1 00:00:00,090 --> 00:00:02,430 The following content is provided under a Creative 2 00:00:02,430 --> 00:00:03,820 Commons license. 3 00:00:03,820 --> 00:00:06,030 Your support will help MIT OpenCourseWare 4 00:00:06,030 --> 00:00:10,120 continue to offer high-quality educational resources for free. 5 00:00:10,120 --> 00:00:12,660 To make a donation or to view additional materials 6 00:00:12,660 --> 00:00:16,620 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:16,620 --> 00:00:17,850 at ocw.mit.edu. 8 00:00:23,875 --> 00:00:25,250 PROFESSOR: I'm really sick today. 9 00:00:25,250 --> 00:00:27,620 And I hope I can make it through this lecture 10 00:00:27,620 --> 00:00:29,570 without disgracing myself. 11 00:00:29,570 --> 00:00:32,330 But this is a favorite topic of mine, too, 12 00:00:32,330 --> 00:00:36,440 so maybe it will be OK. 13 00:00:36,440 --> 00:00:42,020 So we're going to talk about the helium atom, which 14 00:00:42,020 --> 00:00:46,580 is really an amazing thing because hydrogen-- 15 00:00:46,580 --> 00:00:49,610 it was possible to solve it exactly. 16 00:00:49,610 --> 00:00:52,780 All sorts of fantastic things happened. 17 00:00:52,780 --> 00:00:57,763 And it seemed like just a small step from hydrogen to helium. 18 00:00:57,763 --> 00:01:01,910 And nobody can take that step. 19 00:01:01,910 --> 00:01:04,489 It's entirely approximate. 20 00:01:04,489 --> 00:01:08,930 And you can do better and better on the approximations, 21 00:01:08,930 --> 00:01:13,280 but we'd like to have some physical sense of what's 22 00:01:13,280 --> 00:01:15,980 going on, and what are the important terms, 23 00:01:15,980 --> 00:01:20,000 and can we steal things from hydrogen? 24 00:01:20,000 --> 00:01:21,980 The answer is, yeah, we can. 25 00:01:21,980 --> 00:01:26,150 And that was what last lecture was mostly about. 26 00:01:26,150 --> 00:01:35,240 The energies for hydrogen are minus z squared hc times 27 00:01:35,240 --> 00:01:38,750 the Rydberg over n squared. 28 00:01:38,750 --> 00:01:48,570 And we have the ionization energy from the nth level. 29 00:01:48,570 --> 00:01:51,710 And so this actually is the way we 30 00:01:51,710 --> 00:01:55,190 make a link to this concept of electronegativity 31 00:01:55,190 --> 00:01:57,620 because the dominant thing in electronegativity 32 00:01:57,620 --> 00:01:59,910 is the ionization energy. 33 00:01:59,910 --> 00:02:01,980 And this is the ionization energy. 34 00:02:01,980 --> 00:02:07,700 And when you put them together and solve for n, 35 00:02:07,700 --> 00:02:14,630 you get this arrangement hc z squared times the Rydberg 36 00:02:14,630 --> 00:02:18,740 over the ionization energy, square root. 37 00:02:18,740 --> 00:02:23,460 That's n, or the effective principle quantum number. 38 00:02:23,460 --> 00:02:25,850 And then we have all of these formulas that 39 00:02:25,850 --> 00:02:30,410 can be derived from hydrogenic wave functions, 40 00:02:30,410 --> 00:02:33,740 both for diagonal expectation values 41 00:02:33,740 --> 00:02:35,960 or off diagonal matrix elements. 42 00:02:35,960 --> 00:02:37,940 They're all closed form. 43 00:02:37,940 --> 00:02:40,160 And this is wonderful because it then 44 00:02:40,160 --> 00:02:47,750 enables you to use the formulas that are obtained exactly 45 00:02:47,750 --> 00:02:49,920 for hydrogenic wave functions. 46 00:02:49,920 --> 00:02:57,170 And so if we know the dependence of an electronic property on n 47 00:02:57,170 --> 00:03:00,140 and l-- 48 00:03:00,140 --> 00:03:03,260 well, n is experimentally determined-- 49 00:03:03,260 --> 00:03:04,740 distance real ionization. 50 00:03:04,740 --> 00:03:08,210 l is determined from angular momentum. 51 00:03:08,210 --> 00:03:13,980 Then we can determine the expected scaling of everything 52 00:03:13,980 --> 00:03:15,800 on the quantum numbers. 53 00:03:15,800 --> 00:03:18,620 And even if that's wrong, it tells you 54 00:03:18,620 --> 00:03:23,390 the qualitative effects when you get to more complicated systems 55 00:03:23,390 --> 00:03:24,800 in one electron atoms. 56 00:03:24,800 --> 00:03:28,510 And this is really what I meant by structure, 57 00:03:28,510 --> 00:03:34,350 where everything is related to basically one observation. 58 00:03:34,350 --> 00:03:37,920 There is one other thing that I didn't stress. 59 00:03:37,920 --> 00:03:45,480 And it's another concept from 5.112., and that's shielding. 60 00:03:45,480 --> 00:03:48,390 So suppose we have a sodium atom. 61 00:03:48,390 --> 00:03:53,800 Most of the electrons are in a compact core. 62 00:03:53,800 --> 00:03:57,730 The outer electron sees a charge of 1, 63 00:03:57,730 --> 00:04:02,580 even though the charge on the nucleus is larger than 1. 64 00:04:02,580 --> 00:04:09,400 And if you ask, well, suppose we look at this in a region, 65 00:04:09,400 --> 00:04:14,230 and we have an electron inside of it, 66 00:04:14,230 --> 00:04:16,209 the charge that that electron sees 67 00:04:16,209 --> 00:04:21,160 is the amount of electrons inside the sphere defined 68 00:04:21,160 --> 00:04:23,570 by this radius. 69 00:04:23,570 --> 00:04:29,320 And so the z, we can call it the z effective, 70 00:04:29,320 --> 00:04:31,720 is a function of r. 71 00:04:31,720 --> 00:04:34,300 And you have a different function every o. 72 00:04:34,300 --> 00:04:37,105 And so it's a way of putting in more insight. 73 00:04:40,680 --> 00:04:43,660 And I've made a lot of mileage in my career 74 00:04:43,660 --> 00:04:46,760 on these sorts of scaling arguments. 75 00:04:46,760 --> 00:04:52,130 OK, so helium-- the notes are beautiful. 76 00:04:52,130 --> 00:04:53,980 They're written by Troy. 77 00:04:53,980 --> 00:04:55,780 And they're very long. 78 00:04:55,780 --> 00:04:59,830 And they're very correct and incisive. 79 00:04:59,830 --> 00:05:03,700 And even if I weren't sick, I couldn't cover all those notes. 80 00:05:03,700 --> 00:05:08,590 So I'm going to make a rather express trip through the notes. 81 00:05:08,590 --> 00:05:15,050 And the critical thing is we have two electrons. 