1 00:00:00,090 --> 00:00:02,490 The following content is provided under a Creative 2 00:00:02,490 --> 00:00:04,030 Commons license. 3 00:00:04,030 --> 00:00:06,330 Your support will help MIT OpenCourseWare 4 00:00:06,330 --> 00:00:10,690 continue to offer high-quality educational resources for free. 5 00:00:10,690 --> 00:00:13,320 To make a donation or view additional materials 6 00:00:13,320 --> 00:00:17,250 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,250 --> 00:00:18,220 at ocw.mit.edu. 8 00:00:21,270 --> 00:00:26,920 PROFESSOR: I like this lecture because the first half 9 00:00:26,920 --> 00:00:28,810 of my career was entirely devoted 10 00:00:28,810 --> 00:00:31,870 to the electronic structure of diatomic molecules. 11 00:00:31,870 --> 00:00:36,670 And this is how you make sense of diatomic molecules. 12 00:00:36,670 --> 00:00:43,170 Now in 5.111, 5.112, one of the most exciting thing that 13 00:00:43,170 --> 00:00:49,050 attracts people to become chemistry majors 14 00:00:49,050 --> 00:00:53,040 is the global interpretation of the periodic table 15 00:00:53,040 --> 00:00:57,810 in terms of some simple ideas so that those simple ideas 16 00:00:57,810 --> 00:01:01,560 enable you to predict the important properties of all 17 00:01:01,560 --> 00:01:02,910 of the atoms. 18 00:01:02,910 --> 00:01:05,850 And that's a pretty exciting thing to feel like, 19 00:01:05,850 --> 00:01:09,650 yes, it all makes some sense. 20 00:01:09,650 --> 00:01:14,240 And then we go on to other topics in chemistry, 21 00:01:14,240 --> 00:01:16,910 and we forget about the fact that we think we understand 22 00:01:16,910 --> 00:01:18,260 the periodic table completely. 23 00:01:18,260 --> 00:01:20,610 We don't use that information. 24 00:01:20,610 --> 00:01:24,470 And one thing that I've tried to do throughout my career 25 00:01:24,470 --> 00:01:29,660 is to take the simple insights into the periodic table, 26 00:01:29,660 --> 00:01:33,080 and explain molecular properties. 27 00:01:33,080 --> 00:01:37,700 And now, this is really a strange thing because we have 28 00:01:37,700 --> 00:01:41,110 computer programs that can calculate any 29 00:01:41,110 --> 00:01:43,730 of the properties-- any property you want-- 30 00:01:43,730 --> 00:01:46,460 of a small molecule. 31 00:01:46,460 --> 00:01:50,090 So why would we ever care about being 32 00:01:50,090 --> 00:01:54,390 able to predict the properties-- because we can calculate them? 33 00:01:54,390 --> 00:01:58,250 And the answer is we want to understand. 34 00:01:58,250 --> 00:02:01,010 We want to be able, without a computer, 35 00:02:01,010 --> 00:02:05,030 to say this is what we expect for a particular molecule not 36 00:02:05,030 --> 00:02:07,130 just a diatomic molecule. 37 00:02:07,130 --> 00:02:10,160 What are the things that control the bonding 38 00:02:10,160 --> 00:02:13,580 in molecules in general and the properties in molecules 39 00:02:13,580 --> 00:02:14,780 in general? 40 00:02:14,780 --> 00:02:19,580 And if you can understand what happens in diatomic molecules, 41 00:02:19,580 --> 00:02:23,930 there are very few cases which would leave you 42 00:02:23,930 --> 00:02:30,860 at a loss for what will be the properties of this new molecule 43 00:02:30,860 --> 00:02:32,420 that you're looking at. 44 00:02:32,420 --> 00:02:35,000 Or how would you manipulate a molecule 45 00:02:35,000 --> 00:02:38,240 so that you can force it to go into the states that 46 00:02:38,240 --> 00:02:42,590 will expose the critical properties that you want? 47 00:02:42,590 --> 00:02:46,010 So it's this sort of part of a toolkit 48 00:02:46,010 --> 00:02:48,590 that any practicing physical chemist who's 49 00:02:48,590 --> 00:02:51,890 worried about spectra, molecular properties, 50 00:02:51,890 --> 00:02:57,520 electronic properties would have to master. 51 00:02:57,520 --> 00:03:02,930 And this is really hard because, one, you 52 00:03:02,930 --> 00:03:06,710 have a computer-- you can generate the answers completely 53 00:03:06,710 --> 00:03:07,950 accurately. 54 00:03:07,950 --> 00:03:12,500 And two, you're smart MIT students. 55 00:03:12,500 --> 00:03:16,130 And you believe in the truth rather than 56 00:03:16,130 --> 00:03:18,660 estimating the truth. 57 00:03:18,660 --> 00:03:23,190 But I like to maintain that if you can estimate the truth 58 00:03:23,190 --> 00:03:27,030 and find out whether your estimates are wrong or right, 59 00:03:27,030 --> 00:03:30,720 that enables you to have confidence 60 00:03:30,720 --> 00:03:34,710 to deal with other properties, other problems. 61 00:03:34,710 --> 00:03:39,830 And so what I want to do today is 62 00:03:39,830 --> 00:03:43,250 to give a sense of how you begin to understand 63 00:03:43,250 --> 00:03:47,060 the electronic structure of simple molecules. 64 00:03:47,060 --> 00:03:50,120 And I want to tell you that a lot 65 00:03:50,120 --> 00:03:55,520 of the second-order or complicated patches you put 66 00:03:55,520 --> 00:03:59,480 on this simple picture are things 67 00:03:59,480 --> 00:04:03,680 that you develop and criticize throughout your career. 68 00:04:03,680 --> 00:04:06,940 You say, yeah, well, this works, but that doesn't. 69 00:04:06,940 --> 00:04:08,250 And why doesn't it work? 70 00:04:08,250 --> 00:04:10,190 And that's what we do as physical chemists. 71 00:04:10,190 --> 00:04:13,550 We look for when the simple picture breaks. 72 00:04:13,550 --> 00:04:17,620 And we have our own private simple pictures which 73 00:04:17,620 --> 00:04:22,089 we're constantly trying to make better than anybody else's. 74 00:04:22,089 --> 00:04:27,220 OK, so we have the periodic table. 75 00:04:27,220 --> 00:04:29,910 We know the properties of atoms. 76 00:04:29,910 --> 00:04:34,530 We know, especially, how the ionization energy 77 00:04:34,530 --> 00:04:36,310 is supposed to behave. 78 00:04:36,310 --> 00:04:41,910 We know how the low-lying electronic states 79 00:04:41,910 --> 00:04:44,280 are supposed to be ordered. 80 00:04:44,280 --> 00:04:44,910 Excuse me. 81 00:04:48,020 --> 00:04:50,360 We know about orbital sizes. 82 00:04:50,360 --> 00:04:54,860 Everything is related to this idea of atomic-orbital energies 83 00:04:54,860 --> 00:04:56,520 and shielding. 84 00:04:56,520 --> 00:05:00,050 And it's all related to the hydrogen atom. 85 00:05:00,050 --> 00:05:02,000 What we got from the hydrogen atom 86 00:05:02,000 --> 00:05:04,700 was a framework for understanding 87 00:05:04,700 --> 00:05:08,510 electronic properties of atoms. 88 00:05:08,510 --> 00:05:11,360 And what we know about electronic properties of atoms 89 00:05:11,360 --> 00:05:15,410 should be important in explaining 90 00:05:15,410 --> 00:05:18,290 electronic properties of molecules. 91 00:05:18,290 --> 00:05:23,030 So I want you to be able to make toy models, 92 00:05:23,030 --> 00:05:27,440 to draw pictures which are crude, but guiding, 93 00:05:27,440 --> 00:05:30,430 in your insights. 