.630 --> 17:42.490 But notice it's like a p orbital, it has that horizontal 17:42.490 --> 17:45.170 node, because it's higher kinetic energy. 17:45.170 --> 17:45.690 Okay? 17:45.690 --> 17:49.120 And then this one, that's on the carbons, 17:49.123 --> 17:49.813 right? 17:49.809 --> 17:50.379 Mostly. 17:50.380 --> 17:54.040 But it's a p_y, it has a vertical node. 17:54.038 --> 17:57.698 So it's negative say on the fluorine and oxygen and positive 17:57.695 --> 17:58.745 on the carbons. 17:58.750 --> 18:02.340 Or if we rotate this one around a horizontal axis to look at it 18:02.344 --> 18:04.344 from the side, it looks like that, 18:04.337 --> 18:07.307 and you can see that it's a p_y kind of orbital. 18:07.309 --> 18:08.009 Elizabeth? 18:08.009 --> 18:09.689 Student: I'm just asking a point of clarification. 18:09.690 --> 18:11.540 These aren't technically p orbitals, 18:11.538 --> 18:11.888 right? 18:11.890 --> 18:13.980 We're just saying they're analogous to them. 18:13.980 --> 18:15.120 Prof: They're analogous to p orbitals; 18:15.119 --> 18:18.419 because it has to be the same. 18:18.420 --> 18:22.360 The orbitals must go in order of the number of nodes. 18:22.363 --> 18:22.973 Right? 18:22.970 --> 18:25.410 So the same thing is in an atom or in a molecule. 18:25.410 --> 18:27.170 You go from no nodes to one node. 18:27.170 --> 18:29.920 And there are three ways of getting one node: 18:29.924 --> 18:33.684 spherical or distorted sphere, planes, or distorted planes if 18:33.683 --> 18:36.443 the nuclei are pulled around; and so on. 18:36.440 --> 18:39.680 So they're really very much like the atomic orbitals. 18:39.680 --> 18:42.170 Okay, or this one. 18:42.170 --> 18:46.070 Now that's an interesting one because that's like a hybrid 18:46.074 --> 18:46.764 orbital. 18:46.759 --> 18:48.369 It looks like the orbital up here. 18:48.368 --> 18:50.238 It's a mixture of s with p. 18:50.240 --> 18:52.030 Did everybody see how that is? 18:52.029 --> 18:55.289 It's a big sort of blue lobe on the top and a small red one on 18:55.286 --> 18:55.976 the bottom. 18:55.980 --> 18:58.630 And also the next one is the other one, 18:58.630 --> 19:00.720 the hybrid that points the other direction, 19:00.720 --> 19:02.810 another combination of the s and the p, 19:02.809 --> 19:04.859 that points down. 19:04.859 --> 19:06.979 Okay? 19:06.980 --> 19:09.620 And then this one looks like d_xy again. 19:09.619 --> 19:10.079 Right? 19:10.078 --> 19:14.468 It's sort of a cloverleaf with two nodes. 19:14.470 --> 19:16.700 Okay, so that's all I want to do with that, 19:16.701 --> 19:18.721 and we won't go any further with it. 19:18.720 --> 19:21.810 This is just an interesting way of looking at how -- 19:21.808 --> 19:24.868 that molecular orbitals are really just like atomic 19:24.865 --> 19:27.155 orbitals, and have energies for the same 19:27.161 --> 19:28.991 reason, except the potential energy 19:28.993 --> 19:31.873 gets screwed up by breaking the nucleus and pulling pieces 19:31.874 --> 19:34.734 around, but in an understandable way, 19:34.733 --> 19:38.313 and the nodes get distorted because of this. 19:38.308 --> 19:43.128 Okay now, now we're getting into really -- we just looked at 19:43.126 --> 19:46.716 this view, at the single united atom view. 19:46.720 --> 19:49.740 But the other view is the one that's going to be more 19:49.737 --> 19:52.117 generalizable, and that's the one where we 19:52.116 --> 19:53.156 looked at bonding. 19:53.161 --> 19:53.801 Right? 19:53.798 --> 19:57.368 So you have to probe a little harder to get a qualitative 19:57.374 --> 20:00.124 understanding of what chemical bonds are. 20:00.118 --> 20:03.648 And that's what we're going to do now by choosing a higher 20:03.647 --> 20:06.307 contour with which to look at a molecule. 20:06.308 --> 20:10.648 Now, true molecular orbitals, to the extent that orbitals are 20:10.647 --> 20:14.767 true all together -- why aren't they true all together; 20:14.769 --> 20:18.879 why aren't orbitals true all together? 20:18.880 --> 20:20.360 Yes, Alex? 20:20.358 --> 20:21.638 Student: Because you have multiple electrons. 20:21.640 --> 20:22.740 Prof: Because you have many electrons; 20:22.740 --> 20:25.680 you can't have independent electrons, you can't have 20:25.675 --> 20:26.305 orbitals. 20:26.308 --> 20:29.048 But we're approximating things by orbitals, 20:29.048 --> 20:32.128 trying to take electron interaction into account in a 20:32.130 --> 20:35.330 sort of a left-handed way by Self-Consistent Field, 20:35.329 --> 20:36.419 or something like that. 20:36.420 --> 20:39.610 Because it's much easier if we can divide the whole into a 20:39.605 --> 20:41.555 bunch of parts, each of which we can 20:41.560 --> 20:42.400 understand. 20:42.400 --> 20:44.900 So, to the extent that molecular orbitals are true -- 20:44.900 --> 20:46.360 the kinds of things I've just been showing you, 20:46.358 --> 20:50.738 calculated with my laptop -- they extend over the whole 20:50.738 --> 20:54.358 molecule; they're not local. Right? 20:54.358 --> 20:57.878 Except like the 1s of fluorine was local, 20:57.878 --> 21:01.248 but mostly they go over the whole molecule. 21:01.250 --> 21:05.230 But bonds are thought of, and have always been thought 21:05.230 --> 21:08.610 of, as interactions between pairs of atoms. 21:08.608 --> 21:14.378 So we want to divide things completely differently and look 21:14.378 --> 21:18.658 at the bonds now, at pairwise LCAO molecular 21:18.655 --> 21:19.945 orbitals. 21:19.950 --> 21:22.240 Now what's an LCAO? 21:22.240 --> 21:26.030 It's a sum, or a linear combination. 21:26.032 --> 21:26.902 Right? 21:26.900 --> 21:30.220 A weighted sum of atomic orbitals. 21:30.220 --> 21:31.460 So here's an example. 