1 00:00:00,040 --> 00:00:01,480 Hello and welcome back to The Deep Dive. 2 00:00:01,480 --> 00:00:04,640 I'm your host and today we are tackling a subject that I think 3 00:00:06,080 --> 00:00:09,000 really rewires the way you look at the physical world. 4 00:00:09,120 --> 00:00:11,480 It's a fundamental shift in perspective, isn't it? 5 00:00:11,520 --> 00:00:13,720 It really is. You know, we spend so much time 6 00:00:13,720 --> 00:00:17,360 in school, in chemistry class, looking at these diagrams of 7 00:00:17,360 --> 00:00:19,720 molecules. You open a textbook and what do 8 00:00:19,720 --> 00:00:21,920 you see? These ball and stick models. 9 00:00:22,040 --> 00:00:25,360 Right, they're static. They're rigid, frozen in place. 10 00:00:25,360 --> 00:00:26,480 Exactly. They look like little 11 00:00:26,480 --> 00:00:28,440 architectural statues sitting in a void. 12 00:00:28,520 --> 00:00:31,200 Very clean, very peaceful. It's what I like to call the 13 00:00:31,200 --> 00:00:34,440 museum view of chemistry. Everything is perfectly still, 14 00:00:34,440 --> 00:00:37,320 perfectly posed under glass. A museum view. 15 00:00:37,320 --> 00:00:39,640 I love that. But the source material we're 16 00:00:39,640 --> 00:00:43,760 covering today, this is Lecture 4 from Professor Pedro Camargos 17 00:00:43,760 --> 00:00:46,720 course on structural methods in inorganic chemistry. 18 00:00:47,080 --> 00:00:49,360 It basically takes a sledgehammer to that entire 19 00:00:49,360 --> 00:00:50,480 museum. It really does. 20 00:00:50,480 --> 00:00:52,600 It shatters the illusion of stillness. 21 00:00:52,600 --> 00:00:54,880 It tells us that if you could actually shrink yourself down to 22 00:00:54,880 --> 00:00:57,760 the nanoscale and, you know, look at the molecule, you would 23 00:00:57,760 --> 00:00:59,960 not see a statue, you'd see mosh pit. 24 00:01:01,080 --> 00:01:04,360 A mosh pit is a very evocative way to put it, but yes, that is 25 00:01:04,360 --> 00:01:06,800 honestly closer to the truth than the statue. 26 00:01:06,800 --> 00:01:09,600 It feels that way. The atomic world is just defined 27 00:01:09,600 --> 00:01:12,880 by constant, rhythmic, almost chaotic motion. 28 00:01:13,320 --> 00:01:16,960 Bonds are stretching, angles are bending, entire molecules are 29 00:01:17,080 --> 00:01:19,240 tumbling and twisting through space. 30 00:01:19,520 --> 00:01:22,800 It is a frantic dance that never, ever stops. 31 00:01:22,800 --> 00:01:24,880 And that is the hook for today's deep dive. 32 00:01:25,160 --> 00:01:28,040 We're exploring what we're calling the mathematics of the 33 00:01:28,040 --> 00:01:29,120 wiggle. The Wiggle. 34 00:01:29,120 --> 00:01:31,520 I like it. Specifically, how do scientists 35 00:01:31,520 --> 00:01:35,400 actually see this dance? Because let's be real, we don't 36 00:01:35,400 --> 00:01:38,640 have a microscope powerful enough to just record a video of 37 00:01:38,640 --> 00:01:40,400 a silicon atom doing the cha cha. 38 00:01:40,400 --> 00:01:43,200 No, we certainly do not. The wavelength of visible light 39 00:01:43,200 --> 00:01:47,600 is far, far too large to resolve individual atomic bonds moving 40 00:01:47,600 --> 00:01:48,200 in that way. So. 41 00:01:48,200 --> 00:01:49,600 You can't watch the dance directly. 42 00:01:49,600 --> 00:01:51,000 You can't. So we have to act like 43 00:01:51,000 --> 00:01:53,120 detectives. We have to listen to the music, 44 00:01:53,120 --> 00:01:55,040 so to speak. We measure the energy of those 45 00:01:55,040 --> 00:01:58,520 vibrations and this entire field, this science, is called 46 00:01:58,520 --> 00:02:01,640 vibrational spectroscopy. Which sounds incredibly 47 00:02:01,640 --> 00:02:05,600 technical, and I won't lie to you, our listener, We're going 48 00:02:05,600 --> 00:02:07,640 to get into some pretty heavy weeds today. 49 00:02:07,640 --> 00:02:10,800 We've got quantum mechanics, we have matrix algebra, and we have 50 00:02:10,800 --> 00:02:14,760 something called group theory. The trifecta of scary sounding 51 00:02:14,760 --> 00:02:15,680 topics. Right. 52 00:02:15,920 --> 00:02:18,560 But the mission here is to break this down so that by the end of 53 00:02:18,560 --> 00:02:21,800 the hour, you can look at a spectrum, you know, those 54 00:02:21,840 --> 00:02:25,320 squiggly lines on a graph, and understand that it's not just 55 00:02:25,320 --> 00:02:29,600 noise, it's actually a map of symmetry and quantum physics. 56 00:02:29,760 --> 00:02:32,360 That's the goal, and it's a beautiful journey because it 57 00:02:32,360 --> 00:02:33,840 connects two very different worlds. 58 00:02:34,320 --> 00:02:38,600 We start with very simple, intuitive physics, like weights 59 00:02:38,600 --> 00:02:40,920 on a spring, something you can picture in your garage. 60 00:02:40,920 --> 00:02:43,360 OK, very tangible. And we end up with some pretty 61 00:02:43,360 --> 00:02:46,200 abstract mathematics things called character tables and 62 00:02:46,440 --> 00:02:50,480 irreducible representations to predict exactly how light is 63 00:02:50,480 --> 00:02:53,400 going to interact with matter. So let's lay out the road map 64 00:02:53,400 --> 00:02:55,000 for everyone. First, we're going to talk about 65 00:02:55,000 --> 00:02:58,320 the bond itself. Why on earth do we treat atoms 66 00:02:58,320 --> 00:03:01,440 like balls on a spring? What's the physics there? 67 00:03:01,440 --> 00:03:03,960 The foundational analogy. Then we're going to look at what 68 00:03:03,960 --> 00:03:07,040 the source calls the great divide in this field. 69 00:03:07,600 --> 00:03:10,360 This is the 2 main ways we can detect these vibrations. 70 00:03:10,960 --> 00:03:15,480 Infrared spectroscopy or IR and ramen spectroscopy. 71 00:03:15,800 --> 00:03:18,240 The two detectives, they're working the same case, but 72 00:03:18,240 --> 00:03:20,520 they're looking for completely different clues. 73 00:03:20,640 --> 00:03:22,720 I like that. And then we get to the main 74 00:03:22,720 --> 00:03:24,800 event. We're going to walk through, 75 00:03:24,880 --> 00:03:29,880 step by step, the actual algorithm that chemists use to, 76 00:03:29,960 --> 00:03:32,040 well, predict the future. It really is a form of 77 00:03:32,040 --> 00:03:33,720 prediction. We're going to take a specific 78 00:03:33,720 --> 00:03:38,680 molecule, silicon dichloride dihydride, or CCL 2H2, and use 79 00:03:38,680 --> 00:03:41,680 symmetry to calculate exactly what its spectrum should look 80 00:03:41,680 --> 00:03:43,320 like before we even do an experiment. 81 00:03:43,360 --> 00:03:45,800 It's a classic example. It's complex enough to be 82 00:03:45,800 --> 00:03:48,400 interesting, but simple enough that we can actually do the math 83 00:03:48,400 --> 00:03:50,120 together if we just take it slow. 84 00:03:50,280 --> 00:03:52,320 And finally, we'll talk about why this all matters. 85 00:03:52,320 --> 00:03:54,240 I mean, why do we care about the wiggle in the 1st place? 86 00:03:54,800 --> 00:03:56,440 So let's dive in. Section 1. 87 00:03:56,640 --> 00:03:59,480 The bond is a spring. The lecture notes start right 88 00:03:59,480 --> 00:04:02,240 here with this fundamental analogy, the harmonic 89 00:04:02,440 --> 00:04:04,960 oscillator. This is the bedrock of the 90 00:04:04,960 --> 00:04:07,400 entire field of vibrational spectroscopy. 91 00:04:08,320 --> 00:04:12,520 To even begin to understand a chemical bonds vibration, you 92 00:04:12,520 --> 00:04:14,800 have to strip away the complexity for a moment. 93 00:04:15,280 --> 00:04:17,480 Forget about electrons and orbitals and just think about 94 00:04:17,480 --> 00:04:19,120 mechanics. OK, pure physics. 95 00:04:19,160 --> 00:04:24,120 Imagine 2 heavy balls, like 2 steel ball bearings, and they're 96 00:04:24,120 --> 00:04:27,560 connected by a metal spring. OK, I'm visualizing it. 2 steel 97 00:04:27,560 --> 00:04:30,560 balls, 1 spring connecting them. Now if you pull them apart and 98 00:04:30,560 --> 00:04:32,760 just let go, they don't just snap back to the middle and 99 00:04:32,760 --> 00:04:34,320 stop. Right now, they keep going. 100 00:04:34,920 --> 00:04:37,360 They have momentum, they overshoot the center, compress 101 00:04:37,360 --> 00:04:38,960 the spring and they get pushed back out. 102 00:04:39,040 --> 00:04:41,360 And they overshoot again. They oscillate back and forth, 103 00:04:41,360 --> 00:04:43,880 back and forth. And they oscillate at a specific 104 00:04:43,880 --> 00:04:46,080 speed, don't they? A specific frequency? 105 00:04:46,560 --> 00:04:48,240 It wouldn't be random. Precisely. 106 00:04:48,400 --> 00:04:50,800 And that frequency is determined by just two things. 107 00:04:50,800 --> 00:04:54,560 First, the mass of the balls. Heavier atoms are more sluggish, 108 00:04:54,560 --> 00:04:57,400 they move more slowly. It's just harder to reverse 109 00:04:57,400 --> 00:04:58,920 their direction. Mass makes sense. 110 00:04:59,080 --> 00:05:01,520 What's the second thing? The stiffness of the spring. 