94 00:05:30,430 --> 00:05:32,990 And it's all based on what we saw. 95 00:05:38,980 --> 00:05:42,940 And you soon are going to have two lectures from Troy Van 96 00:05:42,940 --> 00:05:50,080 Voorhis, which will lead you to use some of the simple computer 97 00:05:50,080 --> 00:05:54,100 programs that enable you to calculate electronic property 98 00:05:54,100 --> 00:05:56,940 of anything. 99 00:05:56,940 --> 00:06:04,870 OK, so we want to extend what we had for H2+ to be able 100 00:06:04,870 --> 00:06:14,510 to understand H 2 AH, A 2, and AB molecules. 101 00:06:17,210 --> 00:06:23,030 And each of these steps involve extensions 102 00:06:23,030 --> 00:06:25,790 of relatively simple ideas. 103 00:06:25,790 --> 00:06:32,000 And so let's begin. 104 00:06:32,000 --> 00:06:36,570 OK, I told you that I believe the simplest explanation 105 00:06:36,570 --> 00:06:44,250 for bonding is when we have two atomic orbitals that overlap 106 00:06:44,250 --> 00:06:48,870 in the region where this overlap region can 107 00:06:48,870 --> 00:06:50,745 be attracted to both nuclei. 108 00:06:53,540 --> 00:06:58,280 Constructive interference causes the density 109 00:06:58,280 --> 00:07:01,010 in this special region-- 110 00:07:01,010 --> 00:07:04,340 the amplitude to be twice as large, the density 111 00:07:04,340 --> 00:07:06,230 to be four times as large. 112 00:07:06,230 --> 00:07:08,180 And that's a large, energetic effect. 113 00:07:11,110 --> 00:07:15,590 And the amount of overlap is crucial. 114 00:07:15,590 --> 00:07:19,310 You can have two little if the atoms are too far apart-- 115 00:07:19,310 --> 00:07:20,340 no bond. 116 00:07:20,340 --> 00:07:22,970 You could have too much if the atoms are too close together. 117 00:07:25,860 --> 00:07:31,990 OK, so we need to now look at cartoons of orbitals 118 00:07:31,990 --> 00:07:35,690 and learn about the names for these orbitals. 119 00:07:35,690 --> 00:07:39,550 So the simplest orbital is gerade-- 120 00:07:39,550 --> 00:07:44,390 sigma gerade NS orbital. 121 00:07:44,390 --> 00:07:49,090 OK, so "sigma" means there are no nodal planes 122 00:07:49,090 --> 00:07:52,210 containing the internuclear axis. 123 00:07:52,210 --> 00:07:56,110 "Gerade" means that in the body frame, it's even symmetry. 124 00:07:59,390 --> 00:08:11,090 So we have ns A plus ns B. And that looks like this. 125 00:08:11,090 --> 00:08:14,910 We can have an antibonding orbital. 126 00:08:14,910 --> 00:08:21,790 Again, we put the predominant atomic-orbital character 127 00:08:21,790 --> 00:08:23,040 in parentheses. 128 00:08:23,040 --> 00:08:24,970 And for heteronuclear molecules, we also 129 00:08:24,970 --> 00:08:29,470 put which atom is more important to that orbital. 130 00:08:29,470 --> 00:08:33,220 And this will be just ns A minus ns. 131 00:08:42,919 --> 00:08:47,310 And that will look like this. 132 00:08:47,310 --> 00:08:48,670 And there is no overlap here. 133 00:08:48,670 --> 00:08:51,330 In fact, there is negative overlap. 134 00:08:51,330 --> 00:08:54,450 And that's an antibonding orbital. 135 00:08:54,450 --> 00:08:56,760 And you can say, if there's no overlap, well, 136 00:08:56,760 --> 00:08:59,010 then the actual picture will sort of 137 00:08:59,010 --> 00:09:04,200 look like this, where the region in the binding region 138 00:09:04,200 --> 00:09:07,530 is actually depleted, and there's less there. 139 00:09:07,530 --> 00:09:09,381 And that's part of why it's antibonding. 140 00:09:09,381 --> 00:09:09,880 OK. 141 00:09:15,570 --> 00:09:19,920 Now we can also make sigma g np orbitals. 142 00:09:22,560 --> 00:09:28,270 And np z-- "z" is the internuclear axis-- 143 00:09:28,270 --> 00:09:36,409 A minus and np z, B-- 144 00:09:36,409 --> 00:09:38,575 well, that gives us something that looks like this-- 145 00:09:41,720 --> 00:09:44,360 plus, plus, minus, minus. 146 00:09:44,360 --> 00:09:46,840 And the minus sign is needed to turn this around. 147 00:09:50,440 --> 00:09:52,890 But this is overall gerade symmetry. 148 00:09:52,890 --> 00:09:54,785 And so everything is fine. 149 00:09:54,785 --> 00:10:01,530 And then we can have sigma u star np. 150 00:10:01,530 --> 00:10:05,430 And that would just be the plus sign here. 151 00:10:05,430 --> 00:10:08,200 And that would be an antibonding orbital. 152 00:10:08,200 --> 00:10:12,240 Now I put a star on these orbitals 153 00:10:12,240 --> 00:10:17,640 because when we don't have atomic symmetry, when 154 00:10:17,640 --> 00:10:20,640 we don't have g and u symmetry, we still 155 00:10:20,640 --> 00:10:24,360 use the star to imply antibonding. 156 00:10:24,360 --> 00:10:28,670 But in diatomic molecules, this is redundant. 157 00:10:31,960 --> 00:10:35,590 You should be able, just by looking 158 00:10:35,590 --> 00:10:40,420 at the orbital character or looking at a picture, to say, 159 00:10:40,420 --> 00:10:43,750 yes, it's a bonding orbital, and it's either g or u. 160 00:10:48,380 --> 00:10:53,120 Then we have pi u np. 161 00:10:53,120 --> 00:10:56,210 And that would be-- 162 00:10:56,210 --> 00:10:57,800 well, let's do it the opposite order. 163 00:11:00,600 --> 00:11:03,810 So we'd like to have something that's bonding. 164 00:11:03,810 --> 00:11:06,350 So we have to have the phases right. 165 00:11:06,350 --> 00:11:08,930 And that's ungerade because when you reflect 166 00:11:08,930 --> 00:11:15,260 through the center of the origin of coordinates, 167 00:11:15,260 --> 00:11:17,180 we go from plus to minus. 168 00:11:17,180 --> 00:11:18,350 So that's ungerade. 169 00:11:18,350 --> 00:11:20,050 But that's bonding. 170 00:11:20,050 --> 00:11:21,680 And that's a little trick because you 171 00:11:21,680 --> 00:11:26,043 start this think, from sigma, that ungerade is antibonding. 172 00:11:29,210 --> 00:11:34,480 And finally, we have the pi g star np. 173 00:11:34,480 --> 00:11:38,320 And that would be plus, minus, minus, plus. 174 00:11:38,320 --> 00:11:39,080 And that's gerade. 175 00:11:42,740 --> 00:11:45,650 OK, there are a couple other things that we know just 176 00:11:45,650 --> 00:11:47,450 from these pictures. 177 00:11:47,450 --> 00:11:56,280 And that is the p-sigma orbital is much more directional 178 00:11:56,280 --> 00:11:59,460 than the s-sigma orbital. 179 00:11:59,460 --> 00:12:02,100 That's all you get in sigma for s. 180 00:12:02,100 --> 00:12:12,570 And so the bonding interaction between p-sigma orbitals 181 00:12:12,570 --> 00:12:17,760 turns on at longer range than for s. 182 00:12:17,760 --> 00:12:20,350 So there's qualitative differences. 183 00:12:20,350 --> 00:12:24,720 And sometimes, you have a bond distance 184 00:12:24,720 --> 00:12:28,800 determined by favorable overlap between one pair of orbitals. 185 00:12:28,800 --> 00:12:32,100 And it makes things not so good for other orbitals. 186 00:12:32,100 --> 00:12:33,600 So usually, you have a ground state. 