82 00:05:15,050 --> 00:05:19,270 And so we can't simply solve the problem 83 00:05:19,270 --> 00:05:24,750 by using hydrogenic functions, although we would like to. 84 00:05:24,750 --> 00:05:28,020 And the crucial thing that is missing 85 00:05:28,020 --> 00:05:32,580 is if we could ignore the interactions between the two 86 00:05:32,580 --> 00:05:37,120 electrons, which are enormous, then 87 00:05:37,120 --> 00:05:38,980 we would be using hydrogenic functions, 88 00:05:38,980 --> 00:05:41,590 and everything would be fine. 89 00:05:41,590 --> 00:05:43,090 So we can't. 90 00:05:43,090 --> 00:05:47,050 And we use what's called the independent electron 91 00:05:47,050 --> 00:05:52,390 approximation, where we're still using hydrogenic orbitals 92 00:05:52,390 --> 00:05:55,990 and energies, but we're using those 93 00:05:55,990 --> 00:06:00,400 to include the first order perturbation 94 00:06:00,400 --> 00:06:05,160 correction to the energies. 95 00:06:05,160 --> 00:06:09,000 So there's a term, the interelectron repulsion 96 00:06:09,000 --> 00:06:12,360 that leads to this extra correction energy. 97 00:06:12,360 --> 00:06:14,560 And that's most of the difference. 98 00:06:14,560 --> 00:06:19,280 But then, we have identical particles. 99 00:06:19,280 --> 00:06:23,980 And when we have identical particles, 100 00:06:23,980 --> 00:06:27,500 the permutation of those two particles 101 00:06:27,500 --> 00:06:29,560 has to commute with the Hamiltonian 102 00:06:29,560 --> 00:06:31,820 because the Hamiltonian only knows that we're 103 00:06:31,820 --> 00:06:33,270 dealing with all the electrons. 104 00:06:33,270 --> 00:06:35,840 It doesn't put names on the individual electrons. 105 00:06:35,840 --> 00:06:38,090 And so there's a permutation symmetry 106 00:06:38,090 --> 00:06:40,930 that has to be imposed. 107 00:06:40,930 --> 00:06:44,680 And this is very subtle because the permutation symmetry has 108 00:06:44,680 --> 00:06:47,670 nothing to do with the energy, but it has 109 00:06:47,670 --> 00:06:51,110 to do with the energy states. 110 00:06:51,110 --> 00:06:53,480 And so we'll explore that. 111 00:06:53,480 --> 00:06:56,180 Now, since we're going to be dealing 112 00:06:56,180 --> 00:06:59,330 with more than one electron, there's a lot of symbols 113 00:06:59,330 --> 00:07:01,550 that we're going to be carrying around. 114 00:07:01,550 --> 00:07:04,122 And so there's a useful thing called 115 00:07:04,122 --> 00:07:08,900 the atomic units, which sets most of the physical constants 116 00:07:08,900 --> 00:07:11,160 to 1. 117 00:07:11,160 --> 00:07:14,310 And that means all of the rubble that you have to carry around 118 00:07:14,310 --> 00:07:15,780 goes away. 119 00:07:15,780 --> 00:07:19,620 And it's not a big deal because if you 120 00:07:19,620 --> 00:07:26,510 use all of the things in your calculation in atomic units, 121 00:07:26,510 --> 00:07:31,730 namely value 1 or something like 1, then at the end, 122 00:07:31,730 --> 00:07:34,370 you're going to get the energy in atomic units. 123 00:07:34,370 --> 00:07:36,720 And we know what that is in conventional units. 124 00:07:36,720 --> 00:07:40,400 So it's a risky thing because you don't really 125 00:07:40,400 --> 00:07:42,180 know what you're doing. 126 00:07:42,180 --> 00:07:45,410 But at the end, you can probably convert 127 00:07:45,410 --> 00:07:49,680 your number, which is a pure number, into an actual energy. 128 00:07:49,680 --> 00:07:55,390 OK, so let me start with the good news. 129 00:07:58,610 --> 00:08:02,990 Well, maybe I should leave this exposed. 130 00:08:02,990 --> 00:08:07,400 The good news is we have two particles for helium-- 131 00:08:07,400 --> 00:08:16,340 vector r, momentum, momentum p vector for particle 1, 132 00:08:16,340 --> 00:08:20,300 vector, momentum. 133 00:08:20,300 --> 00:08:23,370 OK, so these are the actors. 134 00:08:23,370 --> 00:08:26,580 And when do we have to worry about commutation 135 00:08:26,580 --> 00:08:27,750 or non-commutation? 136 00:08:30,570 --> 00:08:35,090 Well, the answer is the coordinates 137 00:08:35,090 --> 00:08:40,500 associated with different particles, 138 00:08:40,500 --> 00:08:42,799 they contribute to the energy. 139 00:08:42,799 --> 00:08:44,860 But there is no problem of commutation. 140 00:08:44,860 --> 00:08:49,580 The momenta don't also commute. 141 00:08:49,580 --> 00:08:52,140 The coordinate for particle 1 and the momentum 142 00:08:52,140 --> 00:08:55,700 for particle 2, they commute. 143 00:08:55,700 --> 00:09:03,950 The only things that doesn't commute is r1x, p1x-- 144 00:09:03,950 --> 00:09:07,100 those kinds of things. 145 00:09:07,100 --> 00:09:09,530 They're commutators ih bar, and that 146 00:09:09,530 --> 00:09:11,610 means there's some problems. 147 00:09:11,610 --> 00:09:15,770 So most of the variables are commuting variables, 148 00:09:15,770 --> 00:09:17,850 but some are not. 149 00:09:17,850 --> 00:09:21,260 And you have to deal with that. 150 00:09:21,260 --> 00:09:21,760 OK. 151 00:09:24,550 --> 00:09:27,880 We know that if we have, as, for example, 152 00:09:27,880 --> 00:09:30,310 in a three-dimensional harmonic oscillator, 153 00:09:30,310 --> 00:09:33,550 if we have a Hamiltonian that doesn't commute-- 154 00:09:33,550 --> 00:09:37,180 if we can divide the system up into parts, then 155 00:09:37,180 --> 00:09:40,630 the Hamiltonian for each part commutes with the Hamiltonian 156 00:09:40,630 --> 00:09:43,300 for the other part, which is part of this. 157 00:09:43,300 --> 00:09:45,790 Then, you can write the Hamiltonian 158 00:09:45,790 --> 00:09:49,300 as a sum of individual system Hamiltonians 159 00:09:49,300 --> 00:09:53,580 that commute with each other, the energies as the sum, 160 00:09:53,580 --> 00:09:55,817 and the wave functions as the product. 161 00:09:55,817 --> 00:09:56,650 And so we like that. 162 00:09:56,650 --> 00:09:59,770 And we'd like to be able to build a picture for helium that 163 00:09:59,770 --> 00:10:02,090 is like that. 164 00:10:02,090 --> 00:10:05,770 And so if we write the Hamiltonian for helium, 165 00:10:05,770 --> 00:10:13,660 we have p squared for particle 1 over to me with p squared-- 166 00:10:13,660 --> 00:10:14,720 I don't know. 167 00:10:14,720 --> 00:10:15,820 Le me see. 168 00:10:15,820 --> 00:10:17,590 Let me organize better. 169 00:10:23,610 --> 00:10:33,620 Minus zb squared over 4 pi epsilon 0 r1. 170 00:10:36,220 --> 00:10:43,100 And then we have another term like this for particle 2. 171 00:10:43,100 --> 00:10:47,489 And we can call this Hamiltonian 1. 172 00:10:47,489 --> 00:10:50,210 I don't want to do that. 173 00:10:50,210 --> 00:10:54,910 Hamiltonian 1, and this Hamiltonian 2. 