21:31.460 --> 21:34.400 Ψ, which is an orbital -- what's it a function of? 21:34.400 --> 21:35.690 Student: Position. 21:35.690 --> 21:36.650 Prof: Position of what? 21:36.650 --> 21:37.270 Student: One electron. 21:37.269 --> 21:37.939 Prof: One electron. 21:37.940 --> 21:40.220 So it's a function of x_1,y_1,z_1; 21:40.220 --> 21:42.630 we're talking just about electron one. 21:42.630 --> 21:46.640 So the wave function for electron one we say is 21:46.644 --> 21:51.364 1/√2 times one atomic orbital plus another 21:51.359 --> 21:52.669 atomic orbital. 21:52.669 --> 21:53.629 Right? 21:53.630 --> 21:58.170 Now, have you ever seen adding orbitals like that before? 21:58.170 --> 22:01.960 That's what hybridization is; we added s and p. 22:01.960 --> 22:04.880 But this is different, because when we added s 22:04.877 --> 22:08.367 and p before, they were on the same nucleus, 22:08.366 --> 22:12.246 and we did it to get a new orbital for that particular 22:12.248 --> 22:15.368 nucleus for that atom; to distort it one way or the 22:15.367 --> 22:17.297 other, for example, or to rotate a p 22:17.300 --> 22:17.670 orbital. 22:17.669 --> 22:18.129 Right? 22:18.130 --> 22:22.500 But this is very different, because we're adding orbitals 22:22.500 --> 22:26.330 that are on different nuclei: A, nucleus A, 22:26.325 --> 22:27.725 and nucleus B. 22:27.730 --> 22:30.600 See the difference? 22:30.598 --> 22:34.048 Adding is just -- wave functions are numbers, 22:34.049 --> 22:36.089 we just add the numbers. 22:36.088 --> 22:38.318 But in the first case, hybridization, 22:38.323 --> 22:41.243 those two functions were on the same nucleus. 22:41.240 --> 22:44.460 Now they're on different nuclei, what we're adding 22:44.464 --> 22:45.194 together. 22:45.190 --> 22:50.270 Okay, now why is it sensible to think that you might get 22:50.266 --> 22:55.986 pairwise molecular orbitals that can be expressed like this? 22:55.990 --> 22:58.750 How do you interpret an orbital? 22:58.750 --> 23:00.990 Corey? 23:00.990 --> 23:01.860 What good is an orbital? 23:01.859 --> 23:03.709 What do you use it for? 23:03.710 --> 23:06.320 Student: It's a one-electron wave function. 23:06.318 --> 23:08.538 Prof: Well what do you use it for? 23:08.538 --> 23:10.738 Student: For probability. 23:10.740 --> 23:14.110 Prof: And how do you get probability? 23:14.108 --> 23:16.518 From the wave function -- if you have the wave function, 23:16.520 --> 23:18.320 how do you get the probability density? 23:18.319 --> 23:19.019 Student: You square it. 23:19.019 --> 23:20.109 Prof: You square it. 23:20.108 --> 23:25.278 So do you see why we have a 1/√2 in this? 23:25.278 --> 23:28.838 Because when we square it, that's going to be half, 23:28.843 --> 23:32.763 and we're going to get half of atomic orbital A squared, 23:32.763 --> 23:35.333 and of atomic orbital B squared; 23:35.329 --> 23:36.979 so it's a half of each of them. 23:36.980 --> 23:39.050 That's why we have 1/√2. 23:39.049 --> 23:40.649 So let's go on with this. 23:40.650 --> 23:44.570 Suppose we have a hydrogen molecule, and suppose that the 23:44.568 --> 23:48.068 nuclei are at a great distance from one another. 23:48.068 --> 23:52.558 So far they don't interact, or negligible interaction; 23:52.559 --> 23:53.889 they're very far apart. 23:53.890 --> 23:57.180 What would you expect the lowest energy, 23:57.178 --> 24:00.718 one-electron wave function to look like? 24:00.720 --> 24:05.750 One possibility is that the electron could sit exactly 24:05.746 --> 24:08.686 halfway between the two nuclei? 24:08.685 --> 24:09.535 Right? 24:09.538 --> 24:12.568 Is that a reasonable place, is that the low energy place 24:12.565 --> 24:13.385 for it to be? 24:13.390 --> 24:14.830 Lucas, what do you say? 24:14.828 --> 24:17.848 Student: No, because the added probability 24:17.849 --> 24:20.069 density there is not the greatest. 24:20.069 --> 24:20.719 Prof: Why not? 24:20.720 --> 24:22.430 Student: It needs to be one or the other. 24:22.430 --> 24:24.410 Prof: Why shouldn't the electrons sit -- if the two 24:24.407 --> 24:27.167 nuclei are this far apart; and I don't mean two angstroms 24:27.171 --> 24:30.751 apart, I mean two meters apart -- there's a proton here and a 24:30.753 --> 24:31.653 proton here. 24:31.650 --> 24:35.140 Is the electron most likely to be here, halfway between? 24:35.140 --> 24:35.840 Student: No. 24:35.838 --> 24:37.268 Prof: Where would it be most likely? 24:37.269 --> 24:39.329 Student: Probably nearer to one of the two atoms. 24:39.328 --> 24:43.268 Prof: And which of the two? 24:43.269 --> 24:44.539 <> 24:44.538 --> 24:46.958 Prof: Suppose you took the long view. 24:46.960 --> 24:52.480 Suppose you averaged it over eighteen zillion millennia -- 24:52.480 --> 24:54.030 time averaged. 24:54.029 --> 24:57.679 Sometimes it would be near this one, sometimes it would be near 24:57.682 --> 24:58.332 this one. 24:58.329 --> 24:59.089 Okay? 24:59.088 --> 25:01.208 What would it be if you took a really long view? 25:01.210 --> 25:02.210 Student: Both. 25:02.210 --> 25:05.520 Prof: And half here, and half here. 25:05.519 --> 25:08.409 So the wave function, when you square it, 25:08.414 --> 25:11.674 you want it to be half looking like this atom, 25:11.673 --> 25:14.283 and half looking like this atom. 25:14.278 --> 25:16.438 Then you see the time average. 25:16.435 --> 25:16.935 Right? 25:16.940 --> 25:19.640 It might take a long time to achieve that average, 25:19.637 --> 25:22.887 because it'd take a long time for the electron to tunnel two 25:22.887 --> 25:23.877 meters, right? 