187 00:12:33,600 --> 00:12:36,307 You figure out what the ground state of molecule is. 188 00:12:36,307 --> 00:12:38,640 And that tells you something about what the internuclear 189 00:12:38,640 --> 00:12:40,230 distance is. 190 00:12:40,230 --> 00:12:44,640 And then you might grow to excited states of the molecule 191 00:12:44,640 --> 00:12:47,540 or excited configurations. 192 00:12:47,540 --> 00:12:50,040 And these excited configurations-- 193 00:12:50,040 --> 00:12:52,770 well, if you're going to learn about a molecule, 194 00:12:52,770 --> 00:12:54,630 you've got to do spectroscopy. 195 00:12:54,630 --> 00:13:00,070 So if you go from the ground state to an excited state, 196 00:13:00,070 --> 00:13:02,490 and the internuclear distance of the ground state 197 00:13:02,490 --> 00:13:06,180 is wrong for the excited state, well, then, you're 198 00:13:06,180 --> 00:13:09,000 going to have a big change in geometry. 199 00:13:09,000 --> 00:13:11,850 You'll have Frank-Condon factors, which-- 200 00:13:11,850 --> 00:13:14,190 I haven't talked about Frank-Condon factors-- 201 00:13:14,190 --> 00:13:19,290 I will-- that say, OK, the transition 202 00:13:19,290 --> 00:13:22,070 is going to have a lot of vibrational structure. 203 00:13:22,070 --> 00:13:24,550 And it'll tell you all sorts of good stuff about that. 204 00:13:24,550 --> 00:13:27,510 But the important thing is when you 205 00:13:27,510 --> 00:13:29,940 start looking at a molecule, you want 206 00:13:29,940 --> 00:13:32,580 to know what the ground state looks like 207 00:13:32,580 --> 00:13:35,920 and what transitions from the ground state 208 00:13:35,920 --> 00:13:39,270 ought to look like when you're ready to know how 209 00:13:39,270 --> 00:13:41,970 to use the simple structural information you're getting 210 00:13:41,970 --> 00:13:46,930 from these orbitals because there's no point doing 211 00:13:46,930 --> 00:13:49,690 an experiment if you don't have an idea about what 212 00:13:49,690 --> 00:13:50,800 the experiment will yield. 213 00:13:54,150 --> 00:13:56,890 That's what most people do when they do an experiment. 214 00:13:56,890 --> 00:13:59,130 They did the experiment because they 215 00:13:59,130 --> 00:14:01,050 thought it would yield something, 216 00:14:01,050 --> 00:14:03,000 and they wanted to be surprised because they'd 217 00:14:03,000 --> 00:14:04,820 like to show that maybe it didn't 218 00:14:04,820 --> 00:14:06,870 do what they were expecting. 219 00:14:06,870 --> 00:14:10,170 OK, so we have pictures. 220 00:14:10,170 --> 00:14:11,970 And we have notation. 221 00:14:11,970 --> 00:14:13,800 And this notation is what is used 222 00:14:13,800 --> 00:14:20,120 by professional spectroscopies, 223 00:14:20,120 --> 00:14:23,510 So we care about directionality. 224 00:14:28,120 --> 00:14:31,550 And we care about size of orbitals. 225 00:14:31,550 --> 00:14:35,410 And this is definitely something that you're 226 00:14:35,410 --> 00:14:40,880 empowered from 5.111, 5.112. 227 00:14:40,880 --> 00:14:44,480 I'll talk more about that too. 228 00:14:44,480 --> 00:14:47,390 How about right now? 229 00:14:47,390 --> 00:14:52,460 OK, in thermodynamics, when we're 230 00:14:52,460 --> 00:14:59,610 concerned about the enthalpy of formation of a molecule, 231 00:14:59,610 --> 00:15:04,250 we have to set a 0 of enthalpy. 232 00:15:04,250 --> 00:15:11,090 And we set it as separated atoms in their most-stable state 233 00:15:11,090 --> 00:15:15,030 and energy, or enthalpy, is something 234 00:15:15,030 --> 00:15:18,120 where there is no natural 0. 235 00:15:18,120 --> 00:15:22,140 We wanted a 0 which is convenient for all 236 00:15:22,140 --> 00:15:24,870 of our calculations and insights. 237 00:15:24,870 --> 00:15:30,480 And so what we do is we say, OK, every particle 238 00:15:30,480 --> 00:15:31,950 that we're going to be looking at-- 239 00:15:31,950 --> 00:15:35,550 we're going to make a molecule out of two atoms, 240 00:15:35,550 --> 00:15:38,270 like A and B and AB star. 241 00:15:38,270 --> 00:15:46,920 So we have the energy of A plus electron B plus electron, 242 00:15:46,920 --> 00:15:50,460 or AB plus electron. 243 00:15:50,460 --> 00:15:54,072 That's a common 0 of energy. 244 00:15:54,072 --> 00:15:58,520 So we talk about orbital energies 245 00:15:58,520 --> 00:16:00,530 relative to this common 0. 246 00:16:00,530 --> 00:16:04,700 Now in almost every textbook, the orbital diagrams 247 00:16:04,700 --> 00:16:07,030 are given to you. 248 00:16:07,030 --> 00:16:11,310 There's no discussion of, what's the 0 of energy? 249 00:16:11,310 --> 00:16:13,290 If you're going to know anything, 250 00:16:13,290 --> 00:16:17,430 you really want to be able to calculate or to say 251 00:16:17,430 --> 00:16:23,340 all of the actors in this game are acting in a way related 252 00:16:23,340 --> 00:16:26,370 to how far below this ionization energy 253 00:16:26,370 --> 00:16:30,570 they are because that determines their size. 254 00:16:30,570 --> 00:16:32,810 And the size determines the overlap. 255 00:16:32,810 --> 00:16:36,350 And bonding is related to overlap. 256 00:16:36,350 --> 00:16:38,220 OK, so we'll play that game. 257 00:16:42,040 --> 00:16:48,730 So for H 2+, it was very simple. 258 00:16:48,730 --> 00:16:55,620 We just had the 1s at r equals infinity, 259 00:16:55,620 --> 00:17:02,870 and over here, the 1s orbital at r equals infinity. 260 00:17:02,870 --> 00:17:07,859 And then we solved the minimal-basis variational 261 00:17:07,859 --> 00:17:09,119 problem. 262 00:17:09,119 --> 00:17:16,180 And we discovered that we had something like this, where this 263 00:17:16,180 --> 00:17:17,710 is the antibonding orbital. 264 00:17:17,710 --> 00:17:19,510 This is the binding orbital. 265 00:17:19,510 --> 00:17:22,540 The antibonding orbital is more antibonding 266 00:17:22,540 --> 00:17:26,329 than the bonding orbital is bonding. 267 00:17:26,329 --> 00:17:31,520 And, of course, this energy difference 268 00:17:31,520 --> 00:17:34,810 depends on internuclear distance, 269 00:17:34,810 --> 00:17:36,280 also not often talked about. 270 00:17:40,170 --> 00:17:44,970 And so we have the starting point at r equals infinity 271 00:17:44,970 --> 00:17:48,860 for the separated atoms. 272 00:17:48,860 --> 00:17:50,880 Now as we bring these atoms together, 273 00:17:50,880 --> 00:17:54,690 before there's any bonding at all, 274 00:17:54,690 --> 00:17:58,350 say, we have 1s on A. We make the internuclear 275 00:17:58,350 --> 00:18:00,170 distance smaller and smaller. 276 00:18:00,170 --> 00:18:05,230 And the bare nucleus of the other atom 277 00:18:05,230 --> 00:18:10,090 starts to penetrate into the-- 278 00:18:10,090 --> 00:18:14,290 so as you move the positively charged nucleus 279 00:18:14,290 --> 00:18:18,850 towards the static charge on the other atom, 280 00:18:18,850 --> 00:18:20,820 that's a favorable interaction. 281 00:18:20,820 --> 00:18:25,800 But there's a repulsion between the two nuclei. 