111 00:05:01,680 --> 00:05:04,560 In chemistry, we call this the force constant of the bond. 112 00:05:04,720 --> 00:05:07,600 It's just a measure of how strong that bond is. 113 00:05:07,800 --> 00:05:12,080 So let me see if I've got this. A single bond between two atoms 114 00:05:12,320 --> 00:05:16,840 would be like a loose kind of slinky spring, and a triple bond 115 00:05:16,840 --> 00:05:20,440 would be like a really stiff suspension coil from a car. 116 00:05:20,720 --> 00:05:24,000 That is the perfect analogy. A triple bond between 2 carbon 117 00:05:24,000 --> 00:05:27,400 atoms is incredibly stiff. It snaps back and forth very, 118 00:05:27,400 --> 00:05:29,640 very quickly. That's a high frequency 119 00:05:29,640 --> 00:05:31,960 vibration. Well, a single bond is looser, 120 00:05:32,160 --> 00:05:34,000 so it vibrates at a lower frequency. 121 00:05:34,000 --> 00:05:36,400 Exactly. This all makes perfect sense in 122 00:05:36,400 --> 00:05:39,520 a classical physics way. If I have a spring in my garage, 123 00:05:39,640 --> 00:05:42,480 I can understand this intuitively, but the lecture 124 00:05:42,480 --> 00:05:46,400 notes immediately pivot to the quantum reality and this is 125 00:05:46,400 --> 00:05:49,280 where things start to get weird. The garage analogy breaks down. 126 00:05:49,480 --> 00:05:52,680 It has to because in your garage you can pull that spring any 127 00:05:52,680 --> 00:05:54,920 distance you want. You can give it any amount of 128 00:05:54,920 --> 00:06:00,000 energy in a continuous spectrum. You can pull it 1mm or 1.1mm or 129 00:06:00,000 --> 00:06:01,920 1.11. Right, it's analog. 130 00:06:02,000 --> 00:06:04,520 But a molecule is digital. It is quantized. 131 00:06:04,520 --> 00:06:06,600 Meaning you can't vibrate at just any energy level. 132 00:06:06,840 --> 00:06:09,560 It can't. The molecule has specific 133 00:06:09,560 --> 00:06:12,600 discrete energy states that it is allowed to inhabit. 134 00:06:12,800 --> 00:06:15,040 We call these vibrational levels, and we label them with a 135 00:06:15,040 --> 00:06:18,160 quantum number V. So you have the ground state 136 00:06:18,160 --> 00:06:21,640 which is V-0, the first excited state V1, the 2nd V2 and so on 137 00:06:21,640 --> 00:06:22,760 up. So it's a ladder. 138 00:06:22,840 --> 00:06:25,880 You can stand on rung one, or you can stand on rung 2, but you 139 00:06:25,880 --> 00:06:28,160 cannot hover in the space between the rungs. 140 00:06:28,360 --> 00:06:29,320 That's it. You cannot. 141 00:06:29,320 --> 00:06:32,320 And when a molecule absorbs energy to vibrate more 142 00:06:32,320 --> 00:06:35,360 vigorously, it doesn't ramp up at speed gradually. 143 00:06:35,360 --> 00:06:38,560 It essentially teleports from, say, rung 0 to to run one. 144 00:06:38,840 --> 00:06:40,640 It's an instantaneous quantum jump. 145 00:06:40,880 --> 00:06:44,160 Now there is an equation in the source material for the energy 146 00:06:44,160 --> 00:06:48,000 of these rungs. It says EV equals H Omega times 147 00:06:48,240 --> 00:06:51,640 and then in ( v + 12. I want to stop and look at that 148 00:06:51,640 --> 00:06:53,760 part in the parentheses, the V + 1/2. 149 00:06:53,800 --> 00:06:56,400 It's a very specific and a very, very important term. 150 00:06:56,400 --> 00:06:58,960 Because if I'm at the absolute bottom of the ladder, the lowest 151 00:06:58,960 --> 00:07:02,680 possible energy, the ground state, where V is 0, the energy 152 00:07:02,680 --> 00:07:04,080 isn't 0. No, it's not. 153 00:07:04,120 --> 00:07:06,760 The math says the energy is 1/2 times any Omega. 154 00:07:06,880 --> 00:07:09,240 What is that? You've just found the zero point 155 00:07:09,240 --> 00:07:12,320 energy. This is one of the most profound 156 00:07:12,360 --> 00:07:16,200 and frankly bizarre concepts in all of quantum mechanics. 157 00:07:16,200 --> 00:07:18,400 So a molecule can never stop moving. 158 00:07:18,400 --> 00:07:21,480 It can never ever have 0 energy. It can never sit perfectly 159 00:07:21,480 --> 00:07:23,880 still. Not even at absolute 0 if we 160 00:07:23,880 --> 00:07:30,080 could freeze the entire universe down to 200 two 173.15°C. 161 00:07:30,240 --> 00:07:33,600 Even then, if you could somehow cool a single molecule to 162 00:07:33,600 --> 00:07:37,520 absolute zero, it would still be vibrating with that 1/2 unit of 163 00:07:37,520 --> 00:07:39,320 energy. It would still be wiggling. 164 00:07:39,400 --> 00:07:42,160 Why is nature just fundamentally jittery? 165 00:07:42,480 --> 00:07:44,720 It's because of the Heisenberg Uncertainty principle. 166 00:07:45,240 --> 00:07:48,320 Quantum mechanics simply forbids us from knowing both the exact 167 00:07:48,320 --> 00:07:51,600 position and the exact momentum of a particle at the same time. 168 00:07:51,920 --> 00:07:55,200 If a molecule stopped vibrating completely, what would we know? 169 00:07:55,200 --> 00:07:57,720 We know exactly where the atoms were, their position, and we 170 00:07:57,720 --> 00:08:00,080 know their speed. Their momentum was 0. 171 00:08:00,080 --> 00:08:02,000 We would know both things perfectly. 172 00:08:02,200 --> 00:08:04,440 Know too much? We would violate the fundamental 173 00:08:04,440 --> 00:08:07,800 laws of physics. So to keep the universe legal, 174 00:08:07,800 --> 00:08:10,840 nature forces everything to keep moving just a little bit. 175 00:08:11,200 --> 00:08:14,360 It's the zero point wiggle. That is slightly unsettling, the 176 00:08:14,360 --> 00:08:17,560 idea that this chair I'm sitting on, the air I'm breathing, the 177 00:08:17,560 --> 00:08:21,440 very neurons in my brain, nothing is ever truly at rest. 178 00:08:22,320 --> 00:08:25,360 It's all shimmering. It is a shimmering universe. 179 00:08:25,760 --> 00:08:28,800 Now we have to complicate the spring model just a little bit 180 00:08:28,800 --> 00:08:31,080 more. The harmonic oscillator. 181 00:08:31,080 --> 00:08:34,240 That perfect spring is what we call a useful lie A. 182 00:08:34,520 --> 00:08:35,960 Useful lie. I like that. 183 00:08:35,960 --> 00:08:38,919 It works really well for small vibrations right at the bottom 184 00:08:38,919 --> 00:08:43,039 of that energy ladder, but real bonds aren't perfect springs. 185 00:08:43,159 --> 00:08:46,440 The lecture calls this the reality check or the an harmonic 186 00:08:46,440 --> 00:08:47,480 oscillator. Right. 187 00:08:47,720 --> 00:08:49,280 Think about your garage spring again. 188 00:08:49,480 --> 00:08:52,560 If you hook one end to a solid wall and the other into a truck, 189 00:08:52,600 --> 00:08:56,040 and you start to drive the truck away, what happens? 190 00:08:56,120 --> 00:08:58,600 Oh, the spring stretches, then it deforms permanently and 191 00:08:58,600 --> 00:09:00,720 eventually it just snaps. Exactly. 192 00:09:00,840 --> 00:09:03,360 A perfect harmonic oscillator would never snap. 193 00:09:03,360 --> 00:09:06,520 Its energy just goes up and up forever in a perfect parabolic 194 00:09:06,520 --> 00:09:09,480 curve. But real chemical bonds have a 195 00:09:09,480 --> 00:09:11,400 breaking point. And we describe this with a 196 00:09:11,400 --> 00:09:13,880 different curve, the Morse potential. 197 00:09:14,120 --> 00:09:15,840 The Morse potential. It sounds like something from a 198 00:09:15,840 --> 00:09:18,000 spy movie, doesn't it? We've activated the Morse 199 00:09:18,000 --> 00:09:19,080 potential, Mr. Bunk? Yeah. 200 00:09:19,320 --> 00:09:23,080 Exactly, but it's really just a graph of energy versus the 201 00:09:23,080 --> 00:09:26,080 distance between the atoms. On the left side of the graph, 202 00:09:26,080 --> 00:09:29,040 where you try to push the atoms too close together, the energy 203 00:09:29,040 --> 00:09:32,120 just shoots up wildly. Because the nuclei are both 204 00:09:32,120 --> 00:09:34,640 positive, they repel each other. They don't want to touch. 205 00:09:34,680 --> 00:09:36,120 Great. There's a steep wall of 206 00:09:36,120 --> 00:09:38,560 repulsion. But on the right side of the 207 00:09:38,560 --> 00:09:41,560 graph, as you stretch the bond out, the curve doesn't go up 208 00:09:41,560 --> 00:09:43,560 forever. It starts to flatten out. 209 00:09:43,560 --> 00:09:46,680 It approaches a plateau. And that plateau is the bond 210 00:09:46,680 --> 00:09:49,120 breaking. That is the dissociation energy. 211 00:09:49,120 --> 00:09:51,640 We call it day. Once you put more energy into 212 00:09:51,640 --> 00:09:54,360 the bond than that value, the spring snaps. 213 00:09:54,440 --> 00:09:57,040 The atoms just fly apart. They're no longer a molecule. 214 00:09:57,440 --> 00:10:00,480 So the Morse potential is the map of a breakable spring. 215 00:10:00,880 --> 00:10:03,280 What does that do to our ladder of energy levels? 216 00:10:03,760 --> 00:10:06,000 The rungs? Well, in the perfect spring 217 00:10:06,000 --> 00:10:08,640 model, the harmonic oscillator where the rungs on the ladder 218 00:10:08,640 --> 00:10:12,480 are all evenly spaced, the energy jump from step one to two 219 00:10:12,480 --> 00:10:15,440 is the exact same as the jump from step 10:50. 