25:23.880 --> 25:26.110 But in the very, very long time it would look 25:26.111 --> 25:26.721 like that. 25:26.720 --> 25:28.710 So we know what we want it to look like. 25:28.710 --> 25:31.620 What we want is that the probability density, 25:31.618 --> 25:34.308 the square of this one-electron wave function, 25:34.308 --> 25:38.028 should look half of the time like the atomic orbital A 25:38.032 --> 25:42.322 squared and half of the time like atomic orbital B squared. 25:42.318 --> 25:45.348 So on time average it's half of one and half of the other. 25:45.349 --> 25:47.509 Everybody with me? 25:47.509 --> 25:49.529 So that's what the -- yes, Nate? 25:49.529 --> 25:54.649 Student: Why isn't there a 2A -- or a 2AB? 25:54.650 --> 25:56.890 Prof: Oh, because I'm telling you what it 25:56.890 --> 25:57.510 looks like. 25:57.509 --> 26:00.329 It's got to look half like this and half like this. 26:00.327 --> 26:00.777 Right? 26:00.778 --> 26:04.538 So the square of it has to be this. 26:04.538 --> 26:08.238 So all we have to do to find the wave function is what, 26:08.238 --> 26:10.428 if we know what its square is? 26:10.430 --> 26:13.130 All we got to do is take the square root and we've got the 26:13.131 --> 26:13.891 wave function. 26:13.890 --> 26:14.810 Bingo! 26:14.809 --> 26:20.169 There's the square root. Right? 26:20.170 --> 26:25.150 So that is a reasonable way to write the wave function, 26:25.150 --> 26:28.750 1/√2 (AO_A + AO_B). Claire, 26:28.746 --> 26:30.956 you have a question. 26:30.960 --> 26:33.950 Student: From my small understanding, 26:33.951 --> 26:37.491 with the math that I've got -- and this may be wrong, 26:37.486 --> 26:39.386 correct me if I'm wrong. 26:39.390 --> 26:42.820 But don't you, if you have a bracket and two 26:42.819 --> 26:46.169 things inside it, you square outside of it, 26:46.171 --> 26:50.241 don't you have four things that come out of it. 26:50.240 --> 26:51.530 Prof: You want that? 26:51.529 --> 26:52.559 Student: Yes. 26:52.559 --> 26:53.649 > 26:53.650 --> 26:57.350 Prof: Okay, now Claire you're going to help 26:57.349 --> 26:58.029 me out. 26:58.029 --> 27:00.479 How big is -- that's a number. 27:00.480 --> 27:04.570 It's the product of -- atomic orbital A assigns numbers 27:04.566 --> 27:07.816 everywhere in space, everywhere in space. 27:07.818 --> 27:11.268 Atomic orbital B assigns numbers everywhere in space. 27:11.269 --> 27:14.969 So at some point in space atomic orbital B assigns a 27:14.974 --> 27:18.394 number and atomic orbital A assigns a number; 27:18.390 --> 27:22.840 and A times B is the product of those two numbers. 27:22.839 --> 27:25.959 How big is that product? 27:25.960 --> 27:30.130 How big is atomic orbital A here? 27:30.130 --> 27:32.290 This, a meter away from the proton? 27:32.289 --> 27:33.159 Student: Not very big. 27:33.160 --> 27:36.090 Prof: And how big is atomic orbital B there? 27:36.089 --> 27:36.599 Student: Not very big. 27:36.598 --> 27:39.658 Prof: Now how big is atomic orbital A here? 27:39.660 --> 27:40.780 Student: Very big. 27:40.779 --> 27:42.779 Prof: Okay. 27:42.781 --> 27:49.571 So now, how big an error do we make if we neglect A times B? 27:49.569 --> 27:50.999 Where do we make an error? 27:51.000 --> 27:53.120 Do we make an error here? 27:53.119 --> 27:54.719 Student: No. 27:54.720 --> 27:56.590 Prof: Do we make an error here? 27:56.589 --> 27:57.059 Student: No. 27:57.058 --> 27:58.078 Prof: Do we make an error here? 27:58.079 --> 27:58.809 Student: Yes. 27:58.809 --> 28:00.009 Prof: No! 28:00.009 --> 28:01.919 We make an error; yes, sure enough, 28:01.919 --> 28:03.019 we make an error. 28:03.019 --> 28:04.759 How big is the error? 28:04.759 --> 28:05.749 Student: Not very big. 28:05.750 --> 28:07.940 Prof: Ah, it's negligible, 28:07.940 --> 28:10.200 because it's at great distance. 28:10.200 --> 28:13.520 Okay, so at great distance we can forget that. 28:13.519 --> 28:18.619 So now it's easier to take the square root, right? 28:18.619 --> 28:19.969 Okay? 28:19.970 --> 28:21.670 The old fox up here, huh? 28:21.670 --> 28:26.030 > 28:26.028 --> 28:29.948 Okay, now your problem is what happens if H_2 is at the 28:29.949 --> 28:31.169 bonding distance? 28:31.170 --> 28:34.220 What if they're only an Angstrom apart? 28:34.220 --> 28:39.690 Now there should be a problem, because A times B is not going 28:39.690 --> 28:42.790 to be negligible everywhere now. 28:42.788 --> 28:44.768 Okay, so now that's going to come back. 28:44.769 --> 28:47.979 So now we got an error, right? 28:47.980 --> 28:49.830 Or do we? 28:49.829 --> 28:52.589 Let's think what that does. 28:52.588 --> 28:57.358 Okay, so we if approximate the molecular orbital as the sum of 28:57.362 --> 29:00.802 atomic orbitals, this way, then it looks very 29:00.804 --> 29:02.764 good near the nuclei. 29:02.759 --> 29:09.479 Because near A it looks like A; near B it looks like B. 29:09.480 --> 29:14.080 And if we want to square it to find the electron density, 29:14.080 --> 29:15.150 we do this. 29:15.150 --> 29:19.670 But if we then subtract, what the atoms would give for 29:19.669 --> 29:21.289 electron density. 29:21.288 --> 29:25.368 Now what does this remind you of, where we look at the total 29:25.373 --> 29:28.283 electron density and subtract the atoms? 29:28.278 --> 29:29.178 Student: Difference density. 29:29.180 --> 29:31.120 Prof: So we're actually looking for the difference 29:31.123 --> 29:31.473 density. 29:31.470 --> 29:33.040 Everybody with me on this? 29:37.962 --> 29:38.572 Right? 29:38.569 --> 29:42.089 If we subtract, what do we get? 29:42.088 --> 29:45.468 What do we get for a difference density? 29:45.470 --> 29:49.150 We get A times B. 29:49.150 --> 29:52.410 So we get the difference electron density, 29:52.411 --> 29:54.481 which is due to overlap. 29:54.480 --> 29:56.140 And what do I mean by overlap? 29:56.140 --> 30:00.930 I mean that is only important in regions where both of them 30:00.925 --> 30:02.575 have a finite value. 