282 00:18:25,800 --> 00:18:30,690 And when the bare nucleus of one atom 283 00:18:30,690 --> 00:18:34,530 penetrates inside the charge distribution of the other, 284 00:18:34,530 --> 00:18:38,590 it sees less attraction and more repulsion. 285 00:18:38,590 --> 00:18:42,000 And so this is why you get a potential curve that 286 00:18:42,000 --> 00:18:43,340 looks like this. 287 00:18:43,340 --> 00:18:47,310 The repulsion occurs at too-short internuclear 288 00:18:47,310 --> 00:18:48,210 distance. 289 00:18:48,210 --> 00:18:50,910 And from the variational calculation, 290 00:18:50,910 --> 00:18:55,190 you can determine the value of equilibrium internuclear 291 00:18:55,190 --> 00:18:57,670 distance. 292 00:18:57,670 --> 00:18:59,290 And so a lot of these diagrams are 293 00:18:59,290 --> 00:19:03,670 calculated at this particular internuclear distance. 294 00:19:03,670 --> 00:19:06,490 It's especially important because the repulsive states 295 00:19:06,490 --> 00:19:08,710 don't have a minimum. 296 00:19:08,710 --> 00:19:12,070 And so there is no particular internuclear distance 297 00:19:12,070 --> 00:19:20,290 that you care about so that if you want a simple picture, 298 00:19:20,290 --> 00:19:23,310 everything is calculated at that particular internuclear 299 00:19:23,310 --> 00:19:23,810 distance. 300 00:19:27,390 --> 00:19:32,860 OK, so sometimes, you want to have a little bit more insight 301 00:19:32,860 --> 00:19:33,910 in this picture. 302 00:19:33,910 --> 00:19:37,870 And instead of drawing the starting point 303 00:19:37,870 --> 00:19:42,280 at infinite separation, you could say, well, 304 00:19:42,280 --> 00:19:48,300 let's schematically suggest what happens as you-- 305 00:19:57,450 --> 00:20:00,180 so at shorter internuclear distance, 306 00:20:00,180 --> 00:20:04,770 the separated atom energy increases a little bit. 307 00:20:04,770 --> 00:20:06,540 And that could be useful. 308 00:20:06,540 --> 00:20:09,490 But it complicates things, so I won't do that. 309 00:20:09,490 --> 00:20:11,010 But often, we do like to know what 310 00:20:11,010 --> 00:20:14,970 happens as we bring particles together as opposed to starting 311 00:20:14,970 --> 00:20:18,570 at infinity and coming into the end point, which 312 00:20:18,570 --> 00:20:21,320 is the bond at the equilibrium internuclear distance. 313 00:20:21,320 --> 00:20:24,940 So you can put that insight in if you wish. 314 00:20:24,940 --> 00:20:29,700 So we know that this bonding interaction is smaller 315 00:20:29,700 --> 00:20:31,757 than the antibonding interaction. 316 00:20:35,960 --> 00:20:48,290 OK, we can now go from H 2+ to H 2. 317 00:20:51,270 --> 00:20:59,060 And the first thing we know is when we go from H 2+ to H 2, 318 00:20:59,060 --> 00:21:04,580 we can put an electron into the same orbital but with opposite 319 00:21:04,580 --> 00:21:06,520 spin. 320 00:21:06,520 --> 00:21:08,710 And as a result, we don't have to worry 321 00:21:08,710 --> 00:21:11,950 about mysterious things, like overlap repulsion 322 00:21:11,950 --> 00:21:17,290 or poly-repulsion because we're putting electrons 323 00:21:17,290 --> 00:21:19,360 into different spin orbitals. 324 00:21:19,360 --> 00:21:21,700 And that's all you have to do. 325 00:21:21,700 --> 00:21:26,590 If you try to put two electrons into the same spin orbital, 326 00:21:26,590 --> 00:21:28,120 that's really bad. 327 00:21:28,120 --> 00:21:30,730 And once we get beyond hydrogen-- hydrogen 328 00:21:30,730 --> 00:21:36,130 has this neat situation that there is no core. 329 00:21:36,130 --> 00:21:38,550 It's just the nuclei. 330 00:21:38,550 --> 00:21:43,750 But for anything other than hydrides or hydrogen, 331 00:21:43,750 --> 00:21:48,760 we have an inner core, which is filled 332 00:21:48,760 --> 00:21:54,750 with electrons of both spins. 333 00:21:54,750 --> 00:21:59,070 So anytime you have an orbital that 334 00:21:59,070 --> 00:22:04,110 gets too close to the other atom, 335 00:22:04,110 --> 00:22:07,890 there is going to be this mysterious overlap repulsion. 336 00:22:07,890 --> 00:22:15,690 And that makes the inner wall of most potentials other 337 00:22:15,690 --> 00:22:17,860 than hydrogen extremely vertical. 338 00:22:21,010 --> 00:22:27,070 OK, so let me just-- 339 00:22:27,070 --> 00:22:33,490 oh, I know what I wanted to say, but i will get to that 340 00:22:33,490 --> 00:22:40,180 When we go from H 2+ to H 2, we discover that the equilibrium 341 00:22:40,180 --> 00:22:49,950 in our internuclear distance decreases by approximately 30%. 342 00:22:49,950 --> 00:22:58,610 The vibrational frequency increases by 90%. 343 00:23:02,250 --> 00:23:14,160 The dissociation energy increases by about 70%. 344 00:23:14,160 --> 00:23:16,770 These are big effects. 345 00:23:16,770 --> 00:23:18,780 They're not a factor of 2. 346 00:23:18,780 --> 00:23:23,160 Naively, if one electron gives a bond 347 00:23:23,160 --> 00:23:25,860 or gives some sort of a bonding interaction, 348 00:23:25,860 --> 00:23:28,200 two should give twice that much. 349 00:23:28,200 --> 00:23:30,640 But this is pretty close to twice that much. 350 00:23:30,640 --> 00:23:34,770 Even though that's only 30%, it's a big effect. 351 00:23:34,770 --> 00:23:37,230 And it's because we're allowed to put 352 00:23:37,230 --> 00:23:42,880 an electron into an orbital which is bonding. 353 00:23:42,880 --> 00:23:47,570 And there is no magic-overlap repulsion. 354 00:23:47,570 --> 00:23:51,000 There is just the interelectronic-energy 355 00:23:51,000 --> 00:23:53,100 repulsion. 356 00:23:53,100 --> 00:23:55,110 So our picture is-- we're chemists. 357 00:23:55,110 --> 00:23:56,070 We believe in bonds. 358 00:23:56,070 --> 00:23:58,410 The most important thing is bonds. 359 00:23:58,410 --> 00:24:06,300 And in almost every situation, you 360 00:24:06,300 --> 00:24:09,270 want to conserve the number of bonds 361 00:24:09,270 --> 00:24:11,340 or know how much you have to pay for changing 362 00:24:11,340 --> 00:24:13,260 the number of bonds. 363 00:24:13,260 --> 00:24:15,250 That's the course. 364 00:24:15,250 --> 00:24:17,610 But we're more sophisticated. 365 00:24:25,260 --> 00:24:33,070 Now in our picture for H 2+, we had an effective Hamiltonian 366 00:24:33,070 --> 00:24:37,340 or a variational calculation. 367 00:24:37,340 --> 00:24:39,350 I like to talk about effective Hamiltonians. 368 00:24:39,350 --> 00:24:45,410 But the crucial actors in the H 2+ problem, 369 00:24:45,410 --> 00:24:49,930 or the overlap is a function R, the interaction energy 370 00:24:49,930 --> 00:24:55,580 as a function of R, and the orbital energy is a function 371 00:24:55,580 --> 00:25:02,060 of R. And these are the actors in almost all further 372 00:25:02,060 --> 00:25:03,410 calculations. 373 00:25:03,410 --> 00:25:06,350 You're going to want to either know these things 374 00:25:06,350 --> 00:25:08,720 or be able to estimate their sizes so you 375 00:25:08,720 --> 00:25:11,570 know the consequences. 