220 00:10:15,440 --> 00:10:18,280 OK, perfectly regular. But in the Morse world, the real 221 00:10:18,280 --> 00:10:21,520 world, the steps get smaller as you go up. 222 00:10:21,640 --> 00:10:25,120 They get squished together. Exactly as you go higher up the 223 00:10:25,120 --> 00:10:27,640 ladder, as you get closer and closer to that breaking point, 224 00:10:27,840 --> 00:10:30,360 the rungs get squeezed closer and closer together. 225 00:10:30,560 --> 00:10:32,360 This effect is called in harmonicity. 226 00:10:32,480 --> 00:10:34,320 There's not a perfect ladder. It's not. 227 00:10:34,360 --> 00:10:38,120 It complicates things for higher energy vibrations, but for our 228 00:10:38,120 --> 00:10:42,160 purposes today, we usually focus on that very first biggest jump 229 00:10:42,560 --> 00:10:46,160 from the ground state V-0 to the first excited state V1. 230 00:10:46,480 --> 00:10:49,240 That's the fundamental vibration and it's the one we see most 231 00:10:49,240 --> 00:10:51,320 often, OK. So we've established our 232 00:10:51,320 --> 00:10:53,640 physics. We have atoms on breakable 233 00:10:53,640 --> 00:10:57,200 springs, they're vibrating on quantized rungs of a ladder, and 234 00:10:57,200 --> 00:11:00,600 they can never, ever stop. Now let's talk about detection. 235 00:11:00,720 --> 00:11:02,360 How do we actually measure this dance? 236 00:11:02,640 --> 00:11:06,760 The lecture introduces the two methods, infrared or IR and 237 00:11:06,760 --> 00:11:08,400 Roman. The two detectives. 238 00:11:08,560 --> 00:11:10,920 The lecture calls this the Great Divide. 239 00:11:11,600 --> 00:11:13,680 Why do we need two different ways to measure what is 240 00:11:13,680 --> 00:11:15,400 essentially the same thing, a vibration? 241 00:11:15,520 --> 00:11:17,840 Because they don't see the same things, they interact with the 242 00:11:17,840 --> 00:11:20,480 molecule through entirely different physical mechanisms. 243 00:11:20,560 --> 00:11:22,200 They're looking for different clues, so they are 244 00:11:22,200 --> 00:11:24,720 complementary. You need both to solve the case. 245 00:11:24,920 --> 00:11:27,520 Let's start with infrared spectroscopy IR. 246 00:11:28,440 --> 00:11:31,560 I feel like this is the one most people encounter first in a 247 00:11:31,560 --> 00:11:33,640 chemistry course. It's the standard. 248 00:11:33,880 --> 00:11:37,880 The workhorse IR spectroscopy is based on a simple mechanism 249 00:11:38,600 --> 00:11:41,520 absorption. You can think of it as a theft. 250 00:11:41,600 --> 00:11:44,560 A theft of energy. Recisely you shine a beam of 251 00:11:44,560 --> 00:11:47,920 infrared light which is basically just radiant heat 252 00:11:48,040 --> 00:11:49,680 through your sample of molecules. 253 00:11:50,160 --> 00:11:53,600 Now if the frequency of a photon of that light exactly matches 254 00:11:53,600 --> 00:11:55,560 the frequency of a bonds vibration. 255 00:11:55,560 --> 00:11:58,040 The molecule eats the photon. It eats the photon. 256 00:11:58,040 --> 00:12:02,760 It absorbs that specific packet of energy to jump up 1 rung on 257 00:12:02,760 --> 00:12:05,200 its vibrational ladder. OK, that seems straightforward 258 00:12:05,200 --> 00:12:06,840 enough, but there's a catch, isn't there? 259 00:12:06,840 --> 00:12:08,480 The lecture talks about a selection rule. 260 00:12:08,480 --> 00:12:11,760 Yes, the molecule is picky. It can't just absorb any photon 261 00:12:11,760 --> 00:12:13,160 just because the frequency matches. 262 00:12:13,160 --> 00:12:15,840 It needs a handle to grab onto the photon's energy. 263 00:12:16,000 --> 00:12:19,080 And for IR the rule is there must be a change in the 264 00:12:19,080 --> 00:12:21,160 molecules dipole moment. Correct. 265 00:12:21,240 --> 00:12:24,200 A dipole moment is just a measure of charge separation in 266 00:12:24,200 --> 00:12:26,480 a molecule. Is there a positive end and a 267 00:12:26,480 --> 00:12:28,560 negative end? Let's take a simple molecule 268 00:12:28,560 --> 00:12:31,760 like carbon monoxide, CEO. Oh yeah, CEO Oxygen is what we 269 00:12:31,760 --> 00:12:34,760 call very electronegative. It's an electron hog. 270 00:12:35,160 --> 00:12:38,600 It pulls the bonding electrons towards itself, making the 271 00:12:38,600 --> 00:12:42,400 oxygen end slightly negative. Which leaves the carbon end 272 00:12:42,400 --> 00:12:44,040 slightly positive. Exactly. 273 00:12:44,040 --> 00:12:47,800 So the molecule is like a tiny little magnet, but for electric 274 00:12:47,800 --> 00:12:51,560 charge instead of magnetism, it has a permanent dipole. 275 00:12:51,920 --> 00:12:54,560 Now imagine that bond stretching and compressing. 276 00:12:54,680 --> 00:12:57,320 The distance between the positive and negative ends is 277 00:12:57,320 --> 00:12:59,360 changing. Right, the strength of the 278 00:12:59,360 --> 00:13:02,080 dipole is oscillating. It's getting bigger and smaller, 279 00:13:02,080 --> 00:13:04,080 bigger and smaller. And light itself is an 280 00:13:04,120 --> 00:13:06,000 oscillating electromagnetic wave. 281 00:13:06,240 --> 00:13:08,920 Bingo. Because the molecule now has an 282 00:13:08,920 --> 00:13:10,960 oscillating electric field of its own. 283 00:13:10,960 --> 00:13:14,120 That changing dipole it can couple with the oscillating 284 00:13:14,120 --> 00:13:15,600 electric field of the light wave. 285 00:13:16,000 --> 00:13:17,840 The light has a handle to grab onto. 286 00:13:17,840 --> 00:13:21,200 It can transfer its energy. So if a vibration causes the 287 00:13:21,200 --> 00:13:24,040 charge distribution to shift, IR can see it. 288 00:13:24,280 --> 00:13:28,800 But what if there's no handle? Let's consider oxygen gas O22 289 00:13:28,800 --> 00:13:31,360 identical oxygen atoms. They pull on the electrons with 290 00:13:31,360 --> 00:13:33,080 equal strength. Perfectly equal. 291 00:13:33,080 --> 00:13:35,200 There's no positive end, no negative end. 292 00:13:35,440 --> 00:13:38,400 The dipole moment is 0. And if they stretch apart and 293 00:13:38,400 --> 00:13:41,120 come back together? It's still perfectly symmetric. 294 00:13:41,120 --> 00:13:44,120 0 is still zero. There is no change in the dipole 295 00:13:44,120 --> 00:13:46,600 moment, no handle. So you can shine as much 296 00:13:46,600 --> 00:13:50,280 infrared light as you want at a sample of pure oxygen gas and it 297 00:13:50,280 --> 00:13:53,440 will never absorb it. It's just transparent. 298 00:13:53,440 --> 00:13:56,680 Correct, it is completely transparent to IR radiation. 299 00:13:56,680 --> 00:13:59,600 This is actually a very very good thing for life on Earth. 300 00:13:59,880 --> 00:14:05,680 Also, nitrogen N2 and oxygen O2 make up 99% of our atmosphere. 301 00:14:06,240 --> 00:14:09,040 Neither of them has a changing dipole when it vibrates. 302 00:14:09,080 --> 00:14:12,000 Neither absorbs IR. If they did, they would trap 303 00:14:12,000 --> 00:14:15,040 heat like crazy and the greenhouse effect would turn 304 00:14:15,040 --> 00:14:17,600 Earth into Venus. It's the trace gases that are 305 00:14:17,600 --> 00:14:19,760 the problem. It's the lopsided ones, the ones 306 00:14:19,760 --> 00:14:23,160 with dipoles, carbon dioxide, water vapor, methane. 307 00:14:23,160 --> 00:14:25,880 They have the handles to grab that outgoing heat radiation. 308 00:14:25,960 --> 00:14:28,120 That's a fascinating connection to climate science. 309 00:14:28,480 --> 00:14:31,360 So IR is the detective that looks for electrical imbalance. 310 00:14:31,680 --> 00:14:33,920 Now what about the other detective, Ramon? 311 00:14:33,920 --> 00:14:36,920 Spectroscopy. Rahman Rahman is the 312 00:14:36,920 --> 00:14:39,200 sophisticated, slightly more mysterious cousin. 313 00:14:39,200 --> 00:14:40,880 It does not rely on absorption at all. 314 00:14:40,880 --> 00:14:43,520 It relies on scattering. Scattering just means the light 315 00:14:43,520 --> 00:14:44,440 bounces off. Right. 316 00:14:44,560 --> 00:14:47,560 Essentially, yes. In a ramen experiment you don't 317 00:14:47,560 --> 00:14:50,160 use infrared light. You hit the molecule with a 318 00:14:50,160 --> 00:14:53,680 sledgehammer of light, usually a very bright monochromatic laser, 319 00:14:53,800 --> 00:14:56,360 often in the visible range, like a green or red laser. 320 00:14:56,360 --> 00:14:58,080 OK, much more energetic approach. 321 00:14:58,080 --> 00:15:03,440 Much more now 99.999% of those laser photons will simply hit 322 00:15:03,440 --> 00:15:06,680 the molecule and bounce off with the exact same energy they came 323 00:15:06,680 --> 00:15:07,840 in with. That's called Rayleigh 324 00:15:07,840 --> 00:15:09,600 scattering, I think. It is. 325 00:15:09,600 --> 00:15:13,680 It's elastic scattering, it's boring, it's the reason the sky 326 00:15:13,680 --> 00:15:15,280 is blue. But it tells us nothing about 327 00:15:15,280 --> 00:15:17,360 the vibration. Just filter that light out. 328 00:15:17,640 --> 00:15:20,320 We are looking for the one in a million photon that does 329 00:15:20,320 --> 00:15:22,640 something different. The special one, the rum and 330 00:15:22,640 --> 00:15:24,000 photon. Exactly. 