30:02.576 --> 30:03.316 Right? 30:03.318 --> 30:06.458 The product of A and B is negligible if A is zero; 30:06.460 --> 30:09.210 it's negligible if B is very, very small. 30:09.210 --> 30:09.760 Right? 30:09.759 --> 30:13.189 So it's only where they overlap, the two functions have 30:13.194 --> 30:15.554 simultaneous values, that you care. 30:15.548 --> 30:21.358 Okay, now that thing, the thing that's the bonding, 30:21.358 --> 30:25.388 the difference density, is really a byproduct -- 30:25.390 --> 30:28.870 that's a little bit of a pun because it's a product -- 30:28.868 --> 30:32.378 but it's a byproduct of squaring the sum -- 30:32.380 --> 30:34.280 of what Claire didn't like about it. 30:34.279 --> 30:38.149 So the very thing you didn't like is what's going to give 30:38.152 --> 30:39.952 rise to bonding density. 30:39.950 --> 30:44.020 Isn't that neat? 30:44.019 --> 30:47.839 Okay, but notice that here we're multiplying A times B. 30:47.838 --> 30:50.498 But this is a completely different instance of 30:50.500 --> 30:52.570 multiplying from what we had before. 30:52.567 --> 30:53.157 Right? 30:53.160 --> 30:55.600 Before we multiplied two orbitals to try to get a 30:55.596 --> 30:57.066 two-electron wave function. 30:57.068 --> 31:00.028 This has nothing to do with this, because both of these are 31:00.032 --> 31:03.742 functions of the same electron; it's like one electron that 31:03.743 --> 31:05.323 we're squaring here. 31:05.318 --> 31:08.158 So this A times B, this product, 31:08.162 --> 31:11.832 this overlap, comes from the squaring. 31:11.828 --> 31:15.118 It was when we squared it that we got that. 31:15.119 --> 31:18.609 Okay? 31:18.608 --> 31:21.178 Now, because we have this extra term, 31:24.338 --> 31:28.258 which would -- what's the probability of A^2^( )summed 31:28.259 --> 31:30.399 over all space, or integrated, 31:30.401 --> 31:31.561 if it's normalized? 31:31.559 --> 31:32.339 Students: One. 31:32.339 --> 31:33.089 Prof: And this one? 31:33.089 --> 31:33.989 Students: One. 31:33.990 --> 31:35.560 Prof: And what's this whole quantity? 31:35.559 --> 31:36.129 Students: One. 31:36.130 --> 31:37.620 Prof: One, because of the half. 31:37.618 --> 31:40.708 But actually we've got it bigger than that, 31:40.711 --> 31:43.581 because we added the overlap term to it. 31:43.583 --> 31:44.323 Right? 31:44.318 --> 31:46.038 So it's actually not going to be half; 31:46.038 --> 31:48.898 it'll have to be something a little less than half, 31:48.895 --> 31:52.205 so that it'll sum up to one, if we want to normalize it. 31:52.210 --> 31:52.750 Okay? 31:52.750 --> 31:55.640 So there's going to be those less than halves there. 31:55.640 --> 31:56.780 Okay. 31:56.779 --> 31:59.079 Now what does that do? 31:59.078 --> 32:01.388 That shifts electron density, right? 32:01.390 --> 32:05.480 We're taking electron density away from where the nuclei are, 32:05.479 --> 32:08.269 from A^2 and B^2, and where's the electron 32:08.272 --> 32:09.502 density going? 32:09.500 --> 32:12.900 Because we're using less than half to begin with, 32:12.898 --> 32:15.518 we're taking electron density away. 32:15.519 --> 32:18.819 There's going to be -- we're subtracting more than was there 32:18.817 --> 32:19.767 at the beginning. 32:19.767 --> 32:20.267 Right? 32:20.269 --> 32:23.079 Which means that we're going to have negative electron density 32:23.076 --> 32:24.176 in the difference map. 32:24.180 --> 32:26.090 Electrons are going away from the atoms. 32:26.089 --> 32:28.369 Where are they going to? 32:28.369 --> 32:29.719 Student: Into it. 32:29.720 --> 32:31.950 Prof: Into the region where there's overlap. 32:31.952 --> 32:32.312 Right? 32:32.308 --> 32:36.448 So they go away from there, into the overlap region. 32:36.450 --> 32:39.870 So this is just like what we were seeing with X-ray. 32:39.868 --> 32:43.028 Okay, so that overlap, the A times B term, 32:43.032 --> 32:45.042 is what creates bonding. 32:45.039 --> 32:46.769 And we've seen this before. 32:46.769 --> 32:50.489 Remember when you have wells far apart, the wave function is 32:50.493 --> 32:53.793 the sum of the two; we saw this in one-dimension. 32:53.785 --> 32:54.255 Right? 32:54.259 --> 32:56.169 But if they come close together, you get a wave 32:56.174 --> 32:58.744 function that looks like that, which we looked before, 32:58.743 --> 33:01.033 just from the point of view of the energy, 33:01.028 --> 33:03.528 and saw that that would stabilize the particle, 33:03.528 --> 33:05.918 because it's got less curvature, less kinetic energy. 33:05.922 --> 33:06.292 Right? 33:06.288 --> 33:10.738 But also the electron density grows in the middle. 33:10.741 --> 33:11.471 Right? 33:11.470 --> 33:15.820 So from the point of view of the electron distribution, 33:15.816 --> 33:19.596 that was the glue holding the atoms together. 33:19.598 --> 33:22.298 So it's held together, both because the energy goes 33:22.298 --> 33:24.888 down and because you put this glue in the middle, 33:24.892 --> 33:27.432 which is what causes the energy to go down. 33:27.430 --> 33:28.150 Okay? 33:28.150 --> 33:29.460 So that's bonding. 33:29.460 --> 33:31.350 And remember we also had this. 33:31.348 --> 33:35.218 So as the energy went up in the middle one, the energy is lower 33:35.223 --> 33:37.603 here than it was in the atoms apart. 33:37.598 --> 33:41.868 So the nuclei push one another apart now, without the glue in 33:41.865 --> 33:44.705 the middle, and that was anti-bonding. 33:44.710 --> 33:46.570 So we've seen it before in one-dimension, 33:46.567 --> 33:48.607 but it's true in three-dimensions as well. 33:48.608 --> 33:50.178 Now let's think about this again. 