376 00:25:11,570 --> 00:25:15,140 And we know much of the consequences from H 2+. 377 00:25:15,140 --> 00:25:29,570 We know that as R decreases, the overlap increases, 378 00:25:29,570 --> 00:25:38,750 the increases, and the interaction 379 00:25:38,750 --> 00:25:42,560 energy between the two atoms increases. 380 00:25:42,560 --> 00:25:49,400 So these are the factors that we put into a calculation. 381 00:25:49,400 --> 00:25:51,880 We know that there is a bonding region. 382 00:25:51,880 --> 00:26:01,460 And typically, we can say that the bonding region for H 2+ 383 00:26:01,460 --> 00:26:07,680 and for any molecule is going to be roughly between 384 00:26:07,680 --> 00:26:13,800 the atomic-orbital radius and twice the atomic-orbital radius 385 00:26:13,800 --> 00:26:15,490 because we've got two atoms. 386 00:26:15,490 --> 00:26:17,670 And so they want to see it at twice that radius. 387 00:26:17,670 --> 00:26:19,980 And then as we get closer, we get bonding. 388 00:26:19,980 --> 00:26:23,430 And as we get too close, we get antibonding. 389 00:26:23,430 --> 00:26:25,730 And so this is a critical region. 390 00:26:33,320 --> 00:26:36,965 Now we're ready to take the next few steps. 391 00:26:46,620 --> 00:26:53,730 So we want to do H 2+ to H 2 to A 2. 392 00:26:57,210 --> 00:27:00,040 And so the questions we ask is, how many electrons? 393 00:27:04,530 --> 00:27:06,910 That's easy. 394 00:27:06,910 --> 00:27:15,242 Feed the energy electrons into lowest MOs. 395 00:27:15,242 --> 00:27:16,700 Well, in order to do that, you need 396 00:27:16,700 --> 00:27:21,290 to know the energy order of the MOs. 397 00:27:21,290 --> 00:27:27,070 We get a configuration, which is just 398 00:27:27,070 --> 00:27:29,890 a list of the number of electrons in each 399 00:27:29,890 --> 00:27:31,015 of the molecular orbitals. 400 00:27:34,360 --> 00:27:40,150 And these configurations have particular states. 401 00:27:40,150 --> 00:27:46,120 And for a diatonic molecule, we know the projection 402 00:27:46,120 --> 00:27:48,670 of the electron orbital-angular momentum 403 00:27:48,670 --> 00:27:53,200 on the other nuclear axis, lambda, we know the spin. 404 00:27:53,200 --> 00:27:57,050 And sometimes, we know about this thing, omega, 405 00:27:57,050 --> 00:28:02,340 which is the projection of the spin, 406 00:28:02,340 --> 00:28:05,630 as well as the projection of the orbital-angular momentum. 407 00:28:05,630 --> 00:28:09,180 And so we have a notation for electronic states, which 408 00:28:09,180 --> 00:28:15,100 will be lambda 2s plus 1 omega. 409 00:28:15,100 --> 00:28:19,950 And so say we have a triplet. 410 00:28:19,950 --> 00:28:23,850 Well, this triplet will be 2s plus 1 is 3. 411 00:28:23,850 --> 00:28:26,130 And this is pi, say. 412 00:28:26,130 --> 00:28:31,050 So the correct way to say that is not "3 pi." 413 00:28:31,050 --> 00:28:33,980 but "triplet pi." 414 00:28:33,980 --> 00:28:38,150 And we normally don't talk about that in any special notation. 415 00:28:41,050 --> 00:28:46,780 So the splittings of the configurations for H2-- 416 00:28:46,780 --> 00:28:48,510 we have the ground state. 417 00:28:52,220 --> 00:28:59,730 And that's sigma g 1s squared, and the lowest excited 418 00:28:59,730 --> 00:29:06,880 state, triplet sigma u plus-- 419 00:29:06,880 --> 00:29:18,316 and that's sigma g 1s sigma u us. 420 00:29:23,680 --> 00:29:25,520 So we have these pictures. 421 00:29:25,520 --> 00:29:27,460 And this is bound. 422 00:29:27,460 --> 00:29:30,370 This has got a binding orbital and an antibonding orbital. 423 00:29:30,370 --> 00:29:31,640 This is a sigma u 1s. 424 00:29:31,640 --> 00:29:36,190 Sorry-- so antibonding. 425 00:29:36,190 --> 00:29:37,660 So this is repulsive. 426 00:29:37,660 --> 00:29:41,830 Or usually, it's repulsive at all internucleuses. 427 00:29:45,270 --> 00:29:50,930 Then there are higher-energy orbitals that come from 2p-- 428 00:29:50,930 --> 00:29:54,115 2p pi and 2p sigma. 429 00:29:56,790 --> 00:30:00,825 And they give rise to many, many states. 430 00:30:04,020 --> 00:30:10,260 But we know, for hydrogen, when we go from 1s to 2p, 431 00:30:10,260 --> 00:30:13,310 we're at very, very high energy-- 432 00:30:13,310 --> 00:30:16,850 3/4 of a Rydberg. 433 00:30:16,850 --> 00:30:20,690 And that's an energy that's larger than any chemical bond 434 00:30:20,690 --> 00:30:22,940 we know. 435 00:30:22,940 --> 00:30:27,500 So the excited states of H 2 are going 436 00:30:27,500 --> 00:30:32,210 to be a little bit weird because of this tremendous excitation 437 00:30:32,210 --> 00:30:33,080 energy. 438 00:30:33,080 --> 00:30:38,990 And because the excitation energy is so high, 439 00:30:38,990 --> 00:30:42,530 we're probably not prepared to understand 440 00:30:42,530 --> 00:30:44,250 what's going on up there. 441 00:30:44,250 --> 00:30:47,820 And in fact, terrible things happen up at high energy. 442 00:30:47,820 --> 00:30:51,560 We get there are some curves that are double minima. 443 00:30:51,560 --> 00:30:53,250 There are all sorts of things. 444 00:30:53,250 --> 00:30:55,610 And they have to do with-- 445 00:30:55,610 --> 00:31:02,860 you could have H+ H- ion-pair states. 446 00:31:02,860 --> 00:31:06,810 You can have H 2+ plus electron-- 447 00:31:06,810 --> 00:31:12,420 those are Rydberg states as well as the normal valence states. 448 00:31:12,420 --> 00:31:15,420 So normally, we don't want to worry about that. 449 00:31:15,420 --> 00:31:17,220 And there are two places we don't 450 00:31:17,220 --> 00:31:19,260 want to go without preparation. 451 00:31:19,260 --> 00:31:22,750 One is to very high excitation energy. 452 00:31:22,750 --> 00:31:27,080 And the other is to very large internuclear distance 453 00:31:27,080 --> 00:31:29,920 because things can happen. 454 00:31:29,920 --> 00:31:31,960 The weirdnesses are much more important 455 00:31:31,960 --> 00:31:33,010 when there is no core. 456 00:31:43,730 --> 00:31:46,520 So since everything is dependent on R, 457 00:31:46,520 --> 00:31:51,960 we need to know how to estimate the internuclear distance. 458 00:31:51,960 --> 00:31:55,040 And we know, for an atomic orbital, 459 00:31:55,040 --> 00:31:58,910 the ionization energy is hc times the Rydberg 460 00:31:58,910 --> 00:32:00,850 constant over n squared. 461 00:32:11,520 --> 00:32:17,790 And we know that the average r for an orbital 462 00:32:17,790 --> 00:32:25,480 is a0 times n squared over the charge. 463 00:32:25,480 --> 00:32:34,470 And so if we want to relate the size of an orbital 464 00:32:34,470 --> 00:32:36,600 to something that we can measure, 465 00:32:36,600 --> 00:32:41,730 well, we can simply put in this equation for n squared here. 466 00:32:41,730 --> 00:32:47,670 And we get that the average size of an orbital 467 00:32:47,670 --> 00:32:52,500 will be a 0, which is the Bohr radius of the hydrogen atom-- 468 00:32:52,500 --> 00:32:54,360 about half an angstrom-- 469 00:32:54,360 --> 00:32:56,730 times hcR. 470 00:32:56,730 --> 00:33:00,330 The Rydberg is 110,000 wave numbers-- 471 00:33:00,330 --> 00:33:06,310 and over z-- 472 00:33:06,310 --> 00:33:09,340 I'm sorry-- over I nl. 473 00:33:09,340 --> 00:33:13,130 So this ionization energy is measured. 