174 00:10:54,910 --> 00:10:57,670 And these are hydrogen atom Hamiltonians. 175 00:10:57,670 --> 00:11:00,030 And we know everything about them. 176 00:11:00,030 --> 00:11:03,030 So it's a fantastic way to build up 177 00:11:03,030 --> 00:11:08,960 a basis set consisting of energy, zero-order energies 178 00:11:08,960 --> 00:11:11,810 and zero-order wave functions, with which 179 00:11:11,810 --> 00:11:14,240 we can calculate anything. 180 00:11:14,240 --> 00:11:17,270 And so then we have this one other term, 181 00:11:17,270 --> 00:11:27,390 which plus e squared over 4 pi epsilon 0 182 00:11:27,390 --> 00:11:32,070 over 1 minus r minus r2 vectors. 183 00:11:36,580 --> 00:11:37,620 That's the bad news. 184 00:11:41,110 --> 00:11:45,790 This says we have a Hamiltonian, where 185 00:11:45,790 --> 00:11:50,260 we can't use the eigenvalues and eigenfunctions 186 00:11:50,260 --> 00:11:57,430 of the hydrogenic things unless we do-- so we can say, well, 187 00:11:57,430 --> 00:12:02,350 these give the first-order energies for whatever state 188 00:12:02,350 --> 00:12:05,235 we want. 189 00:12:05,235 --> 00:12:10,710 But we can calculate matrix elements of this term 190 00:12:10,710 --> 00:12:15,020 in the basis set associated with hydrogen. 191 00:12:15,020 --> 00:12:16,420 And that's fine. 192 00:12:16,420 --> 00:12:19,250 That's a perfectly good way to calculate 193 00:12:19,250 --> 00:12:21,990 the first-order correction to the energy. 194 00:12:21,990 --> 00:12:24,855 But what about this? 195 00:12:27,550 --> 00:12:29,440 When we do perturbation theory and calculate 196 00:12:29,440 --> 00:12:34,050 the second-order corrections to the energy, 197 00:12:34,050 --> 00:12:35,105 what is that usually? 198 00:12:42,620 --> 00:12:45,350 Maybe I didn't ask the question clearly enough. 199 00:12:45,350 --> 00:12:50,110 What do you have to do to get these second-order energies 200 00:12:50,110 --> 00:12:54,360 in an infinite dimension problem? 201 00:12:54,360 --> 00:12:57,120 I just gave it away. 202 00:12:57,120 --> 00:13:02,260 Somebody has to have the courage to say something here. 203 00:13:02,260 --> 00:13:13,514 Remember, we have a sum of n not equal to n from 0 to infinity. 204 00:13:13,514 --> 00:13:14,410 AUDIENCE: Yeah. 205 00:13:14,410 --> 00:13:15,880 PROFESSOR: That's it. 206 00:13:15,880 --> 00:13:18,280 OK, I encouraged you. 207 00:13:18,280 --> 00:13:24,100 So the way we could use this basis set in a perturbation 208 00:13:24,100 --> 00:13:27,340 theory format is to calculate an infinite number 209 00:13:27,340 --> 00:13:30,100 of off-diagonal matrix elements of the hydrogenic wave 210 00:13:30,100 --> 00:13:32,410 functions. 211 00:13:32,410 --> 00:13:36,630 And that could yield results, but it's not 212 00:13:36,630 --> 00:13:39,070 the way it's done usually. 213 00:13:39,070 --> 00:13:40,740 There's a variational calculation 214 00:13:40,740 --> 00:13:44,080 that you use to get the better wave functions. 215 00:13:44,080 --> 00:13:46,050 But we're not going to do that here. 216 00:13:46,050 --> 00:13:48,240 We're going to go as far as we can 217 00:13:48,240 --> 00:13:53,370 without actually addressing what's wrong if we stop here. 218 00:13:58,730 --> 00:14:07,270 OK, so this guy is h1, the source of all trouble. 219 00:14:07,270 --> 00:14:11,330 And how we deal with it is there are three steps. 220 00:14:11,330 --> 00:14:13,710 One is ignore it. 221 00:14:13,710 --> 00:14:19,200 The other is use our hydrogenic functions 222 00:14:19,200 --> 00:14:26,150 to calculate matrix elements of this and get an h1, an e1. 223 00:14:26,150 --> 00:14:28,730 And the third step would be to get the exact answer 224 00:14:28,730 --> 00:14:33,310 by doing an infinitely difficult calculation, OK? 225 00:14:36,030 --> 00:14:36,530 All right. 226 00:14:39,890 --> 00:14:44,810 So this is going to be a notational voyage, which 227 00:14:44,810 --> 00:14:47,240 is horrible. 228 00:14:47,240 --> 00:14:50,990 Because we have many particles, and particles have coordinates 229 00:14:50,990 --> 00:14:55,440 and spins, we're going to have an explosion of stuff, 230 00:14:55,440 --> 00:14:58,970 even if we get rid of the extraneous units. 231 00:14:58,970 --> 00:15:01,040 But let's get rid of the extraneous units. 232 00:15:01,040 --> 00:15:02,240 And so atomic units-- 233 00:15:07,970 --> 00:15:11,330 we can choose any internally consistent set of units 234 00:15:11,330 --> 00:15:12,430 we want. 235 00:15:12,430 --> 00:15:17,750 And in atomic units, we say that the mass of the electron is 1. 236 00:15:17,750 --> 00:15:20,300 The charge of the electron is 1. 237 00:15:20,300 --> 00:15:23,960 The h bar is 1. 238 00:15:23,960 --> 00:15:27,560 4 pi epsilon 0 is 1. 239 00:15:27,560 --> 00:15:28,520 That's good. 240 00:15:28,520 --> 00:15:31,560 A lot of stuff just going away. 241 00:15:31,560 --> 00:15:37,630 OK, and the energy in atomic units is not 1. 242 00:15:37,630 --> 00:15:43,180 The energy in atomic units is minus 2 times 243 00:15:43,180 --> 00:15:47,860 the energy of the hydrogenic 1s orbital. 244 00:15:47,860 --> 00:15:53,440 And that comes out to be 27.21 electron volts, 245 00:15:53,440 --> 00:15:56,780 in more convenient units. 246 00:15:56,780 --> 00:16:00,890 So you calculate a number, and then you multiply it by this 247 00:16:00,890 --> 00:16:02,270 to get the real energy. 248 00:16:04,830 --> 00:16:10,500 OK, and the unit of length is a0, 249 00:16:10,500 --> 00:16:14,070 which is the radius of the n equals 1 hydrogenic Bohr 250 00:16:14,070 --> 00:16:17,970 orbital, or Bohr orbit. 251 00:16:17,970 --> 00:16:23,580 And that's 0.529 angstroms, with a lot more digits if you 252 00:16:23,580 --> 00:16:26,880 need them, OK? 253 00:16:26,880 --> 00:16:33,490 And the speed of light is 137 in atomic units. 254 00:16:33,490 --> 00:16:37,380 So everything that you're going to need 255 00:16:37,380 --> 00:16:42,560 is either 1 or something that is related to-- 256 00:16:42,560 --> 00:16:46,590 OK, so you can do this, and I don't like it 257 00:16:46,590 --> 00:16:51,141 because it's so easy in a big calculation 258 00:16:51,141 --> 00:16:53,730 to get your units screwed up. 259 00:16:53,730 --> 00:16:56,730 But it's certainly great for lecturing 260 00:16:56,730 --> 00:16:59,005 and perhaps for writing computer programs. 