33:50.180 --> 33:53.780 So here's atom A. 33:53.779 --> 33:58.679 Now where is the square of that function significant? 33:58.680 --> 34:00.950 Is it significant there? 34:00.950 --> 34:01.650 No. 34:01.650 --> 34:04.990 Is it significant there? 34:04.990 --> 34:06.850 Yes, it's a little bit significant at least. 34:06.849 --> 34:09.889 How about there? 34:09.889 --> 34:12.389 A little bit. Right? 34:12.389 --> 34:13.479 Okay. 34:13.480 --> 34:17.430 Now suppose we have another atomic orbital there. 34:17.429 --> 34:23.609 Now, where is the product significant, of A times B? 34:23.610 --> 34:24.880 Okay? 34:24.880 --> 34:28.150 So is the product A times B significant there? 34:28.150 --> 34:30.320 No. 34:30.320 --> 34:39.460 Is it significant there? 34:39.460 --> 34:40.990 Nick, what do you say? 34:40.989 --> 34:41.829 Student: No. 34:41.829 --> 34:44.029 Prof: Why not? 34:44.030 --> 34:44.820 Speak up. 34:44.820 --> 34:46.100 Student: It's very small near A. 34:46.099 --> 34:47.129 Prof: What's very small? 34:47.130 --> 34:49.110 Student: The value of A. 34:49.110 --> 34:50.020 Prof: No, no. 34:50.018 --> 34:52.318 The value of A there is something -- we said that 34:52.315 --> 34:55.085 before, when we were looking only at A -- it's not very big 34:55.090 --> 34:56.910 but there's a significant value. 34:56.909 --> 34:58.629 But how about the product? 34:58.630 --> 34:59.120 Josh? 34:59.119 --> 35:00.229 Student: The value of B is very small. 35:00.230 --> 35:02.170 Prof: The value of B is very small there. 35:02.170 --> 35:04.170 So the product is going to be very small. 35:04.170 --> 35:06.720 How about there? 35:06.719 --> 35:08.129 Student: There's going to be a change. 35:08.130 --> 35:09.970 Prof: Ah, now they're equal; 35:09.969 --> 35:11.629 halfway between they're equal. 35:11.630 --> 35:14.980 So both of them are a little bit small, but their product is 35:14.983 --> 35:16.013 still significant. 35:16.007 --> 35:16.517 Right? 35:16.518 --> 35:19.918 Only in this region, where they overlap, 35:19.922 --> 35:22.542 is that product significant. 35:22.539 --> 35:23.189 Okay. 35:23.190 --> 35:25.780 So at the center, notice that the number 35:25.784 --> 35:29.714 Ψ_A assigns and the number Ψ_B assigns 35:29.711 --> 35:31.311 are the same number. 35:31.309 --> 35:35.529 So 2(Ψ_A Ψ_B) is as large as 35:35.525 --> 35:40.135 (Ψ_A)^2+(Ψ_B)^2; because Ψ_A times 35:40.143 --> 35:42.473 Ψ_B is the same as (Ψ_A)^2. 35:42.469 --> 35:45.849 So the electron density is nearly doubled in the middle 35:45.847 --> 35:49.537 from what it would've been if it had just been two atoms. 35:49.539 --> 35:51.679 So that's the region of significant overlap, 35:51.684 --> 35:53.284 and that's what we care about. 35:53.280 --> 35:55.920 So the overlap integral, summing this produc -- or 35:55.922 --> 35:59.322 integrating it -- over all that space, that's a certain density. 35:59.320 --> 35:59.860 Right? 35:59.860 --> 36:01.490 We squared in order to get that. 36:01.489 --> 36:01.909 Right? 36:01.909 --> 36:03.919 That's part of the density. 36:03.920 --> 36:07.130 So we sum that, the density that comes from 36:07.130 --> 36:09.270 that product, over all space, 36:09.271 --> 36:12.561 and that's called the overlap integral. 36:12.559 --> 36:15.989 If the atoms are very far apart, the overlap integral is 36:15.992 --> 36:17.182 essentially zero. 36:17.179 --> 36:19.569 If the atoms are close together, the overlap orbital 36:19.574 --> 36:23.454 will be finite, and the better the -- the more 36:23.447 --> 36:27.757 the orbitals overlap, the bigger the overlap 36:27.755 --> 36:29.645 integral, obviously. 36:29.650 --> 36:33.890 And that measures the net change that arises on bonding, 36:33.889 --> 36:37.359 the difference density, as we've just seen. 36:37.360 --> 36:42.320 Now let's look at some theoretical examples here. 36:42.320 --> 36:47.240 So let's look at the total electron density as calculated 36:47.240 --> 36:52.340 for two -- adding two 1s orbitals of hydrogen at the 36:52.338 --> 36:55.238 appropriate distance for H_2. 36:55.239 --> 36:57.889 This was this was done forty years ago and published in the 36:57.894 --> 36:59.594 Israel Journal of Chemistry. 36:59.590 --> 37:02.240 Okay now, and here's -- so on the left we have the total 37:02.235 --> 37:04.635 electron density that you'd calculate from that. 37:04.639 --> 37:10.969 A very simple thing: 1/√2(A+B). 37:10.969 --> 37:13.939 And you square it, and you get the density, 37:13.936 --> 37:16.826 and that's the density, contoured at 0.025 37:16.833 --> 37:20.583 electrons/cubic a_o, the unit of distance. 37:20.579 --> 37:24.259 Okay, now, on the right is the difference density. 37:24.260 --> 37:28.130 So from that thing on the left we've subtracted the atomic 37:28.132 --> 37:30.922 orbital, the atomic electron densities. 37:30.920 --> 37:33.380 And you see exactly what you expect. 37:33.380 --> 37:37.400 It builds up in the middle, where there's overlap, 37:37.400 --> 37:40.190 and at the expense of the atoms. 37:40.190 --> 37:40.850 Yes Russell? 37:40.849 --> 37:42.829 Student: Shouldn't it be H_2^+ ion? 37:42.829 --> 37:47.409 Prof: Oh, I think it's the H_2_ 37:47.411 --> 37:52.441 molecule, I'm sure it's the H_2_. 37:52.440 --> 37:55.260 They aren't so fantastically different, 37:55.262 --> 37:58.602 because the two electrons are in the same orbital; 37:58.599 --> 38:00.219 two electrons can be in the same orbital. 38:00.219 --> 38:02.059 So one'll be twice as big as the other. 38:02.059 --> 38:04.259 Qualitatively they'll look very similar. 38:04.260 --> 38:06.270 And I think this is the H_2 molecule; 38:06.268 --> 38:08.488 but it might be the ion, I'm not sure. 38:08.489 --> 38:13.409 Okay, at any rate the contours on the right are much smaller. 