1 00:00:00,090 --> 00:00:02,631 ANNOUNCER: The following content is provided under a Creative 2 00:00:02,631 --> 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 --> 00:00:18,210 at ocw.mit.edu. 8 00:00:22,220 --> 00:00:23,810 ROBERT FIELD: When I was really young, 9 00:00:23,810 --> 00:00:27,350 I used to go to a television repair store 10 00:00:27,350 --> 00:00:32,780 as often as possible to take home one of the dead chassis. 11 00:00:32,780 --> 00:00:34,820 And then I would take it apart. 12 00:00:34,820 --> 00:00:36,770 And I don't know what I was looking for, 13 00:00:36,770 --> 00:00:40,670 but that was sort of the empirical stuff. 14 00:00:40,670 --> 00:00:43,610 I had no idea how a television worked, 15 00:00:43,610 --> 00:00:46,250 but I was really curious about maybe 16 00:00:46,250 --> 00:00:49,490 I could find it if I just did stuff. 17 00:00:49,490 --> 00:00:53,420 And what we've been talking about 18 00:00:53,420 --> 00:00:58,730 are ways in which we not just generate numbers, like 19 00:00:58,730 --> 00:01:03,630 parts of the television chassis, but insight. 20 00:01:03,630 --> 00:01:09,000 And there are we've talked about three ways so far. 21 00:01:09,000 --> 00:01:15,060 And one is Huckel theory, where Huckel theory is just 22 00:01:15,060 --> 00:01:20,130 a bunch of simple rules and simple ideas for how 23 00:01:20,130 --> 00:01:25,080 do you represent a large family of related molecules. 24 00:01:25,080 --> 00:01:29,730 And the Huckel theory is incredibly simple, 25 00:01:29,730 --> 00:01:33,990 but it enables you to make really sophisticated 26 00:01:33,990 --> 00:01:36,570 conclusions about how things work 27 00:01:36,570 --> 00:01:38,970 and what are the important factors. 28 00:01:38,970 --> 00:01:47,670 And so it's basically a procedure 29 00:01:47,670 --> 00:01:53,180 for distilling insight from random observations. 30 00:01:53,180 --> 00:01:54,870 Or maybe not random observations, 31 00:01:54,870 --> 00:01:59,100 but observations of many properties 32 00:01:59,100 --> 00:02:02,280 of many related molecules. 33 00:02:02,280 --> 00:02:03,675 Then you've seen LCAOMO. 34 00:02:06,590 --> 00:02:12,550 Based on the small variations all treatment of H2 plus, 35 00:02:12,550 --> 00:02:16,280 we got the idea of orbitals and what makes a bond. 36 00:02:16,280 --> 00:02:19,940 And then with the idea that we can describe 37 00:02:19,940 --> 00:02:23,720 the size of atomic orbitals by the energy 38 00:02:23,720 --> 00:02:26,840 below the ionization limit, we can 39 00:02:26,840 --> 00:02:31,430 make similar quantitative predictions based 40 00:02:31,430 --> 00:02:34,370 on this minimal basis set. 41 00:02:34,370 --> 00:02:37,700 So we draw molecular orbital diagrams. 42 00:02:37,700 --> 00:02:40,100 And these molecular orbital diagrams 43 00:02:40,100 --> 00:02:53,980 are especially valuable for isoelectronic and homologous 44 00:02:53,980 --> 00:02:55,180 comparisons. 45 00:02:55,180 --> 00:03:06,740 So isoelectronic would be, let's say, nitrogen, CO, BF. 46 00:03:10,250 --> 00:03:16,220 Molecules with the same number of electrons. 47 00:03:16,220 --> 00:03:23,414 And homologous would be something like CO, SIO, GEO, 48 00:03:23,414 --> 00:03:25,900 and so on. 49 00:03:25,900 --> 00:03:31,210 And with the concept of orbital size being related 50 00:03:31,210 --> 00:03:34,150 to the ionization energy of the atoms, 51 00:03:34,150 --> 00:03:37,660 we can make a lot of very useful comparisons. 52 00:03:37,660 --> 00:03:38,980 And so we develop insight. 53 00:03:41,810 --> 00:03:45,530 Now LCAOMO is a variation of method, 54 00:03:45,530 --> 00:03:49,130 and it can be a very large variation of method. 55 00:03:49,130 --> 00:03:51,720 And it could become of an issue. 56 00:03:51,720 --> 00:03:54,380 But it's usually not atomic orbitals. 57 00:03:54,380 --> 00:03:57,110 It's just Gaussian orbitals or something, 58 00:03:57,110 --> 00:03:59,390 which is computationally convenient. 59 00:03:59,390 --> 00:04:09,620 And so it can be a procedure to rationalize experience and make 60 00:04:09,620 --> 00:04:15,350 predictions as you go from the same number of electrons 61 00:04:15,350 --> 00:04:19,399 but increasing polarizability, polarity. 62 00:04:19,399 --> 00:04:23,510 And these-- this is where you develop chemical insight. 63 00:04:23,510 --> 00:04:24,927 And it's really exciting. 64 00:04:24,927 --> 00:04:26,510 And it's what was missing when I would 65 00:04:26,510 --> 00:04:31,760 take these chassis from the repair shop 66 00:04:31,760 --> 00:04:33,980 and chop them up into pieces. 67 00:04:33,980 --> 00:04:38,250 And oh yeah, there's a magnet in there and things like that. 68 00:04:38,250 --> 00:04:41,540 It's much deeper. 69 00:04:41,540 --> 00:04:45,260 And now perturbation theory is a kind of a different thing 70 00:04:45,260 --> 00:04:47,090 altogether. 71 00:04:47,090 --> 00:04:50,480 Perturbation theory says we do an experiment. 72 00:04:50,480 --> 00:04:53,720 We measure something, and we turn it 73 00:04:53,720 --> 00:04:55,730 into something we really wanted. 74 00:04:55,730 --> 00:04:59,090 The Rolling Stones are telling us 75 00:04:59,090 --> 00:05:03,740 that if you want to know how a particular property depends 76 00:05:03,740 --> 00:05:07,190 on internuclear distance, and you can't directly 77 00:05:07,190 --> 00:05:10,580 measure the property as a function of internuclear 78 00:05:10,580 --> 00:05:12,800 distance, well maybe you can measure it 79 00:05:12,800 --> 00:05:16,420 as a function of vibrational and rotational quantum numbers. 80 00:05:16,420 --> 00:05:18,570 And that's what perturbation theory does. 81 00:05:18,570 --> 00:05:21,260 It tells you how to get from what you observe to what 82 00:05:21,260 --> 00:05:22,370 you really want to know. 83 00:05:25,670 --> 00:05:34,010 And it can be horrible in terms of the algebraic exercises you 84 00:05:34,010 --> 00:05:39,120 have to go through in order to get what you really want 85 00:05:39,120 --> 00:05:41,110 or from what you-- 86 00:05:41,110 --> 00:05:41,610 yes. 87 00:05:44,280 --> 00:05:48,580 And today's lecture is going to be an example of not the worst 88 00:05:48,580 --> 00:05:50,710 thing you could ever do with perturbation theory, 89 00:05:50,710 --> 00:05:53,760 but pretty close to the worst. 90 00:05:53,760 --> 00:05:59,730 OK, but all three of the first things are related, 91 00:05:59,730 --> 00:06:04,320 are associated with getting insight 92 00:06:04,320 --> 00:06:11,310 from either a crude calculation refined against observations 93 00:06:11,310 --> 00:06:14,790 or just the observations and reducing those observations 94 00:06:14,790 --> 00:06:17,550 to something really neat about how things work. 95 00:06:20,470 --> 00:06:24,215 Then the next two lectures will be given by Professor Van Van 96 00:06:24,215 --> 00:06:25,690 Voorhis and it's-- 97 00:06:25,690 --> 00:06:31,360 they will be on ab initio theory, where basically you 98 00:06:31,360 --> 00:06:33,010 don't assume anything. 