474 00:33:13,130 --> 00:33:19,180 And so if you know the ionization energy, you know r. 475 00:33:19,180 --> 00:33:22,870 If you are talking about a molecular-orbital diagram, 476 00:33:22,870 --> 00:33:26,010 you have r and ionization energy on it. 477 00:33:26,010 --> 00:33:28,270 And so you can see how it would make everything 478 00:33:28,270 --> 00:33:31,450 relate to each other. 479 00:33:31,450 --> 00:33:42,580 OK, let's do an example. 480 00:33:42,580 --> 00:33:45,120 Let's look at the NH molecule. 481 00:33:48,980 --> 00:33:50,830 This is not a stable molecule, but it's 482 00:33:50,830 --> 00:33:54,560 a molecule that is important in some chemical reactions. 483 00:33:54,560 --> 00:33:59,260 It's easy to make an H by having, say, nitrogen-- 484 00:33:59,260 --> 00:34:02,300 N2-- and hydrogen and doing a discharge. 485 00:34:02,300 --> 00:34:05,140 And so in order to be able to understand H, 486 00:34:05,140 --> 00:34:07,975 you start out with all of the orbital energies. 487 00:34:10,699 --> 00:34:14,270 So let's make a little table. 488 00:34:14,270 --> 00:34:17,650 We have H 1s. 489 00:34:17,650 --> 00:34:20,170 And we have its ionization energy. 490 00:34:20,170 --> 00:34:25,350 And its 13.6 electoral volts. 491 00:34:25,350 --> 00:34:33,420 We have H 2s and 2p for hydrogen. They're degenerate. 492 00:34:33,420 --> 00:34:37,980 And that's 3.4 electron volts-- 493 00:34:37,980 --> 00:34:41,429 for nitrogen, 1s. 494 00:34:41,429 --> 00:34:45,239 This is an obscenely large energy. 495 00:34:45,239 --> 00:34:48,820 And so it's greater than 100 electron volts. 496 00:34:48,820 --> 00:34:56,460 We have N 2s, which is 18 electron volts. 497 00:34:56,460 --> 00:35:04,800 We have N 2p, which is 12 electron volts. 498 00:35:04,800 --> 00:35:11,790 And we're going to have, at the end, the ground state of NH, 499 00:35:11,790 --> 00:35:19,680 which turns out to be a triplet sigma, g minus state. 500 00:35:19,680 --> 00:35:23,550 And that's at 13.6 ev. 501 00:35:23,550 --> 00:35:28,680 So one of the things you want to do is ask, OK, all my actors-- 502 00:35:28,680 --> 00:35:30,840 where are they below the ionization energy? 503 00:35:34,070 --> 00:35:40,100 And the next thing you do is you arrange things in energy order. 504 00:35:40,100 --> 00:35:44,150 And so the lowest one is this. 505 00:35:44,150 --> 00:35:47,540 And it's out of the picture altogether. 506 00:35:47,540 --> 00:35:51,650 The next lowest one is this. 507 00:35:51,650 --> 00:35:57,500 And then we have two. 508 00:35:57,500 --> 00:36:03,080 And let's just say this is 3 to 4, and this is 4 to 3. 509 00:36:03,080 --> 00:36:06,290 They're roughly degenerate. 510 00:36:06,290 --> 00:36:09,560 That's what we're looking for because when 511 00:36:09,560 --> 00:36:11,360 the orbitals are degenerate, they 512 00:36:11,360 --> 00:36:13,350 have about the same energy. 513 00:36:13,350 --> 00:36:15,320 And we know from perturbation theory, 514 00:36:15,320 --> 00:36:18,650 if you want to have a mixture of two different things, 515 00:36:18,650 --> 00:36:22,960 you want the energy to not measure to be small. 516 00:36:22,960 --> 00:36:26,710 So all the actors-- the simple stuff-- 517 00:36:26,710 --> 00:36:28,440 we know what their energies are. 518 00:36:28,440 --> 00:36:31,920 We know that if we have two things that have the same, 519 00:36:31,920 --> 00:36:36,170 or nearly the same, energy, they're going to be mixed. 520 00:36:36,170 --> 00:36:42,190 And so now, with this sort of an annotated table-- 521 00:36:42,190 --> 00:36:46,380 the last one is this guy-- 522 00:36:46,380 --> 00:36:49,710 we can write down the low-lying configurations. 523 00:36:49,710 --> 00:36:51,870 And we can say something about the nature 524 00:36:51,870 --> 00:36:54,835 of the molecular-orbital diagram. 525 00:36:59,080 --> 00:37:08,960 [INAUDIBLE] 526 00:37:08,960 --> 00:37:12,440 But I stress the most important thing 527 00:37:12,440 --> 00:37:17,300 is something that's easily measured for atoms because what 528 00:37:17,300 --> 00:37:19,790 we want to do is build on what we think 529 00:37:19,790 --> 00:37:25,220 we know for the periodic table to have some sort of insight 530 00:37:25,220 --> 00:37:27,560 into molecular-periodic table, although it's 531 00:37:27,560 --> 00:37:31,465 a multidimensional periodic table. 532 00:37:31,465 --> 00:37:33,850 I know that's probably a stupid thing to say. 533 00:37:33,850 --> 00:37:41,430 OK, so here's the molecular orbital diagram for NH. 534 00:37:41,430 --> 00:37:46,590 So over on this side, we have H+. 535 00:37:46,590 --> 00:37:48,540 Over this side, we have N+. 536 00:37:51,550 --> 00:37:53,940 And we put the orbitals on-- 537 00:37:53,940 --> 00:38:05,316 OK, so here is minus 3.4, which is the hydrogen 2s or 2p. 538 00:38:05,316 --> 00:38:10,390 And down here, we have the hydrogen 1s. 539 00:38:10,390 --> 00:38:16,740 And that's at minus 13.6. 540 00:38:16,740 --> 00:38:27,200 And over here, we have the nitrogen 2p. 541 00:38:31,010 --> 00:38:32,300 And that's at minus 12. 542 00:38:36,202 --> 00:38:36,910 So what do we do? 543 00:38:36,910 --> 00:38:41,430 And then down much farther below way down here, 544 00:38:41,430 --> 00:38:45,390 we have the nitrogen 1s. 545 00:38:45,390 --> 00:38:46,440 The nitrogen 2s. 546 00:38:49,260 --> 00:38:50,940 OK, how many electrons do we have? 547 00:38:53,900 --> 00:38:55,722 Well, before I do that-- 548 00:38:55,722 --> 00:38:57,305 so how many electrons are on nitrogen? 549 00:39:05,890 --> 00:39:07,780 I mean, if you don't do that, you don't 550 00:39:07,780 --> 00:39:11,510 know the periodic table at all. 551 00:39:11,510 --> 00:39:12,010 How many? 552 00:39:15,030 --> 00:39:16,566 Did everybody see that? 553 00:39:16,566 --> 00:39:17,690 Well, then I can't ask you. 554 00:39:17,690 --> 00:39:22,920 Yes, it's seven electrons and plus 1 for the hydrogen, right? 555 00:39:22,920 --> 00:39:25,205 So we're dealing with eight electrons. 556 00:39:25,205 --> 00:39:28,530 And so we're going to put eight electrons into orbitals. 557 00:39:28,530 --> 00:39:32,970 And now on this diagram, basically, I 558 00:39:32,970 --> 00:39:40,600 have two orbitals that are roughly at the same energy. 559 00:39:40,600 --> 00:39:43,730 And so we know that all the action is going to be that. 560 00:39:43,730 --> 00:39:53,980 And so we can draw something like this and something 561 00:39:53,980 --> 00:39:57,320 like that. 562 00:39:57,320 --> 00:40:03,280 And now, we use our perturbation ideas. 563 00:40:03,280 --> 00:40:06,980 And this guy is higher in energy. 