38:13.409 --> 38:16.649 Remember, difference density is much smaller than total density. 38:16.650 --> 38:20.320 So what you see is that it's contoured at 0.004. 38:20.320 --> 38:22.880 So you have one, two, three, four, 38:22.878 --> 38:24.118 five contours. 38:24.119 --> 38:26.659 So you get up, in the middle, 38:26.655 --> 38:30.275 to 0.02 electrons per cubic angstrom . 38:30.280 --> 38:33.960 That's how much bonding has changed things, 38:33.960 --> 38:35.450 at the maximum. 38:35.449 --> 38:38.979 Okay, now the energy that's calculated, 38:38.980 --> 38:42.190 with this very, very, very crude wave function, 38:42.190 --> 38:45.620 just one half -- 1/√2 times the sum 38:45.621 --> 38:47.741 of the two atomic orbitals. 38:47.739 --> 38:54.019 The energy, you calculate that, is 92.9% of the true energy. 38:54.019 --> 38:57.079 That's pretty darn good, right? 38:57.079 --> 39:01.009 But almost all of that energy that you calculate was already 39:01.014 --> 39:03.154 present in the separate atoms. 39:03.150 --> 39:05.530 We're not interested in the energy of the separate atoms, 39:05.530 --> 39:08.900 we're interested in how much it changes when you make a bond, 39:08.900 --> 39:12.050 which is a small difference between large numbers. 39:12.050 --> 39:16.020 So it turns out that although we're within 7% of the true 39:16.023 --> 39:18.663 total energy, this simple model only 39:18.664 --> 39:22.634 calculates about 50% of the change in energy that came from 39:22.632 --> 39:24.962 putting them together -- right? 39:24.960 --> 39:28.330 -- which is much smaller. 39:28.329 --> 39:32.199 Okay, so high accuracy is required to calculate a correct 39:32.202 --> 39:34.072 value of the bond energy. 39:34.070 --> 39:35.510 This simple thing won't do it. 39:35.510 --> 39:38.840 Well it's in the right direction, you're halfway there, 39:38.838 --> 39:40.568 so it's a pretty good start. 39:40.565 --> 39:41.115 Right? 39:41.119 --> 39:44.719 But to do the difference, as in the same way you needed 39:44.721 --> 39:47.791 high precision to do X-ray difference maps, 39:47.789 --> 39:49.609 you need better orbitals than this, 39:49.610 --> 39:51.730 if you want to calculate good bond energies. 39:51.730 --> 39:54.830 So you need to make the orbitals better. 39:54.829 --> 39:55.329 Okay? 39:55.329 --> 39:59.719 Now -- so but already we can take heart that the very crudest 39:59.724 --> 40:03.894 model shows most, 52%, of the energy of the bond, 40:03.894 --> 40:08.434 and it shows the electron density building up by 0.02 40:08.434 --> 40:11.494 electrons per cubic bohr radius. 40:11.489 --> 40:15.999 And what we saw qualitatively was there was a shift from the 40:16.001 --> 40:19.061 atom to the bond, of electron density. 40:19.059 --> 40:22.909 Okay, now we can adjust the molecular orbital to get a 40:22.909 --> 40:25.889 better approximation of the true thing. 40:25.889 --> 40:28.669 How will we know when we've adjusted it and it's gotten 40:28.666 --> 40:29.126 better? 40:29.130 --> 40:32.870 If we adjust it and get a lower, calculate a lower average 40:32.867 --> 40:35.107 energy -- I should've said a lower 40:35.106 --> 40:37.966 average energy, because if we don't have a true 40:37.965 --> 40:41.415 wave function we get different values for the total energy; 40:41.420 --> 40:44.570 the kinetic won't exactly offset the potential as you move 40:44.565 --> 40:45.775 from place to place. 40:45.780 --> 40:48.380 But if you get the lowest average energy, 40:48.380 --> 40:51.750 then that is, by definition almost, 40:51.750 --> 40:55.860 more realistic, because you can easily prove 40:55.856 --> 41:00.916 that the true energy is the lowest possible energy; 41:00.920 --> 41:02.860 that makes a certain amount of sense. 41:02.860 --> 41:06.640 The lowest possible calculated energy is the true energy. 41:06.639 --> 41:10.699 So if you change your wave function and get a lower average 41:10.699 --> 41:13.289 energy, you're closer to the truth. 41:13.289 --> 41:14.019 Okay? 41:14.018 --> 41:17.428 That's called the Variational Principle. 41:17.429 --> 41:22.409 Okay, so here we've changed the form of the molecular orbital. 41:22.409 --> 41:23.729 And how did we change it? 41:23.730 --> 41:26.290 What does a 1s orbital look like? 41:26.289 --> 41:30.959 Kate, do you remember the form for a 1s atomic orbital? 41:30.960 --> 41:31.540 I can't hear. 41:31.539 --> 41:32.609 Student: A sphere. 41:32.610 --> 41:34.670 Prof: Angularly it's a sphere. 41:34.670 --> 41:37.740 How does it change as you go out, do you remember? 41:37.739 --> 41:41.259 How does it depend on ρ? 41:41.260 --> 41:44.590 The same way they all depend on ρ. 41:44.590 --> 41:47.120 Anybody remember? 41:47.119 --> 41:49.019 Prof: e^-ρ. 41:49.018 --> 41:52.158 So it falls off exponentially at a certain rate. 41:52.159 --> 41:54.199 And that rate, how fast it falls off, 41:54.197 --> 41:56.347 is determined by the nuclear charge. 41:56.349 --> 41:56.869 Okay? 41:56.869 --> 42:01.509 Now one way to change that shape would be change how fast 42:01.507 --> 42:02.577 it falls off. 42:02.583 --> 42:03.333 Right? 42:03.329 --> 42:05.449 It wouldn't be correct for the atom anymore. 42:05.449 --> 42:07.939 We've already got the correct solution for the atom. 42:07.940 --> 42:11.420 But we could change the shape of the thing by changing how 42:11.423 --> 42:13.323 fast that exponent falls off. 42:13.320 --> 42:17.390 And we could vary that, in the molecule, 42:17.391 --> 42:21.881 using one half -- 1/√2(A+B). 42:21.880 --> 42:24.710 But those A+B are no longer true atomic functions, 42:24.706 --> 42:26.976 they're a little fatter, a little skinnier. 42:26.980 --> 42:27.530 Okay? 42:27.530 --> 42:30.260 And we can change how fat or skinny it is, 42:30.257 --> 42:33.117 until we get the lowest molecular energy. 