99 00:06:33,010 --> 00:06:37,870 You don't do anything except solve for the exact energy 100 00:06:37,870 --> 00:06:41,780 levels and wave functions. 101 00:06:41,780 --> 00:06:44,700 Now this is not possible. 102 00:06:44,700 --> 00:06:49,260 Directly you can do this by making approximations, doing 103 00:06:49,260 --> 00:06:51,240 an enormous variational calculation. 104 00:06:53,930 --> 00:07:00,830 Now many people think, well why bother with approximate methods 105 00:07:00,830 --> 00:07:04,190 when you can get the truth? 106 00:07:04,190 --> 00:07:11,950 And the answer is the truth is no more valuable than the parts 107 00:07:11,950 --> 00:07:15,550 of a disabled television. 108 00:07:15,550 --> 00:07:17,030 It's-- they're stuff. 109 00:07:17,030 --> 00:07:19,340 But there's no insight there. 110 00:07:19,340 --> 00:07:24,290 And quantum chemists who do these calculations 111 00:07:24,290 --> 00:07:26,810 are not just generating numbers. 112 00:07:26,810 --> 00:07:31,710 They're trying to explain how things work. 113 00:07:31,710 --> 00:07:36,050 And it's the same business, it's just the tools are different. 114 00:07:36,050 --> 00:07:38,990 The goals are not just getting-- 115 00:07:38,990 --> 00:07:43,190 in spectroscopy we can measure things to 10 or 11 digits. 116 00:07:43,190 --> 00:07:44,990 And nobody cares. 117 00:07:44,990 --> 00:07:48,080 I mean, how many digits? 118 00:07:48,080 --> 00:07:52,670 It's a little bit challenging to remember telephone numbers. 119 00:07:52,670 --> 00:07:59,180 And having big tables of 10-digit numbers, so what? 120 00:07:59,180 --> 00:08:01,130 But what do the numbers tell you? 121 00:08:01,130 --> 00:08:03,800 And this quest-- the same question 122 00:08:03,800 --> 00:08:06,620 is asked by good quantum chemists 123 00:08:06,620 --> 00:08:09,650 by doing a series of calculations 124 00:08:09,650 --> 00:08:13,610 where they turn on and off certain terms 125 00:08:13,610 --> 00:08:14,990 in the Hamiltonian. 126 00:08:14,990 --> 00:08:16,940 So it's the same thing. 127 00:08:16,940 --> 00:08:23,120 I'm an experimentalist, but many people who are experimentalists 128 00:08:23,120 --> 00:08:26,510 think I'm a theorist because I do weird stuff. 129 00:08:26,510 --> 00:08:27,590 But I'm not a theorist. 130 00:08:27,590 --> 00:08:29,610 Troy is a theorist. 131 00:08:29,610 --> 00:08:34,309 And he does weird stuff too, but he's not an experimentalist. 132 00:08:34,309 --> 00:08:36,549 And we're both after insight. 133 00:08:36,549 --> 00:08:41,140 And OK, so let's just-- 134 00:08:44,500 --> 00:08:47,510 I guess I'll just launch it through the perturbation 135 00:08:47,510 --> 00:08:48,010 theory. 136 00:08:51,460 --> 00:08:51,960 OK. 137 00:08:56,800 --> 00:09:01,150 But you do want to understand the difference in how 138 00:09:01,150 --> 00:09:04,810 these different methods, these approximate methods, work, 139 00:09:04,810 --> 00:09:06,510 and what they're good for, OK? 140 00:09:11,090 --> 00:09:16,700 So the goal of perturbation theory 141 00:09:16,700 --> 00:09:28,370 is to go from molecular constants 142 00:09:28,370 --> 00:09:32,570 to structural constants, or structural parameters. 143 00:09:41,370 --> 00:09:44,360 Molecular constants are like rotational constant, 144 00:09:44,360 --> 00:09:48,500 vibrational constant, stuff that you get by fitting the energy 145 00:09:48,500 --> 00:09:53,600 levels you observe to a dumb empirical expression, a power 146 00:09:53,600 --> 00:09:56,870 series in quantum numbers. 147 00:09:56,870 --> 00:09:59,850 Now there's lots of dumb empirical expressions 148 00:09:59,850 --> 00:10:03,190 you could use, and some are better than others. 149 00:10:03,190 --> 00:10:06,180 And perturbation theory will often tell you 150 00:10:06,180 --> 00:10:09,070 what is the right way to do it. 151 00:10:09,070 --> 00:10:11,160 So you go from molecular constants 152 00:10:11,160 --> 00:10:14,710 to things in the potential. 153 00:10:14,710 --> 00:10:17,790 So this is the displacement from equilibrium, 154 00:10:17,790 --> 00:10:19,650 and we would like to know something 155 00:10:19,650 --> 00:10:21,900 about how a molecule works. 156 00:10:21,900 --> 00:10:25,270 And there also might be other constants, like spin orbit, 157 00:10:25,270 --> 00:10:28,920 and hyperfine, dipole moment. 158 00:10:28,920 --> 00:10:33,480 And so let's just say we have some observable, 159 00:10:33,480 --> 00:10:39,390 and it is also a function of coordinate. 160 00:10:39,390 --> 00:10:44,490 And we'd like to know what that function is. 161 00:10:44,490 --> 00:10:49,770 But what we are able to do is measure energy levels 162 00:10:49,770 --> 00:10:51,930 as a function of quantum numbers. 163 00:10:51,930 --> 00:10:55,800 And so the information that we really want, 164 00:10:55,800 --> 00:10:59,840 the potential or the internuclear 165 00:10:59,840 --> 00:11:03,860 distance dependence of some electronic property, that's 166 00:11:03,860 --> 00:11:07,610 all gotten from what we can observe 167 00:11:07,610 --> 00:11:09,690 via perturbation theory. 168 00:11:09,690 --> 00:11:11,960 It's a very powerful tool. 169 00:11:11,960 --> 00:11:17,570 And it's not pretty, but it always works. 170 00:11:17,570 --> 00:11:21,050 And it's a good basis for insight. 171 00:11:24,310 --> 00:11:31,460 So perturbation theory is a fit model. 172 00:11:36,200 --> 00:11:40,700 The other methods, one doesn't generally do a least-squares 173 00:11:40,700 --> 00:11:43,530 fit to the-- 174 00:11:43,530 --> 00:11:48,520 of the adjustable parameters in Huckel theory 175 00:11:48,520 --> 00:11:53,750 to determine the properties of a molecule. 176 00:11:53,750 --> 00:11:56,780 One just says, OK, I'm going to try these 177 00:11:56,780 --> 00:12:01,460 because I think that attaching an electro-negative atom 178 00:12:01,460 --> 00:12:07,640 to a carbon atom is going to do something that I can predict. 179 00:12:07,640 --> 00:12:11,360 And maybe it's going to tell me some surprises about how 180 00:12:11,360 --> 00:12:14,210 its influence is not just where it's attached, 181 00:12:14,210 --> 00:12:16,250 but other places in the molecule. 182 00:12:16,250 --> 00:12:19,160 You know this from your first organic courses. 183 00:12:19,160 --> 00:12:21,230 You know all sorts of tricks to be 184 00:12:21,230 --> 00:12:25,520 able to predict reactivity and things that are related. 185 00:12:25,520 --> 00:12:28,710 But Huckel theory is not a fit model. 186 00:12:28,710 --> 00:12:32,480 LCAOMO theory is not a fit model. 187 00:12:32,480 --> 00:12:34,130 These are experience-based. 188 00:12:34,130 --> 00:12:37,280 And you integrate all sorts of stuff 189 00:12:37,280 --> 00:12:41,440 that you learn from comparing molecules. 190 00:12:41,440 --> 00:12:43,520 And the comparing of molecules is 191 00:12:43,520 --> 00:12:51,220 what makes up for the deteriable lists of the approximations. 192 00:12:51,220 --> 00:12:53,830 And your job is to hone your insight. 193 00:12:53,830 --> 00:12:59,080 And so you have nothing to protect you except the truth. 