261 00:17:04,730 --> 00:17:13,480 OK, so let's start with a non-interacting electron 262 00:17:13,480 --> 00:17:14,380 approximation. 263 00:17:17,290 --> 00:17:23,359 And in the notes, it's represented by these letters. 264 00:17:23,359 --> 00:17:25,520 I don't know whether that this is 265 00:17:25,520 --> 00:17:31,940 Troy notation or widely-adopted in quantum chemical circles. 266 00:17:31,940 --> 00:17:34,670 But it's easier to write this than that. 267 00:17:34,670 --> 00:17:39,520 And that just means we're going to ignore this. 268 00:17:39,520 --> 00:17:41,890 We just forget about it. 269 00:17:41,890 --> 00:17:45,790 Now, we know that's stupid because the electrons have 270 00:17:45,790 --> 00:17:47,230 charge of 1. 271 00:17:47,230 --> 00:17:49,000 And they repel each other. 272 00:17:49,000 --> 00:17:52,465 And the nucleus has charge of 2, in the case of helium. 273 00:17:55,660 --> 00:18:04,450 And so we know that this term is rather similar to h0. 274 00:18:04,450 --> 00:18:07,960 And we can't expect to get very far completely ignoring it. 275 00:18:07,960 --> 00:18:10,157 But we can do it, and we can do a calculation, 276 00:18:10,157 --> 00:18:11,740 and we can find out how well it works. 277 00:18:15,280 --> 00:18:22,470 So first of all, it's clear in the notes. 278 00:18:22,470 --> 00:18:24,600 I'm just going to skip to the answer. 279 00:18:24,600 --> 00:18:31,160 So the energy for helium in the 1s 1s orbital 280 00:18:31,160 --> 00:18:35,070 or 1s 1s configuration-- 281 00:18:35,070 --> 00:18:37,890 when we're going to use this word configuration, what 282 00:18:37,890 --> 00:18:45,060 that means is the list of orbitals that are occupied. 283 00:18:45,060 --> 00:18:48,330 Now, we're listing principal quantum number 284 00:18:48,330 --> 00:18:50,420 and orbital angular momentum. 285 00:18:50,420 --> 00:18:53,750 There is more stuff which doesn't get listed, 286 00:18:53,750 --> 00:18:55,960 and that leads to very interesting stuff. 287 00:18:55,960 --> 00:18:59,710 But if we have a list of the orbitals are occupied. 288 00:18:59,710 --> 00:19:03,740 That's called the electronic configuration. 289 00:19:03,740 --> 00:19:05,340 And I'm an electronic spectroscopist, 290 00:19:05,340 --> 00:19:08,320 so you can be sure this is something I care a lot about. 291 00:19:12,830 --> 00:19:25,330 So we can calculate this in the hydrogenic basis. 292 00:19:25,330 --> 00:19:27,670 And what you end up getting is-- 293 00:19:27,670 --> 00:19:28,210 where is it? 294 00:19:46,111 --> 00:19:46,610 I'm sorry. 295 00:19:46,610 --> 00:19:48,560 This is the experimental value. 296 00:19:48,560 --> 00:19:51,350 The calculated value is-- 297 00:19:51,350 --> 00:19:53,380 see, I'm trying to improve on my notes. 298 00:19:53,380 --> 00:19:59,960 And in my present state, I make a lot of mistakes. 299 00:19:59,960 --> 00:20:05,970 So that is 108.8 electron volts. 300 00:20:05,970 --> 00:20:12,530 So this is the energy of the 1s 1s state below the ionization 301 00:20:12,530 --> 00:20:14,660 limit. 302 00:20:14,660 --> 00:20:15,980 So that's what we predict. 303 00:20:20,030 --> 00:20:22,640 Except this is not the ionization limit. 304 00:20:22,640 --> 00:20:27,774 This is helium 2+ plus 2 electrons. 305 00:20:33,100 --> 00:20:36,640 Because we use the energy to ionize both hydrogen 306 00:20:36,640 --> 00:20:41,950 atoms, and so this is 108.8. 307 00:20:41,950 --> 00:20:43,780 Now, you can measure this experimentally. 308 00:20:43,780 --> 00:20:50,050 You can do helium to helium plus an electron, 309 00:20:50,050 --> 00:20:57,440 and then helium plus going to helium 2+ plus an electron. 310 00:20:57,440 --> 00:21:04,160 And the observed value is what I wrote before. 311 00:21:04,160 --> 00:21:11,620 Observed is equal to minus 79.0 electrons volts. 312 00:21:11,620 --> 00:21:14,080 Well, you might say, well, that's pretty good. 313 00:21:14,080 --> 00:21:15,930 It's the same order of magnitude. 314 00:21:15,930 --> 00:21:19,510 It's not even off by a factor of 2. 315 00:21:19,510 --> 00:21:23,590 But almost everybody in this room is a chemist. 316 00:21:23,590 --> 00:21:28,630 And this difference between the calculated value 317 00:21:28,630 --> 00:21:34,330 and the observed value is 30 electron volts. 318 00:21:34,330 --> 00:21:38,590 A chemical bond is typically 5 electron volts. 319 00:21:38,590 --> 00:21:40,910 So this is totally ridiculous for chemistry. 320 00:21:40,910 --> 00:21:42,810 We can't use it. 321 00:21:42,810 --> 00:21:45,760 We don't even want to think about it 322 00:21:45,760 --> 00:21:48,430 because it's an order of magnitude 323 00:21:48,430 --> 00:21:49,720 away from what we need. 324 00:21:57,040 --> 00:22:02,870 We improve on it by including e1 of n, 325 00:22:02,870 --> 00:22:08,791 which is related to h1, which is that simple-looking thing 326 00:22:08,791 --> 00:22:09,290 over there. 327 00:22:11,800 --> 00:22:15,990 So the interelectron repulsions can be dealt with, 328 00:22:15,990 --> 00:22:21,450 not exactly, but by calculating diagonal matrix elements 329 00:22:21,450 --> 00:22:27,570 of the interelectron repulsion. 330 00:22:27,570 --> 00:22:34,310 And that's what the independent electron approximation is. 331 00:22:34,310 --> 00:22:36,170 So that's all there is. 332 00:22:36,170 --> 00:22:40,760 But now we bring in something that we hadn't expected 333 00:22:40,760 --> 00:22:41,590 would be a problem. 334 00:22:52,480 --> 00:22:55,530 So we have this mystical, or mythical, 335 00:22:55,530 --> 00:22:57,040 or whatever operator-- 336 00:22:57,040 --> 00:23:04,410 p12 operating on some function of r1 and r2. 337 00:23:04,410 --> 00:23:08,730 And what it does is it permutes the electrons. 338 00:23:11,490 --> 00:23:16,800 We know that if something is operating 339 00:23:16,800 --> 00:23:19,290 on the names of the electrons, it 340 00:23:19,290 --> 00:23:23,100 has to commute with the Hamiltonian. 341 00:23:23,100 --> 00:23:26,310 So that means that this operator has 342 00:23:26,310 --> 00:23:33,120 to have eigenvalues if these states are eigenfunctions 343 00:23:33,120 --> 00:23:35,780 of the Hamiltonian. 344 00:23:35,780 --> 00:23:38,330 And there are two possible eigenvalues 345 00:23:38,330 --> 00:23:46,940 because we know p12 squared on psi is equal to plus psi. 346 00:23:46,940 --> 00:23:49,820 So the only way you can do that is having 347 00:23:49,820 --> 00:23:52,880 the eigenvalue of the permutation operator 348 00:23:52,880 --> 00:23:57,390 be either plus 1 or minus 1. 