331 00:15:24,280 --> 00:15:27,640 Sometimes a photon will hit the molecule and interact with its 332 00:15:27,640 --> 00:15:31,160 electron cloud, and in that brief interaction it loses a 333 00:15:31,160 --> 00:15:33,200 tiny bit of energy to the molecule. 334 00:15:33,520 --> 00:15:37,440 The molecule steals a quantum of energy enough to jump up 1 335 00:15:37,440 --> 00:15:40,480 vibrational rum. So the photon bounces away, but 336 00:15:40,480 --> 00:15:42,720 it's tired. It has less energy than it 337 00:15:42,720 --> 00:15:45,080 started with. Which means it's frequency has 338 00:15:45,080 --> 00:15:47,200 shifted. It's color has shifted slightly. 339 00:15:47,200 --> 00:15:49,960 It becomes redder. This specific process is called 340 00:15:49,960 --> 00:15:53,440 stoke scattering. By measuring exactly how much 341 00:15:53,440 --> 00:15:56,960 energy the photon lost, we know exactly how much energy the 342 00:15:56,960 --> 00:15:59,240 vibration required. It's a very indirect way of 343 00:15:59,240 --> 00:16:00,880 measuring. So what's the selection rule 344 00:16:00,880 --> 00:16:03,280 here? IR needed a changing dipole 345 00:16:03,280 --> 00:16:05,120 moment. What does Rahman need? 346 00:16:05,440 --> 00:16:07,800 The notes say it needs a change in polarizability. 347 00:16:07,920 --> 00:16:09,520 Polarizability. It's a mouthful. 348 00:16:09,520 --> 00:16:11,680 I know. Essentially, it's a measure of 349 00:16:11,680 --> 00:16:14,800 how squishy the electron cloud of the molecule is. 350 00:16:14,800 --> 00:16:17,200 Squishy. Think of the cloud of electrons 351 00:16:17,200 --> 00:16:19,520 around the molecules atoms like a balloon. 352 00:16:19,920 --> 00:16:21,720 Can you distort it? Can you stretch it? 353 00:16:21,720 --> 00:16:23,360 Can you compress it? In an electric field? 354 00:16:23,360 --> 00:16:27,240 That's polarizability. Big floppy molecules with lots 355 00:16:27,240 --> 00:16:30,680 of electrons far from the nuclei are very polarizable. 356 00:16:30,880 --> 00:16:33,600 OK, so how does a vibration change the squishiness? 357 00:16:33,760 --> 00:16:37,160 Imagine a perfectly symmetric molecule like that O2 molecule. 358 00:16:37,160 --> 00:16:41,160 Again, imagine it breathing the bond, stretching and contracting 359 00:16:41,160 --> 00:16:43,840 symmetrically. When it expands, the electrons 360 00:16:43,840 --> 00:16:45,960 are held a bit further from the nuclei. 361 00:16:46,040 --> 00:16:49,240 They're looser, more squishy. And when it contracts, they're 362 00:16:49,240 --> 00:16:53,080 pulled in tighter, less squishy. So the squishiness itself, the 363 00:16:53,080 --> 00:16:56,120 polarizability is oscillating with the vibration. 364 00:16:56,120 --> 00:16:57,560 And that's the handle for Rahman. 365 00:16:57,600 --> 00:17:00,240 That's a handle. So as a general rule, Rahman 366 00:17:00,240 --> 00:17:04,280 spectroscopy loves symmetry. It loves big symmetric breathing 367 00:17:04,280 --> 00:17:07,640 motions that change the overall size or shape of the electron 368 00:17:07,640 --> 00:17:11,240 cloud. IR, on the other hand, loves 369 00:17:11,240 --> 00:17:14,319 asymmetry, lopsided pulling and bending that changes the charge 370 00:17:14,319 --> 00:17:16,040 balance. So that really is the great 371 00:17:16,040 --> 00:17:18,640 divide. IR sees the lopsided stuff, 372 00:17:18,640 --> 00:17:21,839 Rahman sees the symmetric stuff. And that is why we need both. 373 00:17:22,200 --> 00:17:25,520 If you only used IR, you would be completely blind to the 374 00:17:25,520 --> 00:17:28,280 symmetric vibrations of many, many molecules. 375 00:17:28,640 --> 00:17:31,280 If you only used Rahman, you might miss some very important 376 00:17:31,280 --> 00:17:33,920 asymmetric stretches. Together they give you the 377 00:17:33,920 --> 00:17:37,400 complete picture of the dance. OK, so we have our tools, we 378 00:17:37,400 --> 00:17:39,560 know the physics, we have our two detectives. 379 00:17:40,000 --> 00:17:41,880 But now we run into the problem of complexity. 380 00:17:42,080 --> 00:17:46,560 If I have a simple molecule like AO or HCLI, can guess the 381 00:17:46,560 --> 00:17:49,000 vibration, it just stretches 1 vibration. 382 00:17:49,200 --> 00:17:51,160 Easy. But molecules are rarely that 383 00:17:51,160 --> 00:17:51,880 simple. Right. 384 00:17:52,200 --> 00:17:54,560 As soon as you add more atoms, the number of possible wiggles 385 00:17:54,560 --> 00:17:57,000 just explodes. The notes give us a formula for 386 00:17:57,000 --> 00:17:59,880 this three and the six. It's a simple but powerful 387 00:17:59,880 --> 00:18:02,600 counting rule. N is just the number of atoms in 388 00:18:02,600 --> 00:18:05,800 your molecule. Each of those N atoms lives in a 389 00:18:05,800 --> 00:18:09,520 3D World, so it can move in three directions, XY and Z. 390 00:18:09,960 --> 00:18:13,680 That gives us a total of three times north, or three N total 391 00:18:13,680 --> 00:18:17,200 degrees of freedom. Every possible motion is in that 392 00:18:17,200 --> 00:18:19,040 number. But not all of those are 393 00:18:19,040 --> 00:18:20,440 vibrations. That's a key point. 394 00:18:20,440 --> 00:18:23,800 It's a critical .3 of those motions correspond to the whole 395 00:18:23,800 --> 00:18:25,440 molecule just flying through the room. 396 00:18:25,440 --> 00:18:28,360 That's translation along the XY and Z axis. 397 00:18:28,600 --> 00:18:30,600 We don't care about that. It's not a wiggle. 398 00:18:30,720 --> 00:18:34,560 It's not an internal wiggle, and for a non linear molecule 3 more 399 00:18:34,560 --> 00:18:36,960 of those motions correspond to the whole thing tumbling end 400 00:18:36,960 --> 00:18:41,000 over end like a thrown football. That's rotation around the XY 401 00:18:41,000 --> 00:18:42,960 and Z axis. We don't care about that either. 402 00:18:43,160 --> 00:18:46,280 So we start with three in total motions, and we subtract those 403 00:18:46,280 --> 00:18:49,560 six whole body motions. 3 translations, 3 rotations. 404 00:18:49,640 --> 00:18:51,960 And what's left? The three N 6 must be the 405 00:18:51,960 --> 00:18:53,720 internal vibrations, the wiggles. 406 00:18:53,720 --> 00:18:55,760 So let's introduce our case study for today. 407 00:18:56,040 --> 00:18:59,680 The molecule is silicon dichloride dihydride, that's 408 00:18:59,840 --> 00:19:02,480 Secl 2H2. Let's visualize it. 409 00:19:02,800 --> 00:19:04,720 You have a silicon atom sitting in the center. 410 00:19:04,840 --> 00:19:08,240 Attached to it, pointing sort of forward, are two chlorine atoms, 411 00:19:08,480 --> 00:19:11,320 and pointing sort of backward are are two hydrogen atoms. 412 00:19:11,480 --> 00:19:13,800 It's shaped like a distorted tetrahedron. 413 00:19:14,120 --> 00:19:17,560 So we have 5 atoms total, the silicon, 2 chlorines, 2 414 00:19:17,560 --> 00:19:20,840 hydrogens. Let's do the math. 3 * 5 is 15 415 00:19:20,880 --> 00:19:25,600 total motions. 15° of freedom. Subtract the six for translation 416 00:19:25,600 --> 00:19:28,800 and rotation. That leaves 9. 9 distinct 417 00:19:28,840 --> 00:19:31,760 vibrational modes. 9 fundamental wiggles. 418 00:19:31,880 --> 00:19:34,720 And trying to visualize 9 different ways a pyramid can 419 00:19:34,720 --> 00:19:38,160 wiggle is really difficult. I can imagine a stretch, maybe a 420 00:19:38,160 --> 00:19:41,960 bend, but nine of them all happening and coupled together. 421 00:19:42,280 --> 00:19:44,720 It's impossible to rely on intuition alone. 422 00:19:44,840 --> 00:19:47,360 You might guess 3 or 4 the obvious ones, but you will 423 00:19:47,360 --> 00:19:50,080 absolutely miss the others and more importantly, you won't have 424 00:19:50,080 --> 00:19:53,080 any idea which ones are IR active and which are ramen 425 00:19:53,080 --> 00:19:54,360 active. This is where we need the 426 00:19:54,360 --> 00:19:56,880 algorithm. We need a systematic, foolproof 427 00:19:56,880 --> 00:20:00,040 way to predict all nine vibrations and their activity. 428 00:20:00,600 --> 00:20:03,240 This brings us to Section 3, Symmetry and Group theory. 429 00:20:03,440 --> 00:20:06,240 This is the part that usually scares chemistry students. 430 00:20:06,480 --> 00:20:10,320 It involves matrices and Greek letters and character tables, 431 00:20:10,520 --> 00:20:14,640 but at its core it is just a very rigorous way of quantifying 432 00:20:14,640 --> 00:20:16,600 shape. We start by assigning the 433 00:20:16,600 --> 00:20:18,720 molecule to something called a point group. 434 00:20:19,600 --> 00:20:24,000 For our molecule CCL 2H2, the group is called C2V. 435 00:20:24,240 --> 00:20:26,000 What does that label actually mean? 436 00:20:26,320 --> 00:20:29,040 The label C2V is basically shorthand. 