194 00:12:59,080 --> 00:13:00,730 You can observe molecules, but you 195 00:13:00,730 --> 00:13:06,630 can't be guided to something which is true, because there 196 00:13:06,630 --> 00:13:08,530 is no calculation. 197 00:13:08,530 --> 00:13:12,350 There's nothing exact and there's nothing complete. 198 00:13:12,350 --> 00:13:16,640 But in quantum chemistry you can get really close to the truth. 199 00:13:16,640 --> 00:13:19,142 But you don't know anything about why. 200 00:13:19,142 --> 00:13:20,600 And so you're doing the same thing, 201 00:13:20,600 --> 00:13:22,400 but from the opposite ends. 202 00:13:22,400 --> 00:13:25,460 OK, but perturbation theory is special 203 00:13:25,460 --> 00:13:29,360 because you take a huge amount of highly 204 00:13:29,360 --> 00:13:33,380 accurate experimental data of various types and you fit it. 205 00:13:36,960 --> 00:13:40,140 And so I want to talk about this. 206 00:13:40,140 --> 00:13:42,650 So you have the Hamiltonian. 207 00:13:42,650 --> 00:13:44,720 It's a matrix, and it's expressed 208 00:13:44,720 --> 00:13:47,640 in terms of parameters. 209 00:13:47,640 --> 00:13:51,380 Let's just use notation p, p of i. 210 00:13:51,380 --> 00:13:53,270 A set of parameters. 211 00:13:56,310 --> 00:13:59,030 So this is not a quantum chemical, Hamiltonian. 212 00:13:59,030 --> 00:14:02,510 It's a thing where we say there are certain degrees of freedom, 213 00:14:02,510 --> 00:14:07,160 and each one is controlled by a number of parameters. 214 00:14:09,900 --> 00:14:11,150 And we have the energy levels. 215 00:14:14,790 --> 00:14:17,370 And so what you do is you say let's 216 00:14:17,370 --> 00:14:21,070 choose a set of these parameters, 217 00:14:21,070 --> 00:14:26,480 and calculate the energy levels, and compare them 218 00:14:26,480 --> 00:14:29,070 to the observed energy levels. 219 00:14:29,070 --> 00:14:31,970 And it's not-- the first try, it's not going to be good. 220 00:14:31,970 --> 00:14:35,960 And then you say, OK, now I've got to do a least-squares fit. 221 00:14:35,960 --> 00:14:41,680 I have to vary the parameters in this matrix Hamiltonian 222 00:14:41,680 --> 00:14:45,310 to match the energy levels and maybe 223 00:14:45,310 --> 00:14:49,710 to match other things that you can observe in the experiment. 224 00:14:49,710 --> 00:14:52,250 But it's a complicated least-squares fit, 225 00:14:52,250 --> 00:14:55,850 because you have a matrix, and you want to diagonalize it. 226 00:14:55,850 --> 00:14:58,760 How are the parameters related to the eigenvalues 227 00:14:58,760 --> 00:15:00,370 and eigenvectors? 228 00:15:00,370 --> 00:15:03,590 You don't know. 229 00:15:03,590 --> 00:15:06,840 But when you're close, then you can start fitting these things. 230 00:15:06,840 --> 00:15:08,670 Then you can say, yeah, I know. 231 00:15:08,670 --> 00:15:11,720 And then once you've done that, you have not just 232 00:15:11,720 --> 00:15:16,870 the energy levels, but you have the wave functions. 233 00:15:20,510 --> 00:15:23,930 Now again, you don't know that there is something 234 00:15:23,930 --> 00:15:26,750 missing in your Hamiltonian. 235 00:15:26,750 --> 00:15:28,700 But you know that from the data you 236 00:15:28,700 --> 00:15:33,674 input you can match all the energy levels. 237 00:15:33,674 --> 00:15:35,340 But you didn't put it in all the levels. 238 00:15:35,340 --> 00:15:37,890 You put in just the ones that you measured. 239 00:15:37,890 --> 00:15:39,390 But you are arrogant. 240 00:15:39,390 --> 00:15:45,200 And you think, well, maybe I measured more than I-- 241 00:15:45,200 --> 00:15:47,490 I determined more than I measured, 242 00:15:47,490 --> 00:15:51,300 that I can extend the measurements to other things. 243 00:15:51,300 --> 00:15:53,970 And I told you at the beginning of the course you cannot 244 00:15:53,970 --> 00:15:56,500 observe the wave function. 245 00:15:56,500 --> 00:16:01,260 But if you do this, if you do a least-squares fit and match 246 00:16:01,260 --> 00:16:05,640 the energy levels, you have a pretty darn good representation 247 00:16:05,640 --> 00:16:07,980 of the wave function. 248 00:16:07,980 --> 00:16:10,770 And you can use that to calculate other stuff, 249 00:16:10,770 --> 00:16:11,745 especially dynamics. 250 00:16:14,900 --> 00:16:18,200 Remember dynamics, at least if the Hamiltonian is 251 00:16:18,200 --> 00:16:21,410 independent of time, the dynamics is going 252 00:16:21,410 --> 00:16:26,660 to be a function of x and t. 253 00:16:26,660 --> 00:16:32,100 But you start with the initial preparation. 254 00:16:32,100 --> 00:16:34,800 If you know that, then you, because you know the energy 255 00:16:34,800 --> 00:16:43,300 levels, you can calculate the full x of t for all time. 256 00:16:43,300 --> 00:16:44,820 So that's pretty powerful. 257 00:16:44,820 --> 00:16:49,860 If you fitted enough stuff, if your Hamiltonian has 258 00:16:49,860 --> 00:16:52,620 the important things in it, then you 259 00:16:52,620 --> 00:16:57,050 are basically able to do anything you want, 260 00:16:57,050 --> 00:17:00,530 whether it's static or dynamic. 261 00:17:00,530 --> 00:17:04,460 And if the Hamiltonian is time dependent, you can do that too, 262 00:17:04,460 --> 00:17:05,930 I just haven't showed you how yet. 263 00:17:08,980 --> 00:17:11,650 OK, so I really like perturbation theory, 264 00:17:11,650 --> 00:17:17,089 because it directly that deals with whatever data you have. 265 00:17:17,089 --> 00:17:21,130 And out from that, you get this thing 266 00:17:21,130 --> 00:17:24,720 which tells you everything, unless there's something 267 00:17:24,720 --> 00:17:28,560 that you didn't sample, that you didn't know you didn't sample. 268 00:17:28,560 --> 00:17:32,070 And then you discover that your data and the predictions 269 00:17:32,070 --> 00:17:33,660 are not in agreement. 270 00:17:33,660 --> 00:17:34,930 And then that's when you-- 271 00:17:34,930 --> 00:17:36,940 you don't go home and say, oh, I screwed up. 272 00:17:36,940 --> 00:17:38,040 I got to go take a nap. 273 00:17:38,040 --> 00:17:40,500 I don't-- you go home and you say, let's celebrate, 274 00:17:40,500 --> 00:17:44,800 because I discovered something that was really missing. 275 00:17:44,800 --> 00:17:47,730 And that's what we want to do. 276 00:17:47,730 --> 00:17:54,150 OK, so I have to mess around in the mud here. 277 00:17:54,150 --> 00:17:56,790 Because the perturbation theory you've 278 00:17:56,790 --> 00:17:59,880 done so far has been, perhaps, ugly, 279 00:17:59,880 --> 00:18:02,820 but it's been kind of simple because it's 280 00:18:02,820 --> 00:18:05,580 basically vibration. 281 00:18:05,580 --> 00:18:08,200 And now we've got vibration and rotation. 282 00:18:08,200 --> 00:18:10,320 And we have to combine the two, and we 283 00:18:10,320 --> 00:18:14,250 have to see whether there's something special that we 284 00:18:14,250 --> 00:18:15,900 can get from the combination. 285 00:18:15,900 --> 00:18:17,880 And you bet there is. 286 00:18:17,880 --> 00:18:22,170 OK, so we're going to look at the energy 287 00:18:22,170 --> 00:18:27,180 levels of a non-rigid, non-harmonic, or anharmonic 288 00:18:27,180 --> 00:18:28,260 oscillator. 