349 00:23:57,390 --> 00:23:59,360 And this is a fundamental symmetry. 350 00:23:59,360 --> 00:24:03,710 And I can't possibly tell you where it comes from 351 00:24:03,710 --> 00:24:06,150 or what I'm about to write on the board comes from. 352 00:24:06,150 --> 00:24:25,620 But so all half integer spins correspond 353 00:24:25,620 --> 00:24:36,434 to antisymmetry or belong to the eigenvalue minus 1. 354 00:24:36,434 --> 00:24:37,600 And they're called fermions. 355 00:24:42,560 --> 00:24:47,340 And integer spins are symmetric with permutations. 356 00:24:47,340 --> 00:24:58,760 And so integer or symmetric, and they're bosons. 357 00:25:02,140 --> 00:25:04,380 So we have to write wave functions which 358 00:25:04,380 --> 00:25:08,300 are wave functions of the electrons, which 359 00:25:08,300 --> 00:25:11,270 are antisymmetric with respect to the permutation 360 00:25:11,270 --> 00:25:14,280 of every pair of electrons. 361 00:25:14,280 --> 00:25:16,040 So helium, we only have two electrons. 362 00:25:16,040 --> 00:25:18,670 It's not so bad. 363 00:25:18,670 --> 00:25:20,840 But when you think about something 364 00:25:20,840 --> 00:25:23,510 like carbon monoxide, which I always think about, 365 00:25:23,510 --> 00:25:25,640 or some other molecule where there's 366 00:25:25,640 --> 00:25:28,460 more than two electrons, there's a God awful number 367 00:25:28,460 --> 00:25:30,410 of permutations. 368 00:25:30,410 --> 00:25:33,740 And you have to build them all into the wave function. 369 00:25:33,740 --> 00:25:38,510 And the normal feeling when you hear that is, 370 00:25:38,510 --> 00:25:43,590 oh, no, I can't possibly imagine how to do that. 371 00:25:43,590 --> 00:25:47,180 And if I could, there would be a ridiculous number of integrals 372 00:25:47,180 --> 00:25:50,770 that I would have to evaluate. 373 00:25:50,770 --> 00:25:52,850 And it's true. 374 00:25:52,850 --> 00:25:54,890 There are a ridiculous number of integrals. 375 00:25:54,890 --> 00:25:58,280 But there is a simple algebra that enables you 376 00:25:58,280 --> 00:26:00,440 how to deal with them. 377 00:26:00,440 --> 00:26:05,310 And it involves using what is called Slater determinantal 378 00:26:05,310 --> 00:26:06,140 wave functions. 379 00:26:08,639 --> 00:26:09,680 And I'll talk about that. 380 00:26:09,680 --> 00:26:12,430 Slater was an MIT professor. 381 00:26:12,430 --> 00:26:14,120 He wrote a lot of books. 382 00:26:14,120 --> 00:26:17,250 But the thing-- integer. 383 00:26:17,250 --> 00:26:20,820 Oh, they'res sort of a t. 384 00:26:20,820 --> 00:26:23,050 But the most important thing he did 385 00:26:23,050 --> 00:26:26,020 was to develop this way of dealing 386 00:26:26,020 --> 00:26:30,740 with an antisymmetric electronic wave functions. 387 00:26:30,740 --> 00:26:33,640 So the next thing we have to do is discover 388 00:26:33,640 --> 00:26:36,910 how do we build symmetric wave functions for the simplest 389 00:26:36,910 --> 00:26:40,150 possible case where there's two electrons? 390 00:26:40,150 --> 00:26:43,120 And already, we discover some surprising consequences 391 00:26:43,120 --> 00:26:45,311 for this. 392 00:26:45,311 --> 00:26:46,810 And so I'm going to go through this, 393 00:26:46,810 --> 00:26:48,430 but I'm going to skip a lot of steps 394 00:26:48,430 --> 00:26:50,579 because it's so clearly written in the notes 395 00:26:50,579 --> 00:26:52,120 and because we don't really have time 396 00:26:52,120 --> 00:26:55,360 to deal with 15 pages of single-spaced notes. 397 00:26:55,360 --> 00:27:02,567 And I'm not sufficiently at my best to get through maybe 12. 398 00:27:02,567 --> 00:27:03,900 And you don't want that, anyway. 399 00:27:06,410 --> 00:27:10,909 OK, so let us see what happens here. 400 00:27:10,909 --> 00:27:11,950 And so let's get to work. 401 00:27:16,010 --> 00:27:25,510 Suppose we have an electronic state 1s alpha spin. 402 00:27:25,510 --> 00:27:29,110 We know alpha is ms equals plus 1/2. 403 00:27:29,110 --> 00:27:31,390 And beta is ms equals minus 1/2. 404 00:27:36,920 --> 00:27:40,580 So this is the spatial coordinate. 405 00:27:40,580 --> 00:27:43,120 This is the spin coordinate. 406 00:27:43,120 --> 00:27:55,507 And, OK, so here is a simple product of two wave functions. 407 00:27:55,507 --> 00:27:56,590 And we know what they are. 408 00:27:59,830 --> 00:28:18,720 And so if we apply p12 to this, we get psi 1s alpha r2 sigma 409 00:28:18,720 --> 00:28:28,070 2 1s beta r1 sigma 1. 410 00:28:28,070 --> 00:28:30,500 We don't even know what this is relative to that. 411 00:28:37,430 --> 00:28:43,210 So symbolically, these are two unrelated quantities. 412 00:28:43,210 --> 00:28:45,550 So weird, but they are eigenfunctions 413 00:28:45,550 --> 00:28:50,070 of the zero-order Hamiltonian. 414 00:28:50,070 --> 00:28:53,010 So we can take a linear combination of eigenfunctions 415 00:28:53,010 --> 00:28:55,810 that belong to the same eigenvalue, 416 00:28:55,810 --> 00:28:58,250 and it's still the same energy. 417 00:28:58,250 --> 00:28:59,960 So we do that. 418 00:28:59,960 --> 00:29:10,630 And so what we can do is write a combination of this function 419 00:29:10,630 --> 00:29:12,690 and that function. 420 00:29:12,690 --> 00:29:20,450 And if we do that, so now we can have psi 1s 2s. 421 00:29:20,450 --> 00:29:24,050 And what we get is 1 over the square root over 2 422 00:29:24,050 --> 00:29:25,780 for normalization. 423 00:29:25,780 --> 00:29:41,542 And we have psi 1s alpha r1 sigma 1 minus psi 1s beta 424 00:29:41,542 --> 00:29:45,010 r1 r2 sigma 2. 425 00:29:48,030 --> 00:29:50,520 And this, in fact, is anti-asymmetric. 426 00:29:50,520 --> 00:29:53,870 If you apply the permutation operator to this, 427 00:29:53,870 --> 00:29:57,250 it is an antisymmetric function. 428 00:29:57,250 --> 00:29:58,510 So this does the job. 429 00:30:05,060 --> 00:30:11,490 Now, since the Hamiltonian doesn't depend on spin, 430 00:30:11,490 --> 00:30:15,000 the energy of this state and the energy of that state 431 00:30:15,000 --> 00:30:15,710 are the same. 432 00:30:15,710 --> 00:30:16,920 They are hydrogenic. 