437 00:20:29,280 --> 00:20:32,080 It's a code that describes the complete set of symmetry 438 00:20:32,080 --> 00:20:34,280 operations that the molecule possesses. 439 00:20:34,480 --> 00:20:37,840 And a symmetry operation is a move you can make to the 440 00:20:37,840 --> 00:20:39,840 molecule that leaves it looking unchanged. 441 00:20:39,920 --> 00:20:42,600 It's like a magic trick. If you, the observer, close your 442 00:20:42,600 --> 00:20:44,960 eyes, I perform the operation and you open your eyes again. 443 00:20:44,960 --> 00:20:46,480 You can't tell that I did anything. 444 00:20:46,760 --> 00:20:49,120 The molecule looks identical to how it started. 445 00:20:49,200 --> 00:20:51,880 OK, that makes sense. For this C2V group, there are 446 00:20:51,880 --> 00:20:55,120 four specific moves allowed and we need to understand them 447 00:20:55,360 --> 00:20:57,800 because we're going to use them as our tools in the calculation, 448 00:20:57,800 --> 00:20:58,880 OK? Let's walk through the four 449 00:20:58,880 --> 00:21:01,160 moves. The first one listed is EE. 450 00:21:01,160 --> 00:21:02,920 Stands for the identity operation. 451 00:21:02,920 --> 00:21:06,240 It comes from the German word Einheit, which means unity. 452 00:21:06,680 --> 00:21:10,160 It basically means do nothing. Wait, why is do nothing a move? 453 00:21:10,720 --> 00:21:13,520 It seems kind of pointless. It seems that way, but in 454 00:21:13,520 --> 00:21:16,520 mathematics, and especially in Group theory, you always need a 455 00:21:16,520 --> 00:21:17,800 unit. You need a one. 456 00:21:17,920 --> 00:21:20,120 It's the baseline against which everything else is measured. 457 00:21:20,160 --> 00:21:23,760 If you do nothing, every atom by definition stays exactly where 458 00:21:23,760 --> 00:21:25,880 it is. OK, I can accept that move 459 00:21:25,880 --> 00:21:29,120 number 2 is C2. This is a rotation. 460 00:21:29,360 --> 00:21:31,760 Right. Imagine an axis, a skewer 461 00:21:31,920 --> 00:21:34,600 running vertically right through the central silicon atom. 462 00:21:34,600 --> 00:21:38,520 This is our main Z axis. The C2 operation means we rotate 463 00:21:38,520 --> 00:21:41,760 the entire molecule 180° around the skewer. 464 00:21:41,920 --> 00:21:45,560 So if I visualize our molecule, the silicon is on the skewer, so 465 00:21:45,560 --> 00:21:47,240 it just spins in place, it doesn't move. 466 00:21:47,240 --> 00:21:49,600 Correct. But the two chlorine atoms, 467 00:21:49,600 --> 00:21:52,040 let's say 1 is front left and one is front right, they swap 468 00:21:52,040 --> 00:21:55,440 places, and the two hydrogen atoms at the back also swap 469 00:21:55,440 --> 00:21:57,720 places with each other. But because a chlorine atom 470 00:21:57,720 --> 00:22:01,000 looks exactly like another chlorine atom, the molecule as a 471 00:22:01,000 --> 00:22:03,520 whole appears unchanged. Correct. 472 00:22:03,520 --> 00:22:06,560 So C2 is a valid symmetry operation for this molecule. 473 00:22:06,600 --> 00:22:10,240 OK, moves number three and four are both reflections or mirror 474 00:22:10,240 --> 00:22:11,920 planes. They have the symbol Sigma. 475 00:22:12,040 --> 00:22:14,520 We have two of them. The first one is a vertical 476 00:22:14,520 --> 00:22:18,240 mirror plane that slices right through the silicone atom and 477 00:22:18,240 --> 00:22:21,160 the two chlorine atoms. We can call this the subsplane. 478 00:22:21,280 --> 00:22:23,840 So the silicon and the two chlorines are sitting inside the 479 00:22:23,840 --> 00:22:25,120 glass of the mirror. Exactly. 480 00:22:25,120 --> 00:22:28,280 They don't move, but the two hydrogens are sticking out on 481 00:22:28,280 --> 00:22:30,960 either side of the mirror. So when you reflect, the left 482 00:22:30,960 --> 00:22:33,720 hydrogen swaps places with the right hydrogen. 483 00:22:33,840 --> 00:22:35,840 And the second mirror plane. That is the city Slane. 484 00:22:35,840 --> 00:22:38,800 It's also vertical, but it's perpendicular to the first one. 485 00:22:39,080 --> 00:22:42,360 This mirror slices through the silicon atom and the two 486 00:22:42,360 --> 00:22:45,880 hydrogen. OK, so in this case the silicon 487 00:22:45,880 --> 00:22:49,320 and the hydrogen sits still and the two chlorine swap places 488 00:22:49,320 --> 00:22:50,440 with each other. You've got it. 489 00:22:50,440 --> 00:22:55,080 Those are our four tools. Do nothing E spin 180° C two, 490 00:22:55,080 --> 00:22:58,280 mirror one and mirror 2. Now comes the matrix part. 491 00:22:59,120 --> 00:23:01,280 We're going to use these four tools to generate a special 492 00:23:01,280 --> 00:23:05,280 code. This is Section 4 Calculating E3 493 00:23:05,280 --> 00:23:08,480 N gamma 3 N. This sounds intimidating, but 494 00:23:08,480 --> 00:23:10,160 it's really just a counting exercise. 495 00:23:10,680 --> 00:23:13,560 We want to know how the total motion of the molecule, all 496 00:23:13,560 --> 00:23:15,680 fifteen of those degrees of freedom we talked about, 497 00:23:16,000 --> 00:23:18,600 responds to each of these 4 symmetry operations. 498 00:23:18,760 --> 00:23:21,760 We need to generate a set of four numbers, one for each 499 00:23:21,760 --> 00:23:23,840 operation, and to get each number. 500 00:23:24,040 --> 00:23:26,320 The notes say we ask 2 questions for each move. 501 00:23:27,000 --> 00:23:30,400 First, how many atoms stayed in the exact same spot? 502 00:23:31,000 --> 00:23:33,280 We'll call that new for unshifted atoms. 503 00:23:33,280 --> 00:23:36,400 And 2nd, for those atoms that stayed put, what happened to 504 00:23:36,400 --> 00:23:40,200 their XY and Z coordinates? This gives a character checks a 505 00:23:40,200 --> 00:23:42,440 gaze. OK, let's run the numbers first. 506 00:23:42,440 --> 00:23:45,440 Operation E do nothing. How many atoms stay? 507 00:23:45,440 --> 00:23:47,880 Put all five of them, nothing moves, so new is 5. 508 00:23:48,040 --> 00:23:49,640 Easy enough. Now for the coordinates. 509 00:23:49,760 --> 00:23:54,760 If you do nothing, X stays X, that's a + 1 contribution, Y 510 00:23:54,760 --> 00:23:57,720 stays Y that's another plus one, and Z stays Z another plus one. 511 00:23:57,720 --> 00:24:01,240 So the sum is 3. So the calculation is simple, 5 512 00:24:01,240 --> 00:24:03,920 unshifted atoms times the character of 3 = 15. 513 00:24:03,960 --> 00:24:05,720 That's our first number, and it makes sense, right? 514 00:24:05,720 --> 00:24:07,920 It represents the 15 total degrees of freedom that we 515 00:24:07,920 --> 00:24:09,880 calculated with three NS. It's a good sanity check. 516 00:24:10,480 --> 00:24:14,200 OK, next operation C2, the 180° spin. 517 00:24:14,240 --> 00:24:16,680 Who stays put? Well, the chlorine swap and the 518 00:24:16,680 --> 00:24:19,560 hydrogen swap, so only the central silicon atom, it just 519 00:24:19,560 --> 00:24:22,480 spins in place, so new 1. Correct. 520 00:24:22,520 --> 00:24:24,120 Now the coordinates. This is a little harder to 521 00:24:24,120 --> 00:24:27,240 visualize. We're spinning 180° around the Z 522 00:24:27,240 --> 00:24:29,080 axis. So the Z axis itself doesn't 523 00:24:29,080 --> 00:24:32,080 change, it stays put that contributes a + 1. 524 00:24:32,160 --> 00:24:35,360 Right, but think about the X&Y axis if you have a point at 525 00:24:35,360 --> 00:24:38,680 position X and you rotate it 180°. 526 00:24:38,680 --> 00:24:40,640 You end up on the other side at Nikex. 527 00:24:40,760 --> 00:24:44,320 And similarly Y becomes may Y, so the total character 528 00:24:44,320 --> 00:24:48,840 contribution is plus one for Z -, 1 for X and -1 for Y. 529 00:24:48,840 --> 00:24:51,760 So 1 - 1 - 1 = - 1. Exactly. 530 00:24:51,760 --> 00:24:56,240 So the math is 1 unshifted atom times the character of -1, which 531 00:24:56,240 --> 00:24:58,040 equals -1. That's our second number. 532 00:24:58,040 --> 00:25:02,280 OK, our code is 15 -, 1 not not next mirror one. 533 00:25:02,280 --> 00:25:04,320 The one called sucks. This is the one that contains 534 00:25:04,320 --> 00:25:05,840 the silicon and the two chlorines. 535 00:25:06,000 --> 00:25:08,800 So how many atoms are sitting inside the mirror unshifted? 536 00:25:08,800 --> 00:25:10,920 The silicon and the two chlorines, that's three atoms, 537 00:25:10,920 --> 00:25:12,960 so NU, not 3. And the coordinates. 538 00:25:12,960 --> 00:25:16,560 If an atom is sitting in the XE plane, it's X coordinate stays X 539 00:25:16,560 --> 00:25:20,440 + 1 and it's Z coordinate stays Z + 1, but it's Y coordinate 540 00:25:20,440 --> 00:25:22,960 which sticks out from the mirror gets reflected to the other 541 00:25:22,960 --> 00:25:24,640 side. To Max. 542 00:25:24,680 --> 00:25:27,600 So that's a Max 1. So the total character is plus 1 543 00:25:27,600 --> 00:25:32,480 + + 1 -, 1 which is +1. The map is 3 unshifted atoms 544 00:25:32,480 --> 00:25:37,120 times 1 = 3. And finally, mirror 2, this is 545 00:25:37,120 --> 00:25:39,800 the one that holds the silicon and the two hydrogens. 