289 00:18:28,260 --> 00:18:31,260 And we're going to find out how we generate 290 00:18:31,260 --> 00:18:35,550 from an expression of the potential energy 291 00:18:35,550 --> 00:18:37,980 surface or potential energy curve, 292 00:18:37,980 --> 00:18:41,460 because what I'm going to be talking about is diatomics. 293 00:18:41,460 --> 00:18:44,700 But it all extends to polyatomics. 294 00:18:44,700 --> 00:18:50,520 And we're going to calculate the relationships 295 00:18:50,520 --> 00:18:53,980 between what you observed and what you want to know. 296 00:18:53,980 --> 00:18:59,010 So for a diatonic molecule, we have a dumb representation 297 00:18:59,010 --> 00:19:03,090 of the quantum numbers. 298 00:19:03,090 --> 00:19:04,820 And we can write it. 299 00:19:04,820 --> 00:19:09,030 And I'm just really a stubborn person 300 00:19:09,030 --> 00:19:16,280 about spectroscopic notation, so I include this Hc convert 301 00:19:16,280 --> 00:19:19,790 wave number unit, the reciprocal centimeter units to energy. 302 00:19:19,790 --> 00:19:23,220 But you'll never catch me doing it in real life. 303 00:19:23,220 --> 00:19:26,210 But that leads to all sorts of algebraic errors 304 00:19:26,210 --> 00:19:28,760 that I don't know I'm making because I'm 305 00:19:28,760 --> 00:19:30,460 so unused to doing this. 306 00:19:30,460 --> 00:19:37,670 OK, so we have an expression for the vibrational energy levels. 307 00:19:37,670 --> 00:19:39,770 And I can't apolo-- 308 00:19:39,770 --> 00:19:44,060 I can't explain why we use four letters 309 00:19:44,060 --> 00:19:49,350 to represent one symbol, but that's the traditional thing. 310 00:19:49,350 --> 00:19:51,130 And for polyatomic molecules, which 311 00:19:51,130 --> 00:19:53,500 came after diatomic molecules, we 312 00:19:53,500 --> 00:19:55,270 start using only two letters. 313 00:19:55,270 --> 00:19:58,990 Instead of omega XE, we just use XE. 314 00:19:58,990 --> 00:20:03,910 But since I'm a diatomician, I'm going to do this sort of thing. 315 00:20:03,910 --> 00:20:07,780 And that's v plus 1/2 squared. 316 00:20:07,780 --> 00:20:14,560 And then the next term is omega YE v plus a 1/2 cubed. 317 00:20:17,610 --> 00:20:24,630 OK, and so then that's the vibrational part. 318 00:20:24,630 --> 00:20:26,730 And then we have the rotational part. 319 00:20:26,730 --> 00:20:31,170 The equilibrium internuclear distant rotational constant 320 00:20:31,170 --> 00:20:35,880 minus alpha e times v plus 1/2. 321 00:20:40,680 --> 00:20:42,370 Times JJ plus 1. 322 00:20:45,530 --> 00:20:49,340 And then minus DE, the centrifugal distortion 323 00:20:49,340 --> 00:20:57,410 constant, times JJ plus 1 quantity squared. 324 00:20:57,410 --> 00:20:59,750 And we can have more constants here. 325 00:20:59,750 --> 00:21:01,520 We can have additional constants there. 326 00:21:01,520 --> 00:21:03,920 But basically we have constants, which 327 00:21:03,920 --> 00:21:07,850 are known by multiplying a particular combination 328 00:21:07,850 --> 00:21:13,900 of vibrational and rotational constants, quantum numbers. 329 00:21:13,900 --> 00:21:16,630 And so these are the things we can determine, 330 00:21:16,630 --> 00:21:18,640 or sometimes we determine some of these 331 00:21:18,640 --> 00:21:21,640 and we want to get a prediction of the others 332 00:21:21,640 --> 00:21:24,436 because they're outside of our observation. 333 00:21:29,300 --> 00:21:38,210 And the potential is going to be dependent on the displacement 334 00:21:38,210 --> 00:21:42,540 from equilibrium and the rotational constant. 335 00:21:42,540 --> 00:21:46,050 And so there will be terms in the potential, 336 00:21:46,050 --> 00:21:57,690 like 1/2 KQ squared plus 1/6 AQ cubed, et cetera. 337 00:21:57,690 --> 00:21:59,400 So we have another dumb power series. 338 00:22:01,980 --> 00:22:05,880 And fortunately, we know that as we go up 339 00:22:05,880 --> 00:22:09,640 in the displace-- in this displacement coordinate, 340 00:22:09,640 --> 00:22:12,700 the coefficients get to be really small. 341 00:22:12,700 --> 00:22:17,210 And so we don't need to have a lot of these parameters. 342 00:22:17,210 --> 00:22:22,420 But we should have at least two, the harmonic and the cubic one. 343 00:22:22,420 --> 00:22:27,670 And we know for molecules the cubic parameter gives you 344 00:22:27,670 --> 00:22:32,450 a potential curve, which resembles reality. 345 00:22:32,450 --> 00:22:34,960 It's hard wall here, soft wall there. 346 00:22:34,960 --> 00:22:36,790 Molecules break. 347 00:22:36,790 --> 00:22:42,380 Now this has the unfortunate property of doing that. 348 00:22:42,380 --> 00:22:46,880 And so it's got death built into it. 349 00:22:46,880 --> 00:22:50,320 But it doesn't really matter, because you basically 350 00:22:50,320 --> 00:22:54,190 are looking at increasingly wide regions of the potential. 351 00:22:54,190 --> 00:22:56,410 And the fact that it does something terrible 352 00:22:56,410 --> 00:22:59,960 in your simple representation is almost irrelevant. 353 00:22:59,960 --> 00:23:02,380 But you have to be aware that you 354 00:23:02,380 --> 00:23:04,400 can tunnel through barriers. 355 00:23:04,400 --> 00:23:07,470 And when you start being able to tunnel through barriers, 356 00:23:07,470 --> 00:23:10,480 and it depends on how close you are to the top of the barrier 357 00:23:10,480 --> 00:23:14,230 and how wide it is, then you start seeing effects. 358 00:23:14,230 --> 00:23:18,970 And then this term is going to give nonsense. 359 00:23:18,970 --> 00:23:21,000 But there is a domain of goodness 360 00:23:21,000 --> 00:23:23,190 where you don't have to worry about that. 361 00:23:26,670 --> 00:23:29,100 So we have this, and then we have 362 00:23:29,100 --> 00:23:33,330 the rotational constant kind of contributing 363 00:23:33,330 --> 00:23:43,270 to the effect of potential, which is HCV of R JJ plus 1. 364 00:23:43,270 --> 00:23:45,010 Now here I've got-- 365 00:23:45,010 --> 00:23:47,560 here I have coordinates. 366 00:23:47,560 --> 00:23:49,810 Here I have quantum numbers. 367 00:23:49,810 --> 00:23:51,205 How can I do that? 368 00:23:51,205 --> 00:23:58,960 It's because central force problems can be represented 369 00:23:58,960 --> 00:24:00,880 in a universal form. 370 00:24:00,880 --> 00:24:04,150 And so we can just integrate over the angular part 371 00:24:04,150 --> 00:24:06,790 of the problem. 372 00:24:06,790 --> 00:24:09,400 And so now we have a mixed representation. 373 00:24:09,400 --> 00:24:17,200 But this guy is still a function of R or Q. And so this v of R 374 00:24:17,200 --> 00:24:19,090 is an operator. 375 00:24:19,090 --> 00:24:20,740 It's not just a constant. 376 00:24:20,740 --> 00:24:24,521 Some people call it the rotational constant operator. 377 00:24:24,521 --> 00:24:25,145 Kind of stupid. 378 00:24:28,950 --> 00:24:33,500 And so our job is now to figure out 379 00:24:33,500 --> 00:24:38,330 how to get from this parameter rise potential 380 00:24:38,330 --> 00:24:39,270 to the energy levels. 381 00:24:42,309 --> 00:24:44,100 Of course, we start with the energy levels. 