42:33.119 --> 42:36.989 See, that's a way you can vary it and find the best value. 42:36.989 --> 42:40.209 And that's what was done here to optimize the exponent. 42:40.210 --> 42:42.680 And now you get a total electron density that looks 42:42.681 --> 42:44.661 essentially the same as it did before. 42:44.659 --> 42:48.009 And if you look at the difference density, 42:48.007 --> 42:51.027 how has it changed, if you do this? 42:51.030 --> 42:54.140 First, notice that the energy got lower. 42:54.139 --> 42:58.519 We're now to 73% of the lowering of the bond energy. 42:58.518 --> 43:00.818 So the total energy's gotten lower, it's better. 43:00.820 --> 43:04.320 And how has the electron density changed? 43:04.320 --> 43:06.040 It got higher in the middle. 43:06.039 --> 43:09.589 Because what we did was spread the exponent out a little bit, 43:09.592 --> 43:11.962 so you had more overlap in the middle. 43:11.960 --> 43:13.370 Okay? 43:13.369 --> 43:16.209 So this wouldn't have been good for the single atom, 43:16.206 --> 43:18.816 to spread it out, but it gives a better function 43:18.820 --> 43:19.990 for the molecule. 43:19.989 --> 43:21.679 And it's still very, very simple. 43:21.679 --> 43:25.679 And what you see it did is it increases the bonding density 43:25.677 --> 43:27.537 and the bonding strength. 43:27.539 --> 43:30.319 You get a larger shift from the atoms to the bond. 43:30.320 --> 43:35.850 Now, how else could you change the shape of the atomic orbital 43:35.847 --> 43:39.017 in order to increase the overlap; 43:39.018 --> 43:42.128 some way other than making a single exponential and having it 43:42.126 --> 43:43.366 get fatter or thinner? 43:43.369 --> 43:44.699 Can you think of some other way? 43:44.699 --> 43:47.429 Here you have an atomic orbital, a sphere, 43:47.429 --> 43:51.289 and you want to change its shape so that it overlaps better 43:51.291 --> 43:51.961 over here. 43:51.956 --> 43:52.686 Right? 43:52.690 --> 43:56.830 How could you change the shape of an atomic orbital, 43:56.833 --> 44:00.003 without doing really gross damage to it, 44:00.001 --> 44:04.471 making it a cube or something like that, or a line? 44:04.469 --> 44:08.049 How could you change it so it looked pretty much still like an 44:08.052 --> 44:10.762 atom did, has a lot of the virtues of the atom, 44:10.755 --> 44:12.455 but is shifted over here? 44:12.460 --> 44:17.950 Sam? 44:17.949 --> 44:18.609 I can't hear. 44:18.610 --> 44:20.730 Student: Can't you just allow the electrons to shift? 44:20.730 --> 44:22.850 Prof: Yes, how am I going to write a 44:22.851 --> 44:25.321 function that allows the electrons to shift in the 44:25.324 --> 44:26.794 direction I want them to? 44:26.789 --> 44:27.749 Lexy? 44:27.750 --> 44:29.070 Student: You could hybridize it. 44:29.070 --> 44:30.540 Prof: Hybridize it! 44:30.539 --> 44:32.419 We could hybridize to shift the electrons. 44:32.420 --> 44:33.780 So that's the next one. 44:33.780 --> 44:37.500 So here we're going to -- instead of this we're going to 44:37.503 --> 44:38.523 hybridize it. 44:38.518 --> 44:41.038 Now, this particular calculation did the 44:41.041 --> 44:44.661 hybridization and also did a little self-consistent field 44:44.664 --> 44:45.704 calculation. 44:45.699 --> 44:50.719 And the hybridization left it 96.7% 1s. 44:50.719 --> 44:53.939 So essentially it's still a normal 1s orbital. 44:53.940 --> 44:57.480 But they added .6% of 2s, which expanded it a 44:57.480 --> 45:01.370 little bit, because 2s is -- goes further out than 45:01.367 --> 45:02.337 1s. 45:02.340 --> 45:06.350 And they added 2.7% of 2p. 45:06.349 --> 45:10.549 Now why was 2p much more helpful than 2s? 45:10.550 --> 45:12.730 Lucas? 45:12.730 --> 45:14.620 Student: It has those lobes. 45:14.619 --> 45:16.999 Prof: Yes, so it takes density from one 45:17.001 --> 45:18.801 side and shifts it to the other. 45:18.800 --> 45:21.570 In fact, this is precisely what we saw before. 45:21.570 --> 45:29.190 Notice what it did was -- how much it increased the density in 45:29.186 --> 45:30.806 the middle. 45:30.809 --> 45:33.429 And notice now, it's not taking electron 45:33.429 --> 45:37.129 density away from the nuclei, which was a good place for 45:37.126 --> 45:38.466 electrons to be. 45:38.469 --> 45:40.349 Where does it take it away from? 45:40.349 --> 45:41.759 Student: The left one. 45:41.760 --> 45:44.560 Prof: Out beyond the nuclei, and shift it to the 45:44.561 --> 45:45.031 middle. 45:45.030 --> 45:47.420 So even before, when it was an atom -- 45:47.420 --> 45:50.680 here's the nucleus, a certain distance out here, 45:50.679 --> 45:54.029 and a certain distance out here, were the same in energy, 45:54.030 --> 45:55.360 for that atom. 45:55.360 --> 45:59.630 Now we've taken it from a place which is -- out here, 45:59.634 --> 46:00.954 and put it here. 46:00.949 --> 46:01.689 Right? 46:01.690 --> 46:03.640 Great idea. 46:03.639 --> 46:05.839 Well done Lexy. 46:05.840 --> 46:06.780 So there's what it looked like. 46:06.780 --> 46:09.170 Remember, if it's 100 percent at 1s, 46:09.170 --> 46:11.680 it looks like that, and if you change it to be 46:11.684 --> 46:14.834 hybridized that way, with Atom-in-a-Box, 46:14.827 --> 46:16.687 it looks like this. 46:16.686 --> 46:17.466 Right? 46:17.469 --> 46:20.759 It's not much shift, but it shifts from the left to 46:20.759 --> 46:23.259 the right, and gives better overlap. 46:23.260 --> 46:29.230 Okay, but it requires the atom to be a little bit less happy as 46:29.233 --> 46:34.923 an atom, because it's partly 2s and 2p now. 46:34.920 --> 46:38.190 The electrons are further from the nucleus, but you make up for 46:38.186 --> 46:39.816 that by having a better bond. 46:39.820 --> 46:43.430 Okay, now notice that didn't change the energy very much. 46:43.429 --> 46:46.819 It went from 73% to 76%. Right? 