564 00:40:06,980 --> 00:40:11,720 And so it's closer to this other orbital. 565 00:40:11,720 --> 00:40:14,550 So this one is dominantly nitrogen, 566 00:40:14,550 --> 00:40:20,420 and this one is dominantly hydrogen. Make this dotted. 567 00:40:20,420 --> 00:40:22,700 OK, so we have two orbitals. 568 00:40:22,700 --> 00:40:25,310 This one is polarized towards nitrogen. 569 00:40:25,310 --> 00:40:32,390 And this one is polarized towards hydrogen. 570 00:40:32,390 --> 00:40:40,820 And then there is another orbital here. 571 00:40:40,820 --> 00:40:43,010 This is 2p pi. 572 00:40:47,960 --> 00:40:50,320 Well, what about 2p pi? 573 00:40:50,320 --> 00:40:52,200 There is no pi orbital here. 574 00:40:52,200 --> 00:40:55,840 The lowest pi orbital is way up here. 575 00:40:55,840 --> 00:40:59,170 So this guy is nonbinding. 576 00:40:59,170 --> 00:41:02,050 So we have a bonding orbital, a nonbonding orbital, 577 00:41:02,050 --> 00:41:04,330 and an antibonding orbital. 578 00:41:04,330 --> 00:41:06,670 And then up here, we have something 579 00:41:06,670 --> 00:41:09,290 that's so high energy, we might call it a Rydberg 580 00:41:09,290 --> 00:41:14,260 orbital or just something else. 581 00:41:14,260 --> 00:41:16,680 OK, so now we start putting electrons 582 00:41:16,680 --> 00:41:18,855 into orbitals in orbital-order energy. 583 00:41:21,810 --> 00:41:30,440 So we have sigma nitrogen 1s squared. 584 00:41:30,440 --> 00:41:31,585 That's the lowest one. 585 00:41:31,585 --> 00:41:32,460 That's way down here. 586 00:41:32,460 --> 00:41:33,300 I erased it. 587 00:41:33,300 --> 00:41:36,420 It's at minus 100 electron volts. 588 00:41:36,420 --> 00:41:41,130 And then we have nitrogen 2s. 589 00:41:41,130 --> 00:41:43,140 That's pretty far down too. 590 00:41:43,140 --> 00:41:45,780 And so these two-- 591 00:41:45,780 --> 00:41:47,100 we don't have any doubt. 592 00:41:47,100 --> 00:41:52,260 The electrons go into these sigma orbitals. 593 00:41:52,260 --> 00:41:54,390 And there's nobody nearby, and so these are 594 00:41:54,390 --> 00:41:58,080 localized on the nitrogen atom. 595 00:41:58,080 --> 00:41:59,970 We've got four more electrons to deal with. 596 00:42:03,270 --> 00:42:09,550 We have sigma NH sigma NH. 597 00:42:12,410 --> 00:42:14,530 Well, we'll call this sigma. 598 00:42:14,530 --> 00:42:17,360 We got two electrons there. 599 00:42:17,360 --> 00:42:19,040 And then we have-- 600 00:42:26,100 --> 00:42:32,580 so we got two electrons, two electrons in core orbitals. 601 00:42:32,580 --> 00:42:34,602 We have a bond. 602 00:42:34,602 --> 00:42:35,560 Then we have a nonbond. 603 00:42:38,970 --> 00:42:39,750 Yes. 604 00:42:39,750 --> 00:42:41,265 AUDIENCE: How far apart in energy 605 00:42:41,265 --> 00:42:42,965 do the orbitals have to be such that you 606 00:42:42,965 --> 00:42:45,890 consider them non-attracted? 607 00:42:45,890 --> 00:42:48,940 PROFESSOR: It's really up to you. 608 00:42:48,940 --> 00:42:50,070 I mean, at the lowest-- 609 00:42:56,430 --> 00:42:58,570 it's a question of how much detail 610 00:42:58,570 --> 00:43:01,640 you want to be honest about. 611 00:43:01,640 --> 00:43:03,490 There's always interaction. 612 00:43:03,490 --> 00:43:07,030 But the important thing is if the orbital energy 613 00:43:07,030 --> 00:43:16,090 is really low or stable, the orbitals are really compact. 614 00:43:16,090 --> 00:43:19,450 And so at equally internuclear distance, 615 00:43:19,450 --> 00:43:22,270 which is determined by a bonding orbital, 616 00:43:22,270 --> 00:43:24,120 they're just out of play. 617 00:43:24,120 --> 00:43:26,905 So it's not 0. 618 00:43:31,550 --> 00:43:37,800 But you really care about big effects-- 619 00:43:37,800 --> 00:43:40,250 things that are worth, roughly, the energy 620 00:43:40,250 --> 00:43:43,780 of a tenth of a bond. 621 00:43:43,780 --> 00:43:46,450 As you get more and more better at this, 622 00:43:46,450 --> 00:43:50,410 you could say, well, we'll worry about bonding effects that 623 00:43:50,410 --> 00:43:54,100 are worth less than a tenth of a bond or a typical bond. 624 00:43:54,100 --> 00:44:00,340 But you want to organize things in the order of how important 625 00:44:00,340 --> 00:44:03,280 they are, how careful you have to be, 626 00:44:03,280 --> 00:44:08,330 how many subtle effects you have to take into account. 627 00:44:08,330 --> 00:44:12,020 And so to avoid craziness, you want to say, 628 00:44:12,020 --> 00:44:15,070 I'm not going to worry about this because it's too small, 629 00:44:15,070 --> 00:44:19,170 or because their energy difference is too large. 630 00:44:19,170 --> 00:44:22,240 And it's very important to be able to shed 631 00:44:22,240 --> 00:44:28,180 unnecessary factors until you're ready to deal with them. 632 00:44:28,180 --> 00:44:31,120 Or maybe you never will. 633 00:44:31,120 --> 00:44:36,070 OK, so that was a really great question. 634 00:44:36,070 --> 00:44:41,250 All right, so we have the list of-- 635 00:44:41,250 --> 00:44:42,205 where did I put it? 636 00:44:42,205 --> 00:44:43,660 Oh, here it is. 637 00:44:43,660 --> 00:44:48,340 OK, so these are three filled orbitals. 638 00:44:48,340 --> 00:44:51,070 They give rise to a singlet sigma. 639 00:44:51,070 --> 00:44:54,010 But we have two electrons in a pi orbital. 640 00:44:54,010 --> 00:44:58,810 And if you care about this stuff, 641 00:44:58,810 --> 00:45:02,905 you'll learn how to figure out what two electrons in a pi 642 00:45:02,905 --> 00:45:04,060 orbital will give you. 643 00:45:04,060 --> 00:45:09,040 And they give you triplet sigma minus g-- 644 00:45:09,040 --> 00:45:12,520 I'm sorry-- no g because it's a heteronuclear-- triplet 645 00:45:12,520 --> 00:45:20,070 sigma minus singlet delta and singlet sigma plus. 646 00:45:20,070 --> 00:45:30,940 And this is actually much easier than for inorganic chemistry 647 00:45:30,940 --> 00:45:37,510 or for just figuring out the LS states of an atom 648 00:45:37,510 --> 00:45:41,380 because basically, we have sigma, pi, delta. 649 00:45:41,380 --> 00:45:43,450 And you can figure out, well, pi squared 650 00:45:43,450 --> 00:45:44,740 is going to give you this. 651 00:45:44,740 --> 00:45:49,400 Pi cubed is going to give you this again, except this-- 652 00:45:49,400 --> 00:45:51,030 it's going to give you this again. 653 00:45:51,030 --> 00:45:53,830 So you there's a finite number of things 654 00:45:53,830 --> 00:45:57,120 that you don't have to work out the energy levels 655 00:45:57,120 --> 00:45:58,900 by an elaborate procedure. 656 00:45:58,900 --> 00:46:00,430 And so you can sort of memorize them 657 00:46:00,430 --> 00:46:02,320 because there's very few that you actually 658 00:46:02,320 --> 00:46:03,950 are going to deal with. 659 00:46:03,950 --> 00:46:10,510 OK, so now Hund's rules apply to molecules. 660 00:46:10,510 --> 00:46:11,955 So what's the ground state of NH? 661 00:46:17,180 --> 00:46:18,305 Triplet sigma minus. 