382 00:24:44,100 --> 00:24:46,410 And we're going to want to determine the parameters 383 00:24:46,410 --> 00:24:48,480 in the potential. 384 00:24:48,480 --> 00:24:53,450 But you've got to have that connection. 385 00:24:53,450 --> 00:25:00,220 So this rotational operator has some constants in front of it. 386 00:25:09,560 --> 00:25:11,870 And this is in wave number of units, 387 00:25:11,870 --> 00:25:14,000 because I'm a spectroscopist. 388 00:25:14,000 --> 00:25:21,070 And we have the vibrational frequency, 1 over 2 pi C. K 389 00:25:21,070 --> 00:25:24,620 over mu square root. 390 00:25:24,620 --> 00:25:26,970 Also in wave number of units. 391 00:25:26,970 --> 00:25:31,520 Now the first problem is we don't like R, 392 00:25:31,520 --> 00:25:34,370 because harmonic oscillators are expressed 393 00:25:34,370 --> 00:25:37,620 in terms of displacement from equilibrium. 394 00:25:37,620 --> 00:25:43,841 And so we know that Q is equal to R minus RE, 395 00:25:43,841 --> 00:25:49,790 or R is equal to Q plus RE. 396 00:25:49,790 --> 00:25:53,990 So we want to replace this operator R by this, 397 00:25:53,990 --> 00:25:55,250 an operator plus a constant. 398 00:25:58,050 --> 00:26:02,800 So 1 over R squared can be expressed as 1 399 00:26:02,800 --> 00:26:08,930 over RE squared, the equivalent internucleus, plus 1 over Q 400 00:26:08,930 --> 00:26:09,690 over RE. 401 00:26:13,338 --> 00:26:15,650 Plus [INAUDIBLE] squared. 402 00:26:18,580 --> 00:26:20,680 So this is the power series expansion, 403 00:26:20,680 --> 00:26:23,662 but we only keep the first term generally. 404 00:26:26,260 --> 00:26:28,540 And so now we're ready to go to work. 405 00:26:28,540 --> 00:26:32,830 And we can say we have B as a function of Q 406 00:26:32,830 --> 00:26:39,670 is equal to B of E, the equilibrium value 1 minus 2Q 407 00:26:39,670 --> 00:26:47,940 or RE plus 3Q squared over RE squared. 408 00:26:47,940 --> 00:26:51,490 And there's more terms, but this is enough. 409 00:26:51,490 --> 00:26:53,566 You bet it's enough. 410 00:26:53,566 --> 00:26:54,940 By the time this lecture is over, 411 00:26:54,940 --> 00:26:58,250 you're going to want not ever to see this stuff again. 412 00:26:58,250 --> 00:27:02,140 OK, so we have now a representation 413 00:27:02,140 --> 00:27:04,720 of the rotational operator in terms 414 00:27:04,720 --> 00:27:09,390 of a constant term, a linear term, and a squared term in Q. 415 00:27:09,390 --> 00:27:13,880 And we know how to do matrix elements of Qs, right? 416 00:27:13,880 --> 00:27:15,660 So we know we're in business. 417 00:27:15,660 --> 00:27:20,880 And we have the potential as another power series in Qs. 418 00:27:20,880 --> 00:27:22,770 So everything is going to come together. 419 00:27:22,770 --> 00:27:28,050 It's just going to be will the migraine wipe out your ability 420 00:27:28,050 --> 00:27:30,900 to pay any attention to this after we're done. 421 00:27:30,900 --> 00:27:36,810 OK, so I don't want to cover that up yet. 422 00:27:36,810 --> 00:27:42,390 So we know that this displacement operator 423 00:27:42,390 --> 00:27:47,430 can be replaced by the creation and annihilation operators. 424 00:27:47,430 --> 00:27:56,820 And so we have 4 pi C, mu omega E square root. 425 00:27:56,820 --> 00:27:59,190 A plus A dagger are friends. 426 00:27:59,190 --> 00:28:03,140 These guys, we hardly need to take a deep breath 427 00:28:03,140 --> 00:28:04,710 to know what to do with those. 428 00:28:08,660 --> 00:28:14,730 And so we're going to want to express all of the-- 429 00:28:14,730 --> 00:28:18,930 all of the actors here, these guys and those guys, 430 00:28:18,930 --> 00:28:24,660 in terms of expressions in the creation and annihilation 431 00:28:24,660 --> 00:28:26,350 operators. 432 00:28:26,350 --> 00:28:29,040 So for the rotational constant expansion, 433 00:28:29,040 --> 00:28:34,370 we're going to need something like 2Q over RE. 434 00:28:34,370 --> 00:28:36,870 And that, after a little bit of algebra, 435 00:28:36,870 --> 00:28:48,300 is 4BE over omega E square root times A plus A dagger. 436 00:28:48,300 --> 00:28:54,250 And we need 3Q squared over RE squared. 437 00:28:54,250 --> 00:29:05,030 And that's going to be 3BE over omega E, A plus A dagger 438 00:29:05,030 --> 00:29:07,640 squared. 439 00:29:07,640 --> 00:29:11,450 So all of the terms can be expressed, 440 00:29:11,450 --> 00:29:15,500 can be reduced to constants that we care about, 441 00:29:15,500 --> 00:29:19,340 or that are easily measured, and these powers of As 442 00:29:19,340 --> 00:29:21,500 and a daggers. 443 00:29:21,500 --> 00:29:25,340 Now this is-- you have notes, printed notes. 444 00:29:25,340 --> 00:29:27,200 And I'm following them pretty well, 445 00:29:27,200 --> 00:29:34,480 although there are a few typos, which I hope to correct. 446 00:29:34,480 --> 00:29:43,810 And so we do a similar thing for the terms here. 447 00:29:43,810 --> 00:29:45,476 And we're only going to keep-- 448 00:29:45,476 --> 00:29:46,850 we're only going to look at this. 449 00:29:46,850 --> 00:29:48,730 We're not going to go to Q 1/4. 450 00:29:48,730 --> 00:29:51,540 But this is bad enough. 451 00:29:51,540 --> 00:29:56,030 OK, and this is part of the zero order Hamiltonian. 452 00:29:56,030 --> 00:30:00,700 So we're going to have another term, 453 00:30:00,700 --> 00:30:08,960 and that's going to be 1/6 little a Q cubed. 454 00:30:08,960 --> 00:30:13,065 OK, so that's-- little a is the anharmonicity parameter. 455 00:30:13,065 --> 00:30:13,940 And what is its sign? 456 00:30:18,090 --> 00:30:22,560 Can we know its sign from experiment? 457 00:30:22,560 --> 00:30:23,960 You said yes. 458 00:30:23,960 --> 00:30:25,816 Why do you say yes? 459 00:30:25,816 --> 00:30:27,001 AUDIENCE: [INAUDIBLE] 460 00:30:27,001 --> 00:30:28,000 ROBERT FIELD: I'm sorry? 461 00:30:28,000 --> 00:30:29,788 AUDIENCE: It's cubed, so it's odd. 462 00:30:29,788 --> 00:30:32,860 So there's [INAUDIBLE]. 463 00:30:32,860 --> 00:30:34,810 ROBERT FIELD: The only way you know 464 00:30:34,810 --> 00:30:40,730 the sign of an off diagonal matrix element is-- 465 00:30:40,730 --> 00:30:45,950 I'm sorry, is if there is a diagonal element of it that 466 00:30:45,950 --> 00:30:51,030 can be put into the E, the first order correction to the energy. 467 00:30:51,030 --> 00:30:52,590 As soon as you have to square it when 468 00:30:52,590 --> 00:30:56,160 you do second order perturbation theory, you've lost the sign. 469 00:30:56,160 --> 00:31:00,780 But here we know the sign because of physical insight. 470 00:31:00,780 --> 00:31:03,840 We know that this kind of a potential would be nonsense. 471 00:31:06,750 --> 00:31:13,045 So we know that A has a sign and it's negative. 472 00:31:13,045 --> 00:31:23,180 OK, so we have a over 6 times our favorite parameter 473 00:31:23,180 --> 00:31:30,420 here to the 3/2 power times A plus A dagger cubed. 474 00:31:30,420 --> 00:31:32,270 Now this is something-- 475 00:31:32,270 --> 00:31:35,690 you do the operator algebra before you launch 476 00:31:35,690 --> 00:31:37,740 into a horrible calculation. 