433 00:30:16,920 --> 00:30:21,540 And so using this wave function rather than 434 00:30:21,540 --> 00:30:23,820 the individual non-symmetrized functions 435 00:30:23,820 --> 00:30:26,200 doesn't change the energies. 436 00:30:26,200 --> 00:30:29,016 So are we wasting our time? 437 00:30:29,016 --> 00:30:32,780 AUDIENCE: I think you didn't [INAUDIBLE].. 438 00:30:32,780 --> 00:30:36,635 This should be the product of 1s alpha 1s beta. 439 00:30:36,635 --> 00:30:37,510 PROFESSOR: I'm sorry? 440 00:30:37,510 --> 00:30:39,759 AUDIENCE: So the [INAUDIBLE] must be [FINGER SNAPPING] 441 00:30:39,759 --> 00:30:40,860 PROFESSOR: Yes, yes, yes. 442 00:30:40,860 --> 00:30:42,316 AUDIENCE: Minus the other one. 443 00:30:48,730 --> 00:30:52,370 PROFESSOR: I just want to see what I wrote here. 444 00:30:52,370 --> 00:30:56,910 So yeah, what I wrote is wrong. 445 00:30:56,910 --> 00:31:02,490 1s alpha, and we'll just put 1 in here. 446 00:31:05,510 --> 00:31:24,516 1s beta 2 minus 1s alpha 2 1s beta 1. 447 00:31:24,516 --> 00:31:26,140 OK, this is what I should have written. 448 00:31:26,140 --> 00:31:30,570 And this is, in fact, antisymmetric. 449 00:31:30,570 --> 00:31:32,600 Thank you. 450 00:31:32,600 --> 00:31:35,170 I wrote what's in my notes, which is wrong. 451 00:31:37,940 --> 00:31:41,790 And I didn't realize I had done that because I knew the answer. 452 00:31:41,790 --> 00:31:43,420 All right. 453 00:31:43,420 --> 00:31:50,950 OK, so we can use this kind of function. 454 00:31:50,950 --> 00:31:53,920 It's legal because it's an eigenfunction 455 00:31:53,920 --> 00:31:57,640 of the permutation operator with negative symmetry, which 456 00:31:57,640 --> 00:32:00,740 we're told that's what we have to use for electrons. 457 00:32:00,740 --> 00:32:03,170 I can't tell you where it comes from, 458 00:32:03,170 --> 00:32:05,180 although it came from Pauli. 459 00:32:05,180 --> 00:32:06,890 And the exclusion principle really 460 00:32:06,890 --> 00:32:10,330 is this, rather than what you have memorized. 461 00:32:10,330 --> 00:32:13,400 Its consequence is that you can only 462 00:32:13,400 --> 00:32:15,200 put one electron in an orbital. 463 00:32:15,200 --> 00:32:18,080 But this is the exclusion principle, and it is Pauli. 464 00:32:21,320 --> 00:32:25,570 OK, so now we know what to do. 465 00:32:25,570 --> 00:32:31,200 And we can calculate the matrix elements. 466 00:32:31,200 --> 00:32:39,120 psi 1s alpha go in 1s beta. 467 00:32:39,120 --> 00:32:43,890 This is the antisymmetrized product of two electrons. 468 00:32:43,890 --> 00:32:47,820 Matrix element 1 for Hamiltonian 1 469 00:32:47,820 --> 00:33:02,697 plus 1 over r1 minus r2 psi 1s alpha 1s beta. 470 00:33:02,697 --> 00:33:05,030 Though this is a perfectly legitimate thing to calculate 471 00:33:05,030 --> 00:33:09,880 and we know how to do it, or McQuarrie knows how to do it. 472 00:33:09,880 --> 00:33:11,500 And you don't really need to worry 473 00:33:11,500 --> 00:33:14,830 about doing these integrals because this 474 00:33:14,830 --> 00:33:18,280 is at such a low-level approximation, 475 00:33:18,280 --> 00:33:21,100 it does nothing for you except teach you 476 00:33:21,100 --> 00:33:24,010 how to approach these problems. 477 00:33:24,010 --> 00:33:26,350 But we can still do these integrals. 478 00:33:26,350 --> 00:33:27,600 And it's useful. 479 00:33:27,600 --> 00:33:29,970 And so what do you get? 480 00:33:29,970 --> 00:33:33,370 Well, you get minus 4y because these 481 00:33:33,370 --> 00:33:37,390 are hydrogenic with a nucleus of plus 2. 482 00:33:37,390 --> 00:33:43,270 And in the atomic units, the energy 483 00:33:43,270 --> 00:33:51,990 of a hydrogen atom charge of 1 is 1/2, not 1. 484 00:33:51,990 --> 00:33:58,000 And these go as z squared, so we get 4 over 2 plus 4 over 2. 485 00:33:58,000 --> 00:34:01,290 So we get this minus 4 or you get minus-- 486 00:34:01,290 --> 00:34:02,280 yes. 487 00:34:02,280 --> 00:34:03,780 And then we get something else. 488 00:34:07,840 --> 00:34:11,770 And that's this integral that we don't like, 489 00:34:11,770 --> 00:34:21,460 which is integral of 1s alpha 1s beta squared 490 00:34:21,460 --> 00:34:27,790 over r1 minus r2 dr1 dr2. 491 00:34:32,480 --> 00:34:35,960 So we can do this integral, or Mr. McQuarrie and many 492 00:34:35,960 --> 00:34:41,810 of his predecessors can do that integral, and you get a result. 493 00:34:41,810 --> 00:34:50,330 And this integral turns out to be 5z over 8. 494 00:34:50,330 --> 00:35:02,720 And converting that into units, and that's 5/4 au. 495 00:35:02,720 --> 00:35:05,560 And that's-- why is that? 496 00:35:09,850 --> 00:35:12,460 That doesn't seem right. 497 00:35:12,460 --> 00:35:20,350 Anyway, it's 34 electron volts or minus 34 electron-- plus. 498 00:35:20,350 --> 00:35:22,700 Plus 34 electron volts-- 499 00:35:22,700 --> 00:35:26,040 so the interelectronic repulsion for hydrogenic orbitals 500 00:35:26,040 --> 00:35:28,810 on the same nucleus is 34 electron volts. 501 00:35:28,810 --> 00:35:31,160 It's a big number. 502 00:35:31,160 --> 00:35:39,010 And so we can correct our the calculated versus observed 503 00:35:39,010 --> 00:35:39,590 energy. 504 00:35:39,590 --> 00:35:40,765 And instead of having-- 505 00:35:45,230 --> 00:35:47,795 well, the experimental value is 79. 506 00:35:52,270 --> 00:36:00,550 And now, the calculated value is minus 74.8 electron volts. 507 00:36:00,550 --> 00:36:03,720 So this is marginally relevant to chemistry 508 00:36:03,720 --> 00:36:07,930 because the error is about the typical bond energy. 509 00:36:07,930 --> 00:36:10,980 And so we can do stuff with this. 510 00:36:10,980 --> 00:36:13,200 We know that it's not accurate enough 511 00:36:13,200 --> 00:36:14,790 to do anything quantitative. 512 00:36:14,790 --> 00:36:19,650 But you can use it for qualitative arguments. 513 00:36:19,650 --> 00:36:22,230 And we can do much better than this 514 00:36:22,230 --> 00:36:28,830 using what's going to be called the variational principle, 515 00:36:28,830 --> 00:36:29,520 which is coming. 516 00:36:32,190 --> 00:36:39,520 OK, so now, we're going to be dealing with a problem that has 517 00:36:39,520 --> 00:36:45,880 lots of symbols and equations. 518 00:36:45,880 --> 00:36:57,350 And it's best to start using notation called stick diagrams 519 00:36:57,350 --> 00:37:01,950 because it's completely transparent what they mean. 520 00:37:01,950 --> 00:37:04,770 And so it's a shortcut for writing the correct equations. 