46:46.820 --> 46:49.880 We were already about the right energy, but it changed the 46:49.882 --> 46:52.302 density a lot, to be what we want it to be. 46:52.300 --> 46:57.240 Okay, so there's a much bigger shift, and it's now from beyond 46:57.242 --> 46:59.352 the nucleus into the bond. 46:59.349 --> 47:00.079 Right? 47:00.079 --> 47:04.269 And now we're going to do the last thing. 47:04.268 --> 47:07.008 We've done SCF already, but now you have to do a higher 47:07.005 --> 47:09.485 level calculation that'll do correlation, take the 47:09.489 --> 47:11.819 correlation of the electrons into account. 47:11.820 --> 47:16.640 And if you add some correlation calculation to this, 47:16.641 --> 47:19.481 you now get 90% of the bond. 47:19.480 --> 47:21.980 If you did complete correlation, you'd get 100% of 47:21.978 --> 47:22.538 the bond. 47:22.539 --> 47:27.449 What does it mean to do a calculation with complete 47:27.445 --> 47:28.815 correlation? 47:28.820 --> 47:30.900 It means you don't have an error anymore, 47:30.900 --> 47:32.980 you've used a really good calculation. 47:32.980 --> 47:36.580 So already at this level, that was done fifty years ago 47:36.581 --> 47:38.851 almost, you get 90% of the bond. 47:38.849 --> 47:42.139 And the bond density notice, what about the electron 47:42.143 --> 47:42.793 density? 47:42.789 --> 47:44.129 Student: No change. 47:44.130 --> 47:45.570 Prof: It hasn't really changed. 47:45.570 --> 47:48.760 All that happened was that the electrons kept apart from one 47:48.757 --> 47:51.347 another, but the average density was the same. 47:51.349 --> 47:53.849 So already, with just that hybridization, 47:53.849 --> 47:58.149 we got very close to the truth in electron distribution, 47:58.150 --> 48:01.660 and three-quarters of the way to the truth in how strong a 48:01.663 --> 48:02.283 bond is. 48:02.280 --> 48:04.290 So not a bad approximation. 48:04.289 --> 48:06.179 So hybridization, to give better overlap, 48:06.181 --> 48:07.081 is a great thing. 48:07.079 --> 48:10.639 So the density wasn't changed, but you got a much better 48:10.637 --> 48:11.217 energy. 48:11.219 --> 48:13.619 And how so? 48:13.619 --> 48:16.809 Because the electrons kept apart from one another, 48:16.811 --> 48:18.701 when you allow correlation. 48:18.699 --> 48:23.209 Okay, so here's a pairwise atomic orbital: 48:23.210 --> 48:27.790 1/√2(A+B); and you can use hybridized 48:27.788 --> 48:28.278 orbitals. 48:28.280 --> 48:30.860 And the virtues are it's very easy to formulate and to 48:30.856 --> 48:31.486 understand. 48:31.489 --> 48:35.159 And it looks like atoms, especially when you get down 48:35.159 --> 48:36.429 near the nuclei. 48:36.429 --> 48:39.519 And you don't want that to change because that's the main 48:39.516 --> 48:40.726 event for electrons. 48:40.730 --> 48:43.190 You get much more energy forming atoms, 48:43.193 --> 48:46.183 as we saw before, than you do making new bonds, 48:46.175 --> 48:48.505 once you have the atoms already. 48:48.510 --> 48:49.860 Okay. 48:49.860 --> 48:52.520 It builds up electron density between the nuclei, 48:52.518 --> 48:55.288 through overlap, which is the source of bonding. 48:55.289 --> 49:00.629 It smoothes Ψ, to lower the kinetic energy. 49:00.630 --> 49:04.190 And then there's a pedantic footnote here that actually 49:04.190 --> 49:07.160 there's a thing called the Virial Theorem, 49:07.159 --> 49:09.229 which I'm not going to stress you with, 49:09.230 --> 49:12.470 but that little bit there, it happens to be true. 49:12.469 --> 49:16.299 But still we have the proper understanding of what's going 49:16.302 --> 49:16.642 on. 49:16.639 --> 49:20.509 Hybridizing AOs provides flexibility that gives you 49:20.505 --> 49:21.815 better overlap. 49:21.820 --> 49:25.040 And if you use all the H-like Atomic Orbitals, 49:25.043 --> 49:28.993 you have perfect flexibility, you can make any shape you 49:28.985 --> 49:29.625 want. 49:29.630 --> 49:32.150 Okay, but we're going to keep it simple, 49:32.150 --> 49:35.930 use only 2s and 2p orbitals to hybridize, 49:35.929 --> 49:38.559 because that'll get you most of the way there and it's much 49:38.561 --> 49:40.971 simpler, rather than to try to mix 49:40.965 --> 49:43.475 5f orbitals into it also. 49:43.480 --> 49:45.110 Okay, so that's great. 49:45.110 --> 49:46.840 And when we square it, we get this, 49:46.835 --> 49:49.215 which has the overlap part that helps us out. 49:49.219 --> 49:51.639 We have the atoms, plus the bond, 49:51.641 --> 49:54.821 which is the overlap, that product part. 49:54.820 --> 49:59.040 But we could've done the same thing to get the same product, 49:59.039 --> 50:02.229 as far as the atoms go, if we'd used a minus sign 50:02.233 --> 50:05.233 instead of a plus sign in combining things; 50:05.230 --> 50:08.550 although then we would've changed the sign of the overlap 50:08.545 --> 50:09.015 thing. 50:09.019 --> 50:10.429 It would become minus. 50:10.429 --> 50:13.979 So we would change it from being less than to being greater 50:13.981 --> 50:17.231 than, in order to have it be normalized at the bottom. 50:17.226 --> 50:17.836 Right? 50:17.840 --> 50:19.320 And that's the anti-bond. 50:19.320 --> 50:24.030 So we get both the bond and the anti-bond by doing this. 50:24.030 --> 50:26.920 And now we're going to go on, next time, to overlap, 50:26.922 --> 50:29.992 which we've already introduced, and also the concept of 50:29.985 --> 50:30.945 energy-match. 50:30.949 --> 50:33.429 And when you put these two together, you'll really 50:33.434 --> 50:34.504 understand bonding. 50:34.500 --> 50:40.000