477 00:31:37,740 --> 00:31:40,220 And so we know how to do this in principle 478 00:31:40,220 --> 00:31:44,900 to reduce this to a simple expression. 479 00:31:44,900 --> 00:31:48,080 And now we take this part, this thing, 480 00:31:48,080 --> 00:31:52,040 and we're just going to call it capital A. Because we want it-- 481 00:31:52,040 --> 00:31:54,320 we don't want to clutter up what is going 482 00:31:54,320 --> 00:31:57,600 to be a terrible thing anyway. 483 00:31:57,600 --> 00:32:02,960 And we can always convert back to the little a at the end 484 00:32:02,960 --> 00:32:05,850 if we need to. 485 00:32:05,850 --> 00:32:11,450 OK, so now I've set the stage. 486 00:32:11,450 --> 00:32:12,940 I'll leave this for a while. 487 00:32:16,390 --> 00:32:19,420 That really is-- as you know, I really love revision theory, 488 00:32:19,420 --> 00:32:22,750 because it's the tool that I use all the time. 489 00:32:22,750 --> 00:32:29,440 And it's a psychological condition 490 00:32:29,440 --> 00:32:33,100 that is not curable, all right? 491 00:32:33,100 --> 00:32:35,440 So what do we want? 492 00:32:35,440 --> 00:32:41,480 We want E0 as a function of E and J. 493 00:32:41,480 --> 00:32:48,880 And that's just the J H0 VJ. 494 00:32:48,880 --> 00:33:00,540 And we know that HC will make EB plus 1/2 plus HCBEJJ plus 1. 495 00:33:03,290 --> 00:33:06,311 OK, well this goes in the energy denominators. 496 00:33:09,750 --> 00:33:11,800 And so that's good, because we need 497 00:33:11,800 --> 00:33:17,380 energy denominators as well as the zero order energies. 498 00:33:17,380 --> 00:33:23,970 Right now we need to know the bad stuff, the thing that's 499 00:33:23,970 --> 00:33:25,920 outside of the zero order Hamiltonian, 500 00:33:25,920 --> 00:33:28,170 which is everything. 501 00:33:28,170 --> 00:33:38,730 And so we have this is HC times BE JJ plus 1 times-- 502 00:33:41,900 --> 00:33:43,930 I'm sorry, I have to erase this. 503 00:33:47,480 --> 00:33:54,116 Minus BE over 4BE over a big E square 504 00:33:54,116 --> 00:34:03,470 root times A plus A dagger plus 3 BE. 505 00:34:07,320 --> 00:34:18,010 3BE over omega E times A plus A dagger quality squared. 506 00:34:18,010 --> 00:34:20,639 So that's the rotational part. 507 00:34:20,639 --> 00:34:22,840 And then we have the vibrational power, 508 00:34:22,840 --> 00:34:27,330 which we have A, A plus A dagger cubed. 509 00:34:30,460 --> 00:34:32,920 OK, so now we look at this thing and we say, 510 00:34:32,920 --> 00:34:38,679 oh well, this has matrix elements delta V equals 511 00:34:38,679 --> 00:34:40,239 plus or minus 1. 512 00:34:40,239 --> 00:34:47,889 And this has delta V equals plus or minus 2 and 0. 513 00:34:47,889 --> 00:34:55,711 And this has delta V equals plus and minus 3 plus and minus 1. 514 00:34:55,711 --> 00:34:58,660 We always want to sort things according to the selection 515 00:34:58,660 --> 00:35:01,960 rules, because we always combine the things 516 00:35:01,960 --> 00:35:04,830 with the same selection rules. 517 00:35:04,830 --> 00:35:09,040 And it's best to do that at the beginning 518 00:35:09,040 --> 00:35:11,050 rather than somehow trying to do it at the end. 519 00:35:11,050 --> 00:35:14,020 Because if you have the same selection rule, 520 00:35:14,020 --> 00:35:15,190 you have cross terms. 521 00:35:18,560 --> 00:35:19,950 And that's really important. 522 00:35:19,950 --> 00:35:24,210 The cross terms are where a lot of good stuff happens. 523 00:35:24,210 --> 00:35:32,970 OK, so we know E0. 524 00:35:32,970 --> 00:35:37,740 And we would know E1 if there are any off the-- 525 00:35:37,740 --> 00:35:42,930 there are any diagonal matrix elements of this operator. 526 00:35:42,930 --> 00:35:45,430 This should be a plus as well. 527 00:35:45,430 --> 00:35:50,660 So this is H1. 528 00:35:50,660 --> 00:35:54,740 And this doesn't have any diagonal elements. 529 00:35:54,740 --> 00:35:59,750 And this doesn't have any diagonal elements. 530 00:35:59,750 --> 00:36:02,480 But this one does. 531 00:36:02,480 --> 00:36:04,780 And it's actually something you encountered, 532 00:36:04,780 --> 00:36:06,010 I think, on exam two. 533 00:36:06,010 --> 00:36:08,380 I'm not sure, but you've certainly encountered 534 00:36:08,380 --> 00:36:11,470 the diagonal element of this. 535 00:36:11,470 --> 00:36:18,880 And so if we look at A plus A dagger squared, 536 00:36:18,880 --> 00:36:22,840 we get A squared plus A dagger squared 537 00:36:22,840 --> 00:36:30,110 plus 2 number operator plus 1. 538 00:36:30,110 --> 00:36:33,130 And that's diagonal. 539 00:36:33,130 --> 00:36:36,520 So we have a diagonal element, and that gives us E1. 540 00:36:47,670 --> 00:37:03,640 So E1 of E and J is HC6BE squared over omega E times J, J 541 00:37:03,640 --> 00:37:05,930 plus 1, B plus 1/2. 542 00:37:10,460 --> 00:37:16,110 OK, well that looks pretty good, because this 543 00:37:16,110 --> 00:37:18,240 is the coefficient of-- 544 00:37:18,240 --> 00:37:20,250 we have an expression for the energy levels, 545 00:37:20,250 --> 00:37:28,110 which includes the E minus alpha E, B 546 00:37:28,110 --> 00:37:35,020 plus 1/2 of times JJ plus 1. 547 00:37:35,020 --> 00:37:39,360 So we have a term that involves B plus 1/2 and JJ plus 1. 548 00:37:39,360 --> 00:37:40,650 And it has a name. 549 00:37:40,650 --> 00:37:42,450 Alpha minus alpha. 550 00:37:42,450 --> 00:37:46,840 And here we have a term which has those, 551 00:37:46,840 --> 00:37:48,700 that quantum number dependents. 552 00:37:48,700 --> 00:37:51,810 And it has a value. 553 00:37:51,810 --> 00:37:56,660 And so it's telling us that alpha E, 554 00:37:56,660 --> 00:38:00,370 and I'll put on this harmonic oscillator, 555 00:38:00,370 --> 00:38:06,250 is equal to minus HC6BE v squared over omega 556 00:38:06,250 --> 00:38:08,830 E. One constant. 557 00:38:12,360 --> 00:38:26,220 Now it's also telling you that B of E increases with V. 558 00:38:26,220 --> 00:38:29,970 And for a harmonic oscillator, you 559 00:38:29,970 --> 00:38:35,290 have an equal lobe at each turning point, the largest 560 00:38:35,290 --> 00:38:35,790 lobe. 561 00:38:35,790 --> 00:38:39,020 But they're equal in magnitude. 562 00:38:39,020 --> 00:38:41,770 So one can ask, at the inner turning 563 00:38:41,770 --> 00:38:48,340 point is the change in the rotational operator, 564 00:38:48,340 --> 00:38:52,120 or is it larger-- 565 00:38:52,120 --> 00:38:58,210 is this change in BE larger relative to the equilibrium 566 00:38:58,210 --> 00:39:00,760 value, or is this larger? 567 00:39:00,760 --> 00:39:05,980 And the answer is we're talking about 1 over R squared. 568 00:39:05,980 --> 00:39:10,830 And 1 over R squared gets really large at small r as opposed 569 00:39:10,830 --> 00:39:15,450 to getting smaller at large R. And the amount of change 570 00:39:15,450 --> 00:39:18,040 is much greater at small r. 571 00:39:18,040 --> 00:39:23,610 And so that causes the effective rotational 572 00:39:23,610 --> 00:39:26,280 constant to increase. 