521 00:37:07,860 --> 00:37:14,560 So we have for the ground state of hydrogen-- 522 00:37:14,560 --> 00:37:15,070 I'm sorry. 523 00:37:19,530 --> 00:37:28,840 We want to talk about excited states because 1s with 1s 524 00:37:28,840 --> 00:37:32,190 doesn't have enough complexity to attract our attention. 525 00:37:32,190 --> 00:37:40,420 And when we deal with excited states, 1s and 2s, 526 00:37:40,420 --> 00:37:44,290 we discover something important that's new. 527 00:37:44,290 --> 00:37:45,760 And it's the last thing we really 528 00:37:45,760 --> 00:37:49,300 need in order to understand electronic structure 529 00:37:49,300 --> 00:37:52,940 or how the electronic states that arise from a configuration 530 00:37:52,940 --> 00:37:55,040 are distinguished. 531 00:37:55,040 --> 00:38:05,990 And so we have for the 1s 2s configuration, 532 00:38:05,990 --> 00:38:10,150 there are four ways of writing this in stick diagrams. 533 00:38:10,150 --> 00:38:20,390 So we have 1s 2s, and we can have spins up, alpha alpha. 534 00:38:20,390 --> 00:38:26,170 And we can have 2s 1s. 535 00:38:26,170 --> 00:38:30,020 We can have spins up and down. 536 00:38:30,020 --> 00:38:35,960 2s 1s, and spins down and up. 537 00:38:35,960 --> 00:38:40,840 And 2s 1s spins down. 538 00:38:40,840 --> 00:38:43,810 So it's much easier to write these things 539 00:38:43,810 --> 00:38:47,470 because they have an equation to associate with them, 540 00:38:47,470 --> 00:38:51,820 and you can play with these symbols. 541 00:38:51,820 --> 00:39:00,300 So the first thing you notice is this symbol 542 00:39:00,300 --> 00:39:06,470 can be written as an antisymmetrized product. 543 00:39:06,470 --> 00:39:09,180 And so this symbol-- 544 00:39:09,180 --> 00:39:13,930 we just take the alpha alpha. 545 00:39:13,930 --> 00:39:15,960 And so let me just write it out. 546 00:39:15,960 --> 00:39:31,270 1 over square root of 2 times 1s r1 2s r2 minus psi 1s r2 547 00:39:31,270 --> 00:39:35,810 psi 2s r1. 548 00:39:35,810 --> 00:39:40,190 And then times a spin part, alpha-- 549 00:39:40,190 --> 00:39:42,770 and we can just abbreviate the 1-- 550 00:39:42,770 --> 00:39:43,640 and alpha. 551 00:39:47,330 --> 00:39:50,350 OK, so we have factored the wave function 552 00:39:50,350 --> 00:39:54,430 to a spatial part and a spin part. 553 00:39:54,430 --> 00:39:57,090 Now, I'll make an outrageous statement. 554 00:39:57,090 --> 00:39:59,400 Doesn't matter how many electrons. 555 00:39:59,400 --> 00:40:01,860 You'll always be able to factor it 556 00:40:01,860 --> 00:40:05,980 into a spatial part and a spin part. 557 00:40:05,980 --> 00:40:11,410 Now, what is necessary is always this function 558 00:40:11,410 --> 00:40:13,740 has to be antisymmetric. 559 00:40:13,740 --> 00:40:15,670 And in this case, it's the spatial part 560 00:40:15,670 --> 00:40:18,720 that's antisymmetric and the spin part that's symmetric. 561 00:40:21,360 --> 00:40:24,095 Now, when we try to do these two stick diagrams. 562 00:40:27,540 --> 00:40:32,290 we cannot write a correct equation for this guy. 563 00:40:32,290 --> 00:40:34,630 We have to do both of them. 564 00:40:34,630 --> 00:40:36,940 We have to combine the two. 565 00:40:36,940 --> 00:40:39,820 And that will give us a separated spin part and spatial 566 00:40:39,820 --> 00:40:41,260 part. 567 00:40:41,260 --> 00:40:43,180 And it's antisymmetric. 568 00:40:43,180 --> 00:40:48,170 You can't make in an asymmetric function of this just switching 569 00:40:48,170 --> 00:40:49,565 the coordinates. 570 00:40:52,300 --> 00:40:54,770 And what you find when you do this 571 00:40:54,770 --> 00:40:59,660 is going to be that we get functions 572 00:40:59,660 --> 00:41:04,370 which look like alpha beta plus or minus beta alpha. 573 00:41:07,180 --> 00:41:11,750 OK, so spin 1 alpha spin 2 beta plus or minus 574 00:41:11,750 --> 00:41:14,200 spin 1 beta spin 2 alpha. 575 00:41:17,920 --> 00:41:29,040 And it turns out that there are three spin functions that we 576 00:41:29,040 --> 00:41:31,800 get, which are symmetric. 577 00:41:31,800 --> 00:41:34,920 And all three of these are associated 578 00:41:34,920 --> 00:41:37,860 with this antisymmetric spatial function. 579 00:41:37,860 --> 00:41:40,547 They have the same energy. 580 00:41:40,547 --> 00:41:42,880 And since they're three of them, we call them a triplet. 581 00:41:46,930 --> 00:41:50,020 And since we're talking about spins, 582 00:41:50,020 --> 00:41:53,080 this triplet corresponds to s equals 1. 583 00:41:53,080 --> 00:41:56,680 You use s equals 1 and creation and annihilation operators 584 00:41:56,680 --> 00:41:58,420 or raising and lowering operators 585 00:41:58,420 --> 00:42:04,245 to show that these are, in fact, the three eigenstates of s sub 586 00:42:04,245 --> 00:42:04,745 z. 587 00:42:07,600 --> 00:42:10,460 So we call it a triplet because they're three of them. 588 00:42:10,460 --> 00:42:14,140 And it also corresponds to s equals 1. 589 00:42:14,140 --> 00:42:19,570 And there's one, alpha beta minus beta alpha, 590 00:42:19,570 --> 00:42:22,680 which is antisymmetric and is associated 591 00:42:22,680 --> 00:42:27,430 with a different spatial function, which is symmetric. 592 00:42:27,430 --> 00:42:29,830 And it has to have a different energy 593 00:42:29,830 --> 00:42:33,640 because the Hamiltonian operates on the spatial part and not 594 00:42:33,640 --> 00:42:34,930 on the spin part. 595 00:42:34,930 --> 00:42:37,690 So antisymmetry has forced you to have 596 00:42:37,690 --> 00:42:41,020 two different spatial wave functions, 597 00:42:41,020 --> 00:42:43,090 and that guarantees that these guys 598 00:42:43,090 --> 00:42:45,640 will have different energies. 599 00:42:45,640 --> 00:42:51,070 And since there's only one of them, we call it a singlet, 600 00:42:51,070 --> 00:42:52,690 and it corresponds to s equals 0. 601 00:42:55,790 --> 00:42:59,420 So now, we want to know, which is higher in energy, 602 00:42:59,420 --> 00:43:02,180 the singlet or the triplet? 603 00:43:02,180 --> 00:43:04,050 And how much higher? 604 00:43:04,050 --> 00:43:06,380 And so, again, we take the wave functions 605 00:43:06,380 --> 00:43:08,900 that are correct as far as permutation is concerned, 606 00:43:08,900 --> 00:43:13,100 but only approximate as far as using hydrogenic functions, 607 00:43:13,100 --> 00:43:17,166 and we evaluate the relevant terms in the Hamiltonian. 