546 00:25:39,800 --> 00:25:41,760 Same logic, right? The silicon and the two 547 00:25:41,760 --> 00:25:44,520 hydrogens sit in the mirror, so that's three unshifted atoms. 548 00:25:45,160 --> 00:25:47,280 The character for reflection is plus one. 549 00:25:47,520 --> 00:25:51,480 The result is 3 * 1 which is 3. So we've done it. 550 00:25:51,480 --> 00:25:53,000 We have generated our code vector. 551 00:25:53,160 --> 00:25:57,800 It is 15 Nanish 133. This feels like we've found a 552 00:25:57,800 --> 00:26:01,120 secret key, but what does it unlock? 553 00:26:01,280 --> 00:26:03,720 What is this code? Right now, that vector is what 554 00:26:03,720 --> 00:26:05,360 we call a reducible representation. 555 00:26:05,480 --> 00:26:07,560 It's a jumble. Imagine you made a smoothie. 556 00:26:07,560 --> 00:26:10,200 You've blended up translations, rotations, and vibrations all 557 00:26:10,200 --> 00:26:12,440 together into this one messy list of numbers. 558 00:26:12,440 --> 00:26:15,480 So our job now is to unblend this movie to figure out what 559 00:26:15,480 --> 00:26:18,360 the original ingredients were. Exactly this is Section 5, the 560 00:26:18,360 --> 00:26:20,000 reduction. This is where we find the 561 00:26:20,000 --> 00:26:23,040 fundamental symmetry species. And to do this we need a recipe 562 00:26:23,040 --> 00:26:26,160 book, the character table for the C to V point group. 563 00:26:26,400 --> 00:26:29,520 The character table is our key. It lists all the possible 564 00:26:29,640 --> 00:26:32,040 fundamental ingredients for this specific symmetry. 565 00:26:32,040 --> 00:26:36,040 We call them irreducible representations, or ear reps for 566 00:26:36,040 --> 00:26:37,920 short. For the C2V group, the 567 00:26:37,920 --> 00:26:42,200 ingredients have simple labels A1A2B1 and B2. 568 00:26:42,520 --> 00:26:44,600 In these labels, they aren't just random, right? 569 00:26:44,760 --> 00:26:48,120 They describe different types of symmetry, like a one is usually 570 00:26:48,120 --> 00:26:51,120 the totally symmetric one where nothing changes sign. 571 00:26:51,120 --> 00:26:53,320 Correct. A2 usually involves something 572 00:26:53,320 --> 00:26:56,880 antisymmetric with respect to rotation, and B1 and B2 are 573 00:26:56,880 --> 00:26:59,320 antisymmetric with respect to the different mirror planes. 574 00:26:59,600 --> 00:27:01,480 They're just labels for fundamental patterns of 575 00:27:01,480 --> 00:27:03,240 symmetry. So our goal is to figure out how 576 00:27:03,240 --> 00:27:07,720 many units of A1A2B1 and B2 are hidden inside our messy 577 00:27:07,720 --> 00:27:10,680 smoothie. Vector 15 make it 133. 578 00:27:10,880 --> 00:27:12,720 And for that we use the reduction formula. 579 00:27:12,920 --> 00:27:15,400 It looks scary in a textbook, but it's essentially just a 580 00:27:15,400 --> 00:27:18,400 systematic way of comparing our vector to the vectors for each 581 00:27:18,400 --> 00:27:19,880 ingredient in the character table. 582 00:27:19,880 --> 00:27:22,000 I want to walk through just one of these calculations so we can 583 00:27:22,000 --> 00:27:24,480 all feel the mechanism. Let's try to find how many A1 584 00:27:24,480 --> 00:27:26,800 units are in there. OK, first we look at the 585 00:27:26,800 --> 00:27:29,520 character table. The row for a one has its own 586 00:27:29,680 --> 00:27:33,280 code vector. It's 1111. 587 00:27:33,480 --> 00:27:37,280 OK, so the code for A1 is 1111. Our code for the molecule is 588 00:27:37,280 --> 00:27:39,160 15/9. It was 133. 589 00:27:39,520 --> 00:27:42,160 The formula says we multiply them together pair by pair. 590 00:27:42,280 --> 00:27:44,920 Let's do it first. Pair 15 from our vector times 591 00:27:44,920 --> 00:27:48,440 one from the A1 vector is 15. Second pair -1 from our vector 592 00:27:48,440 --> 00:27:52,160 times one from A1 is -1. Third pair 3 * 1 is 3. 593 00:27:52,480 --> 00:27:57,920 The fourth pair 3 * 1 is also 3. Now we sum those results up 15 + 594 00:27:58,080 --> 00:28:03,040 -, 1 + 3 + 3. That's 15 -. 1 is 14 + 3 is 17 + 595 00:28:03,040 --> 00:28:04,320 3 is 20. Perfect. 596 00:28:04,440 --> 00:28:07,640 The final step of the formula is to divide that sum by the order 597 00:28:07,640 --> 00:28:10,280 of the group, which is just a fancy way of saying the number 598 00:28:10,280 --> 00:28:14,280 of operations we used. We use 4 operations EC2 and the 599 00:28:14,280 --> 00:28:17,680 two Sigma, so we divide by 4. And 20 / 4 is 5. 600 00:28:17,800 --> 00:28:19,720 A clean integer that is deeply satisfying. 601 00:28:19,720 --> 00:28:22,080 If we'd gotten 5.2 or something, we'd know we made a mistake 602 00:28:22,080 --> 00:28:23,280 somewhere. Exactly. 603 00:28:23,360 --> 00:28:26,160 Quantum mechanics and group theory count in whole numbers. 604 00:28:26,280 --> 00:28:29,320 So there it is. Inside our messy smoothie of 15 605 00:28:29,320 --> 00:28:33,160 motion, there are exactly 5 distinct units that have A1 606 00:28:33,160 --> 00:28:34,400 symmetry. That's amazing. 607 00:28:34,560 --> 00:28:37,320 So if we were to repeat this exact process for the other 608 00:28:37,320 --> 00:28:39,400 ingredients. You would you'd take our vector 609 00:28:39,560 --> 00:28:45,280 15 monotis 133 and compare it to the vectors for A2B1 and B2 from 610 00:28:45,280 --> 00:28:47,240 the character table. I won't make us do all the 611 00:28:47,240 --> 00:28:49,560 multiplication out loud, but the source material shows the 612 00:28:49,560 --> 00:28:52,240 result. For a 2 the map gives us 2, for 613 00:28:52,240 --> 00:28:55,520 B1 it gives us four, and for B2 it also gives us 4. 614 00:28:56,000 --> 00:28:57,760 That's right. So the full recipe of our 615 00:28:57,760 --> 00:29:01,960 smoothie, the total motion of the molecule is described AS5A1 616 00:29:01,960 --> 00:29:06,480 plus 2A2 plus 4B1 plus 4B2. If you add up the coefficients 5 617 00:29:06,480 --> 00:29:10,040 + 2 + 4 + 4, it equals 15. It all checks out. 618 00:29:10,120 --> 00:29:12,280 But we are not done. Remember, we only want the 619 00:29:12,280 --> 00:29:14,400 vibrations. We've successfully categorized 620 00:29:14,400 --> 00:29:16,520 everything, but now we have to subtract the junk. 621 00:29:16,680 --> 00:29:20,400 This is section 6, isolating the vibrations, the junk being the 622 00:29:20,400 --> 00:29:23,400 whole molecule flying through space, translation and tumbling 623 00:29:23,400 --> 00:29:25,920 end over end rotation. To do this, we go back to our 624 00:29:26,000 --> 00:29:27,360 recipe book, the character table. 625 00:29:27,720 --> 00:29:30,160 On the right hand side of most tables, there are columns that 626 00:29:30,160 --> 00:29:32,760 tell us explicitly which symmetry labels correspond to 627 00:29:32,760 --> 00:29:36,160 the simple XYZ axis, which represent translation and the 628 00:29:36,160 --> 00:29:40,200 rotation axis RXYRZ. OK, I'm looking at the C2V 629 00:29:40,200 --> 00:29:43,400 character table right now. It shows that translation along 630 00:29:43,400 --> 00:29:46,960 the Z axis T's is listed in the row for A1. 631 00:29:47,000 --> 00:29:50,880 So one of our 5A1 motions is just the whole molecule moving 632 00:29:50,880 --> 00:29:53,080 up and down the Z axis. We need to throw it out. 633 00:29:53,080 --> 00:29:57,200 Translation along the X axis TX corresponds to B1 and 634 00:29:57,200 --> 00:29:59,880 translation along Y tie corresponds to B2. 635 00:29:59,920 --> 00:30:02,880 So we have to throw away one of our B ones and one of our B twos 636 00:30:02,880 --> 00:30:04,280 as well. That takes care of the flying. 637 00:30:04,680 --> 00:30:07,520 Now for the spinning. The table shows rotation around 638 00:30:07,520 --> 00:30:10,560 the Z axis. RZ corresponds to A2. 639 00:30:10,600 --> 00:30:13,560 So one of our 2A2 motions is just the whole molecule spinning 640 00:30:13,560 --> 00:30:15,520 like a top. Useless for vibration. 641 00:30:15,680 --> 00:30:18,840 Get rid of it. And rotation around Y rye is B1 642 00:30:18,880 --> 00:30:22,680 and rotation around XRX is B2. B1 and B2 each lose another 643 00:30:22,680 --> 00:30:25,040 motion to rotation. OK, let's do the final tally. 644 00:30:25,040 --> 00:30:26,680 Let's subtract everything and see what's left. 645 00:30:26,680 --> 00:30:28,880 We started with five units. We subtracted one for 646 00:30:28,880 --> 00:30:32,160 translation along Z. We are left with 4A1 vibrational 647 00:30:32,160 --> 00:30:33,800 modes. For a two, we started with two 648 00:30:33,800 --> 00:30:36,560 units, we subtracted one for rotation around Z we are left 649 00:30:36,560 --> 00:30:40,400 with one A2 vibrational mode. For B1, started with four, we 650 00:30:40,400 --> 00:30:44,040 subtract 1 for translation TX and one for rotation rye. 651 00:30:44,440 --> 00:30:46,480 That leaves 2B1 vibrational modes. 652 00:30:46,600 --> 00:30:49,120 And for B2 we also started with four. 653 00:30:49,320 --> 00:30:52,880 We subtract 1 for translation tie and one for rotation RX. 654 00:30:53,160 --> 00:30:55,320 That leaves 2B2 vibrational modes. 