573 00:39:26,280 --> 00:39:30,610 But we know for a harmonica-- for an anharmonic oscillator, 574 00:39:30,610 --> 00:39:34,580 we have a small lobe here and a big lobe here. 575 00:39:34,580 --> 00:39:37,100 And so we expect that there's going 576 00:39:37,100 --> 00:39:41,870 to be a battle between the harmonic contribution to alpha 577 00:39:41,870 --> 00:39:44,490 and the anharmonic contribution. 578 00:39:44,490 --> 00:39:47,980 And we expect that this is going to win. 579 00:39:47,980 --> 00:39:48,710 Why? 580 00:39:48,710 --> 00:39:53,180 Because every time anybody measures alpha, 581 00:39:53,180 --> 00:39:54,570 it's a positive number. 582 00:39:54,570 --> 00:40:00,830 And that's why it was expressed in the formula, which I've 583 00:40:00,830 --> 00:40:07,220 concealed, with a negative sign, to take into account that alpha 584 00:40:07,220 --> 00:40:08,460 is always-- 585 00:40:08,460 --> 00:40:10,580 the contribution is always negative, 586 00:40:10,580 --> 00:40:13,584 and so alpha is always positive. 587 00:40:13,584 --> 00:40:14,750 Yeah, and these things are-- 588 00:40:17,960 --> 00:40:19,150 it's historical. 589 00:40:19,150 --> 00:40:22,400 But anyway, so we have a contribution 590 00:40:22,400 --> 00:40:24,380 which has the wrong sign. 591 00:40:24,380 --> 00:40:27,170 And we know we're going to get another contribution 592 00:40:27,170 --> 00:40:31,350 from the interaction between the rotation and the vibration, 593 00:40:31,350 --> 00:40:35,360 and that this is going to make things right OK. 594 00:40:35,360 --> 00:40:38,950 And it comes from a term. 595 00:40:38,950 --> 00:40:39,450 OK. 596 00:40:46,380 --> 00:40:48,475 All right, there's no way I can make this simpler. 597 00:40:53,400 --> 00:40:55,860 Just have to bear with me and I will-- 598 00:40:59,480 --> 00:41:06,170 so we have delta V equals plus and minus 1 matrix 599 00:41:06,170 --> 00:41:16,080 elements from both the A plus A dagger and the A 600 00:41:16,080 --> 00:41:19,170 plus A dagger cubed terms. 601 00:41:22,500 --> 00:41:30,240 And we have-- we have this in both the anharmonic expression 602 00:41:30,240 --> 00:41:33,570 and in the rotational constant operator. 603 00:41:33,570 --> 00:41:35,970 We know what these-- 604 00:41:35,970 --> 00:41:37,810 how these things work out. 605 00:41:37,810 --> 00:41:48,160 And so delta V equals plus and minus 1 terms from the A 606 00:41:48,160 --> 00:42:00,270 plus A dagger cubed V plus 1 and V, V minus 1. 607 00:42:00,270 --> 00:42:04,130 So we can work these things out. 608 00:42:04,130 --> 00:42:10,720 And this one is 3V plus 1/2 to the 3/2. 609 00:42:10,720 --> 00:42:17,170 And this is 3V plus V to the 3/2. 610 00:42:17,170 --> 00:42:19,440 So all of this stuff, you want to simplify it 611 00:42:19,440 --> 00:42:21,780 as you go but it. 612 00:42:21,780 --> 00:42:29,002 And these come from the A term, the A, 613 00:42:29,002 --> 00:42:33,820 A plus A dagger cubed' term. 614 00:42:33,820 --> 00:42:37,390 We don't have a term that's linear at A plus A dagger 615 00:42:37,390 --> 00:42:41,470 from the anharmonicity, but we do-- 616 00:42:41,470 --> 00:42:44,500 and we do have a squared term, but that's 617 00:42:44,500 --> 00:42:45,620 the harmonic correction. 618 00:42:45,620 --> 00:42:48,130 So we start here, with A plus A dagger cubed. 619 00:42:52,610 --> 00:42:54,800 And then we're going to want to look 620 00:42:54,800 --> 00:42:59,120 at the delta V equals plus or minus 2, and delta V 621 00:42:59,120 --> 00:43:03,290 equals plus and minus 3 terms. 622 00:43:03,290 --> 00:43:05,340 And there's lots of algebra. 623 00:43:05,340 --> 00:43:06,580 You can do all these things. 624 00:43:06,580 --> 00:43:08,580 There's nothing challenging here. 625 00:43:08,580 --> 00:43:11,600 It's just how do you keep this stuff that you derived 626 00:43:11,600 --> 00:43:13,760 that you're going to need in a place 627 00:43:13,760 --> 00:43:16,760 that you can find it again, because the pages just 628 00:43:16,760 --> 00:43:18,290 get filled with garbage. 629 00:43:18,290 --> 00:43:18,790 OK. 630 00:43:22,150 --> 00:43:26,620 So the best way for me to do this is to write the results 631 00:43:26,620 --> 00:43:31,150 in the next-to-the-final step, and then in the final step, 632 00:43:31,150 --> 00:43:33,670 and show what goes to what. 633 00:43:40,990 --> 00:43:45,620 So the second order correction is a function of EJ. 634 00:43:49,180 --> 00:44:00,070 We're going to have terms that involve the delta V equals 635 00:44:00,070 --> 00:44:01,810 plus or minus 1. 636 00:44:01,810 --> 00:44:06,540 And we're going to get that from the cubic correction 637 00:44:06,540 --> 00:44:10,270 through the harmonic oscillator and the linear correction 638 00:44:10,270 --> 00:44:12,354 to the rotational concept. 639 00:44:12,354 --> 00:44:14,020 So we're going to get an expression that 640 00:44:14,020 --> 00:44:14,710 looks like this. 641 00:44:14,710 --> 00:44:23,170 B squared minus 2AB plus A squared. 642 00:44:26,330 --> 00:44:27,310 So this is a B term. 643 00:44:27,310 --> 00:44:29,150 This is the anharmonic term. 644 00:44:29,150 --> 00:44:30,130 We have a cross term. 645 00:44:30,130 --> 00:44:33,540 There are going to be three terms in this expression. 646 00:44:33,540 --> 00:44:35,450 OK, and so let's do that. 647 00:44:39,555 --> 00:44:40,430 How much time I have? 648 00:44:40,430 --> 00:44:41,390 Not very much time. 649 00:44:41,390 --> 00:44:43,056 Well that's good, because you won't have 650 00:44:43,056 --> 00:44:44,560 to watch much more of this. 651 00:44:44,560 --> 00:44:51,020 OK, so we have the HcBe JJ plus 1. 652 00:44:53,960 --> 00:44:56,180 And this guy is squared, because we're 653 00:44:56,180 --> 00:44:58,160 doing second-order perturbation theory. 654 00:44:58,160 --> 00:45:01,880 This is just the B term squared. 655 00:45:01,880 --> 00:45:08,000 And then we have 4BE be over omega E. 656 00:45:08,000 --> 00:45:15,380 And then we have the delta V of 1 and minus 1 matrix elements. 657 00:45:15,380 --> 00:45:25,520 V plus 1 over minus HC omega, and V plus V over HC omega. 658 00:45:25,520 --> 00:45:27,980 We have the energy denominators, two energy denominators 659 00:45:27,980 --> 00:45:31,400 of opposite side and similar identical magnitude. 660 00:45:31,400 --> 00:45:35,230 And we always do that because then the algebra is simple. 661 00:45:35,230 --> 00:45:38,360 If we do-- we fail to do that, you might as well just 662 00:45:38,360 --> 00:45:40,610 go to a booby hatch right away, because there's just 663 00:45:40,610 --> 00:45:42,950 no way you're going to retain your sanity if you 664 00:45:42,950 --> 00:45:48,150 don't combine the terms of delta V plus 1 and delta V minus 1. 665 00:45:50,970 --> 00:45:54,100 OK, and this is the first term. 666 00:45:54,100 --> 00:45:59,230 And there is the next term in this sum, which is the AB term. 667 00:45:59,230 --> 00:46:06,190 And so we have a minus 2 times HCV JJ plus 1. 