662 00:46:22,230 --> 00:46:25,372 So that's nice because you're going to be doing spectroscopy, 663 00:46:25,372 --> 00:46:27,330 and you're going to be starting with a molecule 664 00:46:27,330 --> 00:46:35,990 in a triplet state and you've got a pi bond. 665 00:46:35,990 --> 00:46:38,850 And so you know a typical internuclear distance for a pi 666 00:46:38,850 --> 00:46:41,120 bond, or you can figure it out. 667 00:46:41,120 --> 00:46:44,045 And we're only going to be seeing triplet states, 668 00:46:44,045 --> 00:46:45,420 and we're only going to be seeing 669 00:46:45,420 --> 00:46:49,770 triplet-sigma-minus states or triplet-pi states because 670 00:46:49,770 --> 00:46:50,750 of the selection rules. 671 00:46:50,750 --> 00:46:51,985 Now I'm telling you this. 672 00:46:51,985 --> 00:46:52,860 You could learn that. 673 00:46:52,860 --> 00:46:54,300 These are extra things. 674 00:46:54,300 --> 00:46:56,610 You've got to know the most-important stuff. 675 00:47:01,080 --> 00:47:05,340 So we have an intentionally naive 676 00:47:05,340 --> 00:47:08,440 primitive-molecular orbital diagram. 677 00:47:08,440 --> 00:47:15,350 OK, now-- my chance to say terrible things 678 00:47:15,350 --> 00:47:16,250 about textbooks. 679 00:47:22,620 --> 00:47:27,420 Every one of you has seen the molecular-orbital diagram 680 00:47:27,420 --> 00:47:32,100 for this mystical A2 molecule. 681 00:47:32,100 --> 00:47:33,690 Everybody's seen it. 682 00:47:33,690 --> 00:47:41,010 And it's presented either dishonestly or semihonestly 683 00:47:41,010 --> 00:47:43,150 with an asterisk. 684 00:47:43,150 --> 00:47:45,550 So let's just understand this. 685 00:47:45,550 --> 00:47:48,870 Again we have the ionization energy. 686 00:47:48,870 --> 00:47:52,770 We have A+ and A+. 687 00:47:52,770 --> 00:47:56,670 And we plot the energies. 688 00:47:56,670 --> 00:48:00,720 And naively, we say, OK, here's 2s-- 689 00:48:00,720 --> 00:48:02,910 same energy 2s. 690 00:48:02,910 --> 00:48:04,490 And here is 2p-- 691 00:48:04,490 --> 00:48:12,665 2p-- for atoms other than hydrogen, 2p is above 2s. 692 00:48:15,950 --> 00:48:20,020 And so we know, well, we can draw some kind 693 00:48:20,020 --> 00:48:23,405 of a diagram like this. 694 00:48:26,860 --> 00:48:35,040 So these are the sigma star 2s and sigma 2s. 695 00:48:35,040 --> 00:48:36,660 No problem. 696 00:48:36,660 --> 00:48:43,052 And then we draw something like this for the p orbitals. 697 00:48:43,052 --> 00:48:44,385 And we have something like this. 698 00:48:48,310 --> 00:48:49,210 That makes sense. 699 00:48:49,210 --> 00:48:53,340 Now, the lowest-energy orbital has 700 00:48:53,340 --> 00:48:55,830 to be the more directional p orbital. 701 00:48:55,830 --> 00:48:58,890 So this is going to be the p-sigma orbital. 702 00:48:58,890 --> 00:49:10,345 And so we'll have a sigma 2p, a pi 2p, a pi star 2p, 703 00:49:10,345 --> 00:49:12,440 and a sigma star 2p. 704 00:49:16,130 --> 00:49:19,530 That's what you get using common sense. 705 00:49:19,530 --> 00:49:22,560 And it's wrong. 706 00:49:22,560 --> 00:49:25,310 And most of the textbooks say, well, 707 00:49:25,310 --> 00:49:28,140 if we have to correct this-- it turns out 708 00:49:28,140 --> 00:49:32,730 that the energy order of these two orbitals 709 00:49:32,730 --> 00:49:35,910 is frequently reversed. 710 00:49:35,910 --> 00:49:39,000 And most textbooks don't tell you why. 711 00:49:39,000 --> 00:49:42,060 They just say, well, this is what you would expect. 712 00:49:42,060 --> 00:49:46,020 And the canonical molecular-orbital diagram 713 00:49:46,020 --> 00:49:50,430 for homonuclear diatomics has the pi below sigma 714 00:49:50,430 --> 00:49:57,910 except for oxygen and fluorine. 715 00:49:57,910 --> 00:50:01,000 Well, that's not something that builds confidence. 716 00:50:06,040 --> 00:50:09,330 And that's something you learned in 5.111, 5.112. 717 00:50:09,330 --> 00:50:18,370 The energy gap between 2s and 2p depends on shielding. 718 00:50:18,370 --> 00:50:29,250 And so as you start out with lithium, 2s and 2p are-- 719 00:50:29,250 --> 00:50:32,060 lithium is more like hydrogen. So 2s and 2p 720 00:50:32,060 --> 00:50:35,260 are nearly degenerate. 721 00:50:35,260 --> 00:50:42,090 When you get to fluorine, the energy gap between 2s and 2p 722 00:50:42,090 --> 00:50:51,900 is enormous because 2s shields 2p. 723 00:50:51,900 --> 00:50:55,200 And as a result, the-- 724 00:50:59,500 --> 00:51:05,680 well, the shielding arguments explain this energy gap. 725 00:51:05,680 --> 00:51:09,370 And it gets large because 2s is unshielded, 726 00:51:09,370 --> 00:51:16,210 and it really gets very stable, whereas 2p is increasingly 727 00:51:16,210 --> 00:51:23,440 affected by the shielding by s, and as a charge, increases 728 00:51:23,440 --> 00:51:24,430 the orbital energy. 729 00:51:24,430 --> 00:51:27,920 OK, I thought I had a better explanation for it. 730 00:51:27,920 --> 00:51:28,950 But I do have one. 731 00:51:28,950 --> 00:51:31,120 It's just not handy. 732 00:51:31,120 --> 00:51:32,420 And so what happens? 733 00:51:32,420 --> 00:51:39,580 Well, what happens, as you go from lithium, where 734 00:51:39,580 --> 00:51:42,610 this and this are close together, 735 00:51:42,610 --> 00:51:47,320 you have interaction between orbitals of the same symmetry. 736 00:51:54,660 --> 00:51:58,350 Now the Hamiltonian is totally symmetric. 737 00:51:58,350 --> 00:52:00,750 Oh, my goodness. 738 00:52:00,750 --> 00:52:05,020 So because the Hamiltonian is totally symmetric, 739 00:52:05,020 --> 00:52:08,070 you can have interactions between orbitals 740 00:52:08,070 --> 00:52:12,000 of the same symmetry. 741 00:52:12,000 --> 00:52:18,630 And as a result, this guy gets pushed up. 742 00:52:18,630 --> 00:52:21,270 And this guy gets pushed up. 743 00:52:21,270 --> 00:52:23,770 And this guy gets pushed down. 744 00:52:23,770 --> 00:52:26,290 But these are so far out of the picture you don't care. 745 00:52:26,290 --> 00:52:29,070 So it's really just the relative energies of these two. 746 00:52:29,070 --> 00:52:33,070 And it's all due to the relative energies of 2s and 2p. 747 00:52:33,070 --> 00:52:35,160 And then you get the correct energy-level diagram. 748 00:52:35,160 --> 00:52:39,062 And you say, well, maybe as we went across the periodic table, 749 00:52:39,062 --> 00:52:40,020 there will be a switch. 750 00:52:40,020 --> 00:52:41,400 And there is a switch. 751 00:52:41,400 --> 00:52:47,550 And since I've lived most of my early life in CO 752 00:52:47,550 --> 00:52:50,490 and N 2 and O 2. 753 00:52:50,490 --> 00:52:52,000 I've encountered that. 754 00:52:52,000 --> 00:52:53,460 And I thought about it a lot. 755 00:52:53,460 --> 00:52:55,770 OK so that's pretty much all I'm going 756 00:52:55,770 --> 00:52:58,800 to say about the molecular-orbital diagram. 757 00:52:58,800 --> 00:53:03,660 I think next lecture is on Huckel theory. 758 00:53:03,660 --> 00:53:05,210 OK.