655 00:30:55,560 --> 00:30:59,160 So the final true vibrational fingerprint of this molecule is 656 00:30:59,240 --> 00:31:04,040 4A1 plus one A 2 + 2 B 1 + 2 B2. Let's do our sanity check. 657 00:31:04,040 --> 00:31:07,160 Let's add them up. 4 + 1 + 2 + 2 of equals 9. 658 00:31:07,320 --> 00:31:08,960 And our three and six prediction was 9. 659 00:31:09,720 --> 00:31:12,840 The math works perfectly. We have successfully found and 660 00:31:12,840 --> 00:31:16,240 categorized every single one of the 9 wiggles of this molecule. 661 00:31:16,240 --> 00:31:18,200 This is amazing. We've gone from a shape to a 662 00:31:18,200 --> 00:31:20,040 list of abstract symmetry labels. 663 00:31:20,200 --> 00:31:23,200 This brings us to Section 7. The final verdict, The selection 664 00:31:23,200 --> 00:31:25,480 rules. We know we have 9 vibrations, 665 00:31:25,480 --> 00:31:27,240 but can our detectives actually see them? 666 00:31:27,320 --> 00:31:29,360 This is where it all pays off. We bring back our two 667 00:31:29,360 --> 00:31:32,200 detectives, IR and Ramen. Let's start with IR 668 00:31:32,200 --> 00:31:33,440 spectroscopy. OK. 669 00:31:33,520 --> 00:31:36,800 The rule for IR, as we said, is that the vibration must cause a 670 00:31:36,800 --> 00:31:39,480 change in the dipole moment. And in the language of group 671 00:31:39,480 --> 00:31:41,880 theory, that means the symmetry of the vibration must be the 672 00:31:41,880 --> 00:31:45,200 same as the symmetry of one of the linear axis XY or Z. 673 00:31:45,320 --> 00:31:47,200 So we just have to look at the character table again. 674 00:31:47,360 --> 00:31:51,960 The Z axis we said belongs to A1, the X axis belongs to B1 and 675 00:31:51,960 --> 00:31:55,160 the Y axis belongs to B2. So any vibration that has one of 676 00:31:55,160 --> 00:31:58,160 those 3 symmetry labels will be IR active. 677 00:31:58,160 --> 00:32:00,160 It will show up as a peak on the IR spectrum. 678 00:32:00,240 --> 00:32:03,720 Let's look at our list. We have 4A1 vibrations since a 679 00:32:03,720 --> 00:32:07,120 one matches the Z axis. All four of those are IR active. 680 00:32:07,120 --> 00:32:08,040 That's. Four peaks. 681 00:32:08,040 --> 00:32:11,800 We have two B1 vibrations. B1 matches the X axis, so both 682 00:32:11,800 --> 00:32:14,320 of those are also IR active. That's another two peaks, 6 683 00:32:14,320 --> 00:32:17,760 total so far. And we have two B2 vibrations. 684 00:32:17,760 --> 00:32:20,560 B2 matches the Y axis, so those two are also active. 685 00:32:20,600 --> 00:32:25,200 So our final prediction, 4 + 2 + 2, we expect to see 8 686 00:32:25,320 --> 00:32:29,600 fundamental bands or peaks in the IR spectrum of this 687 00:32:29,600 --> 00:32:32,080 molecule. But wait, we have 9 vibrations 688 00:32:32,080 --> 00:32:33,840 in total. What happened to the A2 689 00:32:33,840 --> 00:32:35,920 vibration? Look at the character table 690 00:32:35,920 --> 00:32:38,080 again. In the row for A2, does it list 691 00:32:38,080 --> 00:32:39,680 XY or Z next to it? No. 692 00:32:39,880 --> 00:32:42,400 It just has arza for rotation. That's the key. 693 00:32:42,400 --> 00:32:44,680 That means the A2 vibration, which is a kind of twisting 694 00:32:44,680 --> 00:32:47,800 motion of the hydrogens against the chlorines, does not create a 695 00:32:47,800 --> 00:32:52,160 net change in the dipole moment along any of the XY or Z axis. 696 00:32:52,160 --> 00:32:55,600 So it's invisible to IR. It is IR inactive. 697 00:32:56,200 --> 00:32:59,200 The IR detective walks right past that clue, completely blind 698 00:32:59,200 --> 00:33:01,440 to it. OK, so if I run an IR scan of 699 00:33:01,440 --> 00:33:03,480 this molecule, I should see eight peaks. 700 00:33:03,560 --> 00:33:06,040 Now what about our other detective ramen? 701 00:33:06,400 --> 00:33:09,320 Ramen looks for a change in polarizability, the squishiness. 702 00:33:09,760 --> 00:33:12,680 In the language of the character table, this corresponds to the 703 00:33:12,680 --> 00:33:17,160 quadratic functions, things like XEQUZWECDDD double. 704 00:33:17,200 --> 00:33:18,320 So we just checked the list again. 705 00:33:18,320 --> 00:33:22,080 I'm looking at the table. The A1 row has X Phi 1Y row and 706 00:33:22,080 --> 00:33:24,920 Z of fire. So check A1 is ramen active. 707 00:33:25,080 --> 00:33:28,440 The B1 row has XC. Check B2 row has YZ. 708 00:33:28,440 --> 00:33:30,400 Check B2. Is ramen active? 709 00:33:30,400 --> 00:33:32,120 And what about our missing child A2? 710 00:33:32,280 --> 00:33:36,400 The A2 row it has guy Z listed next to it, so check it's active 711 00:33:36,400 --> 00:33:39,040 too. In the C2V point group, every 712 00:33:39,040 --> 00:33:41,360 single one of the symmetry species is ramen active. 713 00:33:41,400 --> 00:33:43,680 So ramen sees all nine vibrations. 714 00:33:43,760 --> 00:33:44,480 Exactly. Exactly. 715 00:33:44,680 --> 00:33:47,480 So here's our final concrete prediction based on nothing but 716 00:33:47,480 --> 00:33:50,000 the molecule shape. The IR spectrum will show 8 717 00:33:50,000 --> 00:33:51,880 bands. The Ramen Spectrum will show 9 718 00:33:51,880 --> 00:33:53,840 bands. And that one missing band in the 719 00:33:53,840 --> 00:33:55,480 IR, The difference between 8:00 and 9:00? 720 00:33:55,480 --> 00:33:57,920 That's the smoking gun. It's the fingerprint of the 721 00:33:57,920 --> 00:34:00,560 geometry. Imagine you're a chemist and you 722 00:34:00,560 --> 00:34:05,280 have a vial of some unknown gas. You suspect it might be CCL 2H2. 723 00:34:05,760 --> 00:34:08,920 You run both Spectra. If you see that 8 versus 9 724 00:34:08,920 --> 00:34:12,120 pattern, you can be extremely confident that the molecule has 725 00:34:12,120 --> 00:34:15,040 C2V symmetry. And if the molecule had a 726 00:34:15,040 --> 00:34:18,199 different shape, a different arrangement of atoms. 727 00:34:18,400 --> 00:34:20,800 The math would have spit out completely different numbers. 728 00:34:20,800 --> 00:34:22,800 The symmetry would be different, the character table would be 729 00:34:22,800 --> 00:34:24,440 different, the selection rules would be different. 730 00:34:24,760 --> 00:34:27,880 The pattern of peaks is a direct consequence of the shape. 731 00:34:28,360 --> 00:34:31,600 That is incredibly powerful. We essentially just derive the 732 00:34:31,600 --> 00:34:35,159 blueprint for identifying matter using light without ever 733 00:34:35,159 --> 00:34:37,960 touching the substance itself. That is the power of group 734 00:34:37,960 --> 00:34:40,920 theory. It turns these qualitative ideas 735 00:34:40,920 --> 00:34:45,120 we have about shape into hard quantitative predictions about 736 00:34:45,120 --> 00:34:47,719 experimental data. Before we move on, I want to 737 00:34:47,719 --> 00:34:50,600 touch on one last concept from the notes, the rule of mutual 738 00:34:50,600 --> 00:34:52,920 exclusion. It sounds very definitive. 739 00:34:53,000 --> 00:34:54,880 It is. It's a very useful shortcut for 740 00:34:54,880 --> 00:34:57,200 chemists. It applies only to molecules 741 00:34:57,200 --> 00:34:59,960 that have a very specific type of symmetry element called a 742 00:34:59,960 --> 00:35:02,520 center of inversion. A center of inversion? 743 00:35:02,520 --> 00:35:04,640 What is that? It means that if you started any 744 00:35:04,640 --> 00:35:07,560 atom in the molecule, draw a straight line through the exact 745 00:35:07,560 --> 00:35:10,160 center of the molecule, and keep going that same distance out the 746 00:35:10,160 --> 00:35:12,480 other side, you hit an identical atom. 747 00:35:12,720 --> 00:35:15,840 Like carbon dioxide, CO2, you have oxygen on the left, carbon 748 00:35:15,840 --> 00:35:17,280 in the middle, oxygen on the right. 749 00:35:17,480 --> 00:35:19,960 If I start at the left oxygen and go through the central 750 00:35:19,960 --> 00:35:22,360 carbon, I hit the right oxygen. Exactly. 751 00:35:22,640 --> 00:35:27,880 CO2 has a center of inversion. Our molecule sexy L2H2 does not. 752 00:35:28,240 --> 00:35:31,520 If you start at a chlorine and go through the silicon, you hit 753 00:35:31,520 --> 00:35:33,280 empty space between the hydrogens. 754 00:35:33,600 --> 00:35:36,760 OK, So what is the rule for molecules that have this 755 00:35:36,760 --> 00:35:38,440 feature? The rule of mutual exclusion 756 00:35:38,440 --> 00:35:41,960 states if a molecule has a center of inversion, no 757 00:35:41,960 --> 00:35:45,040 vibrational mode can be active in both IR and Roman. 758 00:35:45,040 --> 00:35:48,960 It's strictly either no overlap. Strictly, if the IR detective 759 00:35:48,960 --> 00:35:51,280 sees it, the ramen detected is blind to it. 760 00:35:51,400 --> 00:35:54,920 If ramen sees it, IR is blind. They are mutually exclusive. 761 00:35:54,960 --> 00:35:58,320 So let's connect that back to our case for sexy L2H2. 