668 00:46:09,280 --> 00:46:14,820 And it's not squared, because it's one B and one A term. 669 00:46:14,820 --> 00:46:26,630 And so we get 4BE over omega E square root. 670 00:46:26,630 --> 00:46:27,700 And then the A part. 671 00:46:30,470 --> 00:46:38,200 And we get 3B plus 1/2 to the 3/2 times 672 00:46:38,200 --> 00:46:48,500 V plus 1/2 to the 1/2, or minus HC omega E. 673 00:46:48,500 --> 00:46:55,220 And then the other term, which is 3V to the 3/2, V to 1/2, 674 00:46:55,220 --> 00:46:59,100 over HC omega E. 675 00:46:59,100 --> 00:47:01,010 We have to combine these two terms. 676 00:47:01,010 --> 00:47:02,720 And the combination is easy. 677 00:47:02,720 --> 00:47:03,590 And then there is-- 678 00:47:03,590 --> 00:47:05,990 so this term number two. 679 00:47:05,990 --> 00:47:10,580 And number three is the squared anharmonic term. 680 00:47:10,580 --> 00:47:14,570 So I'm going to skip it, since I'm basically done. 681 00:47:14,570 --> 00:47:17,070 There's just a lot of garbage like this. 682 00:47:17,070 --> 00:47:23,840 And at the end, we have an expression, term by term, 683 00:47:23,840 --> 00:47:25,670 which simplifies. 684 00:47:25,670 --> 00:47:32,480 And we get the second-order correction to the energy, VJ, 685 00:47:32,480 --> 00:47:42,520 is equal to minus HcBe cubed over HC omega. 686 00:47:45,100 --> 00:47:46,200 No, not HC. 687 00:47:46,200 --> 00:47:58,960 Just over omega E squared times JJ plus 1 squared. 688 00:47:58,960 --> 00:48:02,290 Well this is nice, because we have a term that just involves 689 00:48:02,290 --> 00:48:08,530 J, and it's negative, that says as a molecule rotates, 690 00:48:08,530 --> 00:48:15,050 the molecule stretches, the rotational constant decreases. 691 00:48:15,050 --> 00:48:17,440 This is called centrifugal distortion. 692 00:48:17,440 --> 00:48:19,960 And this is a very famous expression 693 00:48:19,960 --> 00:48:32,440 that DE is equal to 4BE cubed over omega squared. 694 00:48:32,440 --> 00:48:35,500 I left out a 4. 695 00:48:35,500 --> 00:48:37,150 So this is the Kratzer relationship. 696 00:48:37,150 --> 00:48:40,930 And that turns out to be of the most valuable ones, 697 00:48:40,930 --> 00:48:45,590 because the centrifugal distortion 698 00:48:45,590 --> 00:48:50,760 constant is extremely sensitive to contagion. 699 00:48:50,760 --> 00:48:54,200 If there is a perturbation, if there is something missing, 700 00:48:54,200 --> 00:48:57,280 this centrifugal distortion constant 701 00:48:57,280 --> 00:49:01,050 will have a crazy value. 702 00:49:01,050 --> 00:49:03,750 And if it doesn't ever-- if there's nothing wrong, 703 00:49:03,750 --> 00:49:06,000 then it will say, well, OK, I can-- 704 00:49:06,000 --> 00:49:09,600 if I know something about B and omega, 705 00:49:09,600 --> 00:49:15,720 I can determine how the rotational energy levels get 706 00:49:15,720 --> 00:49:18,960 closer together as you go up in V. 707 00:49:18,960 --> 00:49:22,080 And if you have only a pure rotation spectrum, 708 00:49:22,080 --> 00:49:23,910 then you actually-- 709 00:49:23,910 --> 00:49:25,530 where you're not sampling anything 710 00:49:25,530 --> 00:49:28,050 having to do with vibration, you actually 711 00:49:28,050 --> 00:49:32,250 can determine the vibrational constant 712 00:49:32,250 --> 00:49:35,790 from the centrifugal distortion. 713 00:49:35,790 --> 00:49:38,380 And I should tell you that spectroscopists come 714 00:49:38,380 --> 00:49:41,110 in families, or they used to. 715 00:49:41,110 --> 00:49:45,670 Pure rotation, infrared vibration rotation, 716 00:49:45,670 --> 00:49:48,400 electronic vibration rotation, electronic. 717 00:49:48,400 --> 00:49:51,910 And so the microwavers, who really have only rotation 718 00:49:51,910 --> 00:49:54,610 to look at, are kind of deprived. 719 00:49:54,610 --> 00:49:57,970 And they just don't know anything 720 00:49:57,970 --> 00:49:59,470 about electronic degrees of freedom. 721 00:49:59,470 --> 00:50:02,530 But they know how to get the vibrational frequency 722 00:50:02,530 --> 00:50:07,530 from the rotational spectrum, which is kind of nice. 723 00:50:07,530 --> 00:50:09,210 Gives them something to do. 724 00:50:09,210 --> 00:50:10,930 And there's more terms. 725 00:50:10,930 --> 00:50:13,800 And we're really done, but it's all in the notes. 726 00:50:13,800 --> 00:50:24,700 And so what we get is for the constant OAEXE, and center 727 00:50:24,700 --> 00:50:29,190 of distortion, and alpha E and beta, which 728 00:50:29,190 --> 00:50:34,290 is the vibrational distortion of the centrifugal distortion 729 00:50:34,290 --> 00:50:40,800 constant, we get all these in terms of omega and B. 730 00:50:40,800 --> 00:50:47,050 So it's a nice closed set where you get the-- 731 00:50:47,050 --> 00:50:51,780 you're able to predict how certain constants depend 732 00:50:51,780 --> 00:50:56,370 on the most fundamental ones, omega E and BE. 733 00:50:56,370 --> 00:50:57,900 That's kind of neat. 734 00:50:57,900 --> 00:51:03,310 And it's both a way of extending your observations 735 00:51:03,310 --> 00:51:05,230 and telling well, there's something missing, 736 00:51:05,230 --> 00:51:09,330 because these constants aren't coming out right. 737 00:51:09,330 --> 00:51:12,210 It's also a subject of great consternation. 738 00:51:12,210 --> 00:51:16,860 Because if you're a quantum chemist, what you do 739 00:51:16,860 --> 00:51:24,520 is you determine a lot of stuff by derivatives at equilibrium. 740 00:51:24,520 --> 00:51:27,320 The derivatives, the potential at equilibrium, 741 00:51:27,320 --> 00:51:29,810 are basically telling you the shape of the potential 742 00:51:29,810 --> 00:51:33,500 at moderately low energy. 743 00:51:33,500 --> 00:51:35,940 And it also turns out to be something with modern quantum 744 00:51:35,940 --> 00:51:36,440 chemistry. 745 00:51:36,440 --> 00:51:39,195 You can get all these derivatives pretty accurately. 746 00:51:42,050 --> 00:51:45,990 The trouble is measuring derivatives here 747 00:51:45,990 --> 00:51:49,160 and using the energy formulas gives you 748 00:51:49,160 --> 00:51:51,050 slightly different values from what you 749 00:51:51,050 --> 00:51:52,590 get from perturbation theory. 750 00:51:52,590 --> 00:51:55,100 So what we have is two communities 751 00:51:55,100 --> 00:51:59,410 who are very sophisticated using the same names 752 00:51:59,410 --> 00:52:02,370 for different quantities. 753 00:52:02,370 --> 00:52:04,360 It's dangerous. 754 00:52:04,360 --> 00:52:09,790 And it's also not understood by the two communities. 755 00:52:09,790 --> 00:52:11,650 And so people are struggling and struggling 756 00:52:11,650 --> 00:52:14,850 to get agreement between experiment and theory. 757 00:52:14,850 --> 00:52:18,340 And it's apples and oranges, but it's a subtle thing. 758 00:52:18,340 --> 00:52:22,180 OK, so that's basically all I want to say 759 00:52:22,180 --> 00:52:23,771 about perturbation theory. 760 00:52:23,771 --> 00:52:26,020 I will have one more lecture dealing with perturbation 761 00:52:26,020 --> 00:52:29,020 theory, and that lecture is on Van der Waals 762 00:52:29,020 --> 00:52:33,200 interactions between molecules and why we get liquids. 763 00:52:33,200 --> 00:52:34,750 OK.