762 00:35:58,320 --> 00:36:02,040 We saw that eight of its vibrations, all the A1B1 and B2 763 00:36:02,040 --> 00:36:05,640 modes, appeared in both lists. They were active in both IR and 764 00:36:05,640 --> 00:36:07,680 Roman. Which immediately confirms what 765 00:36:07,680 --> 00:36:08,920 we already knew from looking at it. 766 00:36:09,520 --> 00:36:11,040 It does not have a center of inversion. 767 00:36:11,600 --> 00:36:15,000 The fact that the peaks overlap proves the lack of that specific 768 00:36:15,000 --> 00:36:17,400 symmetry element. It's a binary check that 769 00:36:17,400 --> 00:36:19,840 chemists use all the time. It's like a logic puzzle. 770 00:36:20,320 --> 00:36:24,760 I see a peak at 500 wave numbers in both my IR and ramen Spectra. 771 00:36:24,800 --> 00:36:28,480 Therefore I can definitively say my molecule cannot be Centro 772 00:36:28,480 --> 00:36:29,560 symmetric. Precisely. 773 00:36:29,560 --> 00:36:31,800 It's an immediate piece of structural information. 774 00:36:31,960 --> 00:36:34,920 So this brings us to the final section, why this matters. 775 00:36:35,360 --> 00:36:37,920 We've done a lot of math. We've calculated gammas and 776 00:36:37,920 --> 00:36:41,200 subtracted rotations. But practically speaking on the 777 00:36:41,200 --> 00:36:44,520 lab bench, what does this actually tell a working chemist? 778 00:36:44,840 --> 00:36:47,000 At the most fundamental level, it tells us structure. 779 00:36:47,520 --> 00:36:50,800 This whole process is one of our primary tools for identifying 780 00:36:50,800 --> 00:36:53,320 what things are made of. Because different bonds wiggle 781 00:36:53,320 --> 00:36:54,960 at different speeds. Exactly. 782 00:36:55,480 --> 00:36:58,000 Every type of bond has a characteristic frequency. 783 00:36:58,440 --> 00:37:02,080 For example, a carbon oxygen double bond, a CEO carbonyl 784 00:37:02,080 --> 00:37:04,840 group is a very stiff, strong spring. 785 00:37:05,120 --> 00:37:08,320 It almost always vibrates somewhere around 1700 inverse 786 00:37:08,320 --> 00:37:10,000 centimeters. It's a dead giveaway. 787 00:37:10,040 --> 00:37:12,720 And an OH bond, like in water or an alcohol. 788 00:37:12,920 --> 00:37:15,800 That has a very light hydrogen on the end, so it vibrates much, 789 00:37:15,800 --> 00:37:19,080 much faster. You see a big broad peak way up 790 00:37:19,080 --> 00:37:22,600 around 3400 inverse centimeters. So the Spectra is like a parts 791 00:37:22,600 --> 00:37:25,240 list for the molecule. I see a peak at 1700. 792 00:37:25,240 --> 00:37:27,400 OK, I have a carbonyl group. I see a peak at 3400. 793 00:37:27,400 --> 00:37:30,560 I must have an alcohol. Yes, but the group theory part 794 00:37:30,560 --> 00:37:32,520 we just did takes it to the next level. 795 00:37:32,800 --> 00:37:35,880 It tells us how those parts are arranged in 3D space. 796 00:37:36,360 --> 00:37:40,240 Are the 20 bonds on a water molecule symmetric? 797 00:37:40,600 --> 00:37:42,360 Are they bent? Are they linear? 798 00:37:42,920 --> 00:37:46,080 The number of peaks and which ones are IR active versus ramen 799 00:37:46,080 --> 00:37:48,040 active tells us the architecture. 800 00:37:48,120 --> 00:37:52,280 So it connects the parts list to the actual blueprint of the 801 00:37:52,280 --> 00:37:54,120 molecule. That is a perfect way to put it. 802 00:37:54,320 --> 00:37:56,440 You know, when we started this section, the lecture notes 803 00:37:56,440 --> 00:37:58,440 mentioned something called symmetry coordinates. 804 00:37:58,680 --> 00:38:01,200 This is the idea that the atoms don't really vibrate on their 805 00:38:01,200 --> 00:38:02,440 own. That's right, it's a crucial 806 00:38:02,440 --> 00:38:04,400 concept. We found that the vibrations 807 00:38:04,400 --> 00:38:08,600 were things we labeled A1 or B2. That label doesn't say the 808 00:38:08,600 --> 00:38:11,640 silicon chlorine bond vibrates, it says the entire molecule 809 00:38:11,640 --> 00:38:13,800 vibrates in a way that has B2 symmetry. 810 00:38:13,800 --> 00:38:15,040 It's teamwork. It is. 811 00:38:15,400 --> 00:38:18,640 The two chlorine atoms might stretch together in phase. 812 00:38:18,920 --> 00:38:21,680 At the same time the two hydrogens might bend together in 813 00:38:21,680 --> 00:38:24,040 phase. It's a collective motion. 814 00:38:24,520 --> 00:38:27,000 The atoms are mechanically coupled through the bonds. 815 00:38:27,440 --> 00:38:29,720 Like a series of connected pendulums, if you start one 816 00:38:29,720 --> 00:38:31,560 swinging, it affects all the others. 817 00:38:31,560 --> 00:38:34,400 It's a great analogy. If one chlorine tried to move 818 00:38:34,400 --> 00:38:37,200 completely independently, it would drag the silicon atom, 819 00:38:37,520 --> 00:38:40,240 which would in turn drag the other fluorine and both 820 00:38:40,240 --> 00:38:43,400 hydrogens. The only stable, sustainable 821 00:38:43,400 --> 00:38:46,600 ways for the molecule to move are these normal modes of 822 00:38:46,600 --> 00:38:49,920 vibration, where everyone moves in a perfectly synchronized 823 00:38:49,920 --> 00:38:51,480 pattern. So the molecule is a 824 00:38:51,480 --> 00:38:54,040 synchronized swim team. And group theory is just the 825 00:38:54,040 --> 00:38:55,520 choreography notes for the routine. 826 00:38:55,600 --> 00:38:57,400 It tells you who moves where and when. 827 00:38:57,560 --> 00:39:00,000 Well, we have covered a massive amount of ground today. 828 00:39:00,000 --> 00:39:02,440 We started with a simple spring in a garage. 829 00:39:02,520 --> 00:39:05,640 We found out that the entire universe is shivering with 0 830 00:39:05,640 --> 00:39:08,160 point energy, that nothing is ever truly still. 831 00:39:08,360 --> 00:39:11,840 We met our two detectives, IR and Rahman, and learned that one 832 00:39:11,840 --> 00:39:14,520 looks for electrical lopsidedness and the other looks 833 00:39:14,520 --> 00:39:17,960 for molecular squishiness. We entered the matrix, so to 834 00:39:17,960 --> 00:39:22,320 speak, to calculate the degrees of freedom first Segal to H2 and 835 00:39:22,320 --> 00:39:24,280 generated that reducible representation. 836 00:39:24,440 --> 00:39:28,240 We then unblended the smoothie, breaking that complex data down 837 00:39:28,240 --> 00:39:30,480 into its simple symmetry ingredients. 838 00:39:30,480 --> 00:39:33,720 We subtracted the flying and the spinning to isolate the nine 839 00:39:33,720 --> 00:39:35,640 true vibrations, the actual wiggles. 840 00:39:35,680 --> 00:39:39,440 And finally, we use the rules of to predict exactly which 841 00:39:39,440 --> 00:39:43,040 detective would see which dance move and which ones would be 842 00:39:43,040 --> 00:39:45,240 invisible. It is a complete journey from 843 00:39:45,280 --> 00:39:49,480 abstract concept like shape all the way to a concrete observable 844 00:39:49,480 --> 00:39:51,160 prediction you could test in a lab. 845 00:39:51,480 --> 00:39:54,040 It really does change how I look at my coffee cup now, the 846 00:39:54,040 --> 00:39:57,120 caffeine molecules in there, they are doing this complex 847 00:39:57,120 --> 00:40:00,120 three and six dance right now billions and billions of times a 848 00:40:00,120 --> 00:40:01,880 second and. The selection rules we just 849 00:40:01,880 --> 00:40:06,040 worked out are determining right now which photons of light from 850 00:40:06,040 --> 00:40:08,840 the room are being absorbed by your coffee and which are being 851 00:40:08,840 --> 00:40:10,760 scattered. It's happening constantly. 852 00:40:10,760 --> 00:40:14,080 Symmetry is nature's filter. That is the ultimate take away. 853 00:40:14,520 --> 00:40:17,760 Symmetry isn't just about things looking pretty or balanced. 854 00:40:18,160 --> 00:40:21,200 In physics and chemistry, symmetry is the rule book. 855 00:40:21,640 --> 00:40:24,640 It dictates what interactions are allowed to happen and which 856 00:40:24,640 --> 00:40:27,840 ones are strictly forbidden. I think that is a perfect place 857 00:40:27,840 --> 00:40:31,440 to leave it and a provocative thought for you, our listener, 858 00:40:31,600 --> 00:40:35,640 to take away. If symmetry is the rule book for 859 00:40:35,640 --> 00:40:39,960 how light and matter interact, what other fundamental rules of 860 00:40:39,960 --> 00:40:43,840 the universe are secretly just consequences of a deeper, hidden 861 00:40:43,840 --> 00:40:45,800 symmetry? It's a question physicists are 862 00:40:45,800 --> 00:40:47,720 still wrestling with. Professor Camargo's lecture 863 00:40:47,720 --> 00:40:50,120 notes were a beast, but I think we managed to tame them. 864 00:40:50,320 --> 00:40:51,880 We certainly gave it our best shot. 865 00:40:52,240 --> 00:40:54,640 The math holds up. Thank you so much for guiding us 866 00:40:54,640 --> 00:40:56,600 through the matrices and the character tables. 867 00:40:56,800 --> 00:40:59,880 It was a pleasure to explore the molecular dance with you. 868 00:41:00,320 --> 00:41:02,760 And thanks to you, the listener, for sticking with us through the 869 00:41:02,760 --> 00:41:05,200 group theory. Keep vibrating, keep learning, 870 00:41:05,200 --> 00:41:06,960 and we'll see you on the next deep dive.