608 00:43:21,140 --> 00:43:24,980 OK, so we can forget about the spin parts now. 609 00:43:24,980 --> 00:43:27,740 We know we're going to have two different spatial wave 610 00:43:27,740 --> 00:43:28,250 functions. 611 00:43:28,250 --> 00:43:30,776 And we're going to want to calculate-- yes? 612 00:43:30,776 --> 00:43:32,650 AUDIENCE: For the top state, [INAUDIBLE] 613 00:43:32,650 --> 00:43:35,737 both spins [INAUDIBLE]? 614 00:43:35,737 --> 00:43:36,820 PROFESSOR: That should be? 615 00:43:36,820 --> 00:43:39,160 AUDIENCE: Would that also be s equals 0? 616 00:43:39,160 --> 00:43:40,160 PROFESSOR: I can't hear. 617 00:43:40,160 --> 00:43:42,030 I understand the last word. 618 00:43:42,030 --> 00:43:46,070 AUDIENCE: With the statement, both spin be also s equals 0? 619 00:43:48,670 --> 00:43:50,060 PROFESSOR: No. 620 00:43:50,060 --> 00:43:51,170 Both spins up. 621 00:43:51,170 --> 00:43:54,620 The vector sum of 1/2 and 1/2 is 1. 622 00:43:54,620 --> 00:43:57,155 And so the total spin is going to be 1. 623 00:44:00,920 --> 00:44:03,940 And so we have 3-- 624 00:44:03,940 --> 00:44:07,060 I mean, if we're applying s minus-- 625 00:44:07,060 --> 00:44:08,290 suppose we start with-- 626 00:44:08,290 --> 00:44:11,100 AUDIENCE: I meant that the top spin [INAUDIBLE] both spins. 627 00:44:15,382 --> 00:44:16,090 PROFESSOR: Sorry. 628 00:44:18,730 --> 00:44:20,800 I can't hear properly. 629 00:44:20,800 --> 00:44:23,216 AUDIENCE: The topmost state with both spins on it? 630 00:44:23,216 --> 00:44:23,840 PROFESSOR: Yes. 631 00:44:23,840 --> 00:44:27,270 AUDIENCE: Would that have s equals 0 or s equals 1? 632 00:44:27,270 --> 00:44:29,630 PROFESSOR: It has s equals 1. 633 00:44:29,630 --> 00:44:31,600 OK. 634 00:44:31,600 --> 00:44:35,920 And now, I don't want to go into the angular momentum matrix 635 00:44:35,920 --> 00:44:36,420 element. 636 00:44:36,420 --> 00:44:39,100 That would be the last thing in this lecture. 637 00:44:39,100 --> 00:44:44,590 So yes, the projection quantum numbers 638 00:44:44,590 --> 00:44:48,640 are not uniquely associated with the total spin. 639 00:44:48,640 --> 00:44:52,680 But if you have alpha alpha, you can show that it's s equals 1. 640 00:44:52,680 --> 00:44:56,250 And alpha beta plus beta alpha is 1. 641 00:44:56,250 --> 00:44:58,600 And beta beta is 1. 642 00:44:58,600 --> 00:45:02,740 But alpha beta minus beta alpha is spin 0. 643 00:45:02,740 --> 00:45:05,740 You can do that using creation and annihilation operators. 644 00:45:05,740 --> 00:45:08,860 It's clear, OK? 645 00:45:08,860 --> 00:45:11,470 So you have to take it on faith until you 646 00:45:11,470 --> 00:45:13,520 prove what I've just said. 647 00:45:13,520 --> 00:45:16,510 And it takes about two minutes. 648 00:45:16,510 --> 00:45:22,240 OK, so we can evaluate the integrals. 649 00:45:22,240 --> 00:45:27,330 And something very beautiful occurs. 650 00:45:27,330 --> 00:45:30,830 And it's done very, very clearly in the notes. 651 00:45:30,830 --> 00:45:33,710 And it would take me more time than we have left to do it, 652 00:45:33,710 --> 00:45:35,450 and I didn't plan on doing it anyway. 653 00:45:35,450 --> 00:45:40,880 So we get the energies for the singlet and triplet states, 654 00:45:40,880 --> 00:45:47,720 which are E 1s plus E 2s. 655 00:45:50,240 --> 00:45:53,730 Now, remember, these are for a 1s orbital on helium, 656 00:45:53,730 --> 00:45:57,230 so it's going to be four times the energy 657 00:45:57,230 --> 00:46:00,710 of the 1s orbital on hydrogen. But we know what they are. 658 00:46:00,710 --> 00:46:08,390 Plus what we call J 1s2s plus or minus K 1s2s. 659 00:46:11,480 --> 00:46:13,910 So this is clear. 660 00:46:13,910 --> 00:46:16,130 And the integral associated with this 661 00:46:16,130 --> 00:46:18,810 is taking the square of the wave function. 662 00:46:18,810 --> 00:46:21,740 And basically, we're evaluating the interaction 663 00:46:21,740 --> 00:46:31,070 between an electron 1 in the 1s orbital and electron 2 664 00:46:31,070 --> 00:46:32,130 in the 2s orbital. 665 00:46:32,130 --> 00:46:34,220 So that's a perfectly doable thing. 666 00:46:34,220 --> 00:46:35,610 It's a classical quantity. 667 00:46:35,610 --> 00:46:37,660 We have two charged distributions, 668 00:46:37,660 --> 00:46:40,640 and we're calculating the energy interaction 669 00:46:40,640 --> 00:46:44,480 between two electrons in these two charged distributions. 670 00:46:44,480 --> 00:46:46,670 So this is called the Coulomb integral. 671 00:46:49,490 --> 00:46:53,440 And that is an entirely classical quantity. 672 00:46:53,440 --> 00:46:56,550 And this is called the exchange integral. 673 00:46:59,340 --> 00:47:02,520 And the reason we call it an exchange integral is, 674 00:47:02,520 --> 00:47:04,260 basically, we're switching the electrons 675 00:47:04,260 --> 00:47:06,300 between the two orbitals. 676 00:47:06,300 --> 00:47:10,050 And this is something which is entirely quantum mechanical. 677 00:47:10,050 --> 00:47:13,890 It comes about because of the permutation symmetry 678 00:47:13,890 --> 00:47:16,650 that's required for fermions. 679 00:47:16,650 --> 00:47:19,560 And so this is the surprise. 680 00:47:19,560 --> 00:47:23,250 And it also depends on whether you 681 00:47:23,250 --> 00:47:25,740 have a symmetric, or an antisymmetric, 682 00:47:25,740 --> 00:47:27,960 or a single, or a triplet. 683 00:47:27,960 --> 00:47:29,760 And so you end up getting an energy level 684 00:47:29,760 --> 00:47:32,880 diagram for the state of a configuration, which 685 00:47:32,880 --> 00:47:33,970 looks like this. 686 00:47:33,970 --> 00:47:37,570 So we start out with E 0 1s2s. 687 00:47:40,710 --> 00:47:44,700 And then we allow the charged distributions to interact. 688 00:47:44,700 --> 00:47:46,980 And the electrons are going to repel each other. 689 00:47:46,980 --> 00:47:49,030 It's a positive energy. 690 00:47:49,030 --> 00:47:58,870 And so this is E 0 plus J. And then, we include the exchange 691 00:47:58,870 --> 00:48:00,970 interaction. 692 00:48:00,970 --> 00:48:03,520 And so this is the difference 2K. 693 00:48:03,520 --> 00:48:13,880 And this is E0 plus J plus K. And this is E0 plus J minus K. 694 00:48:13,880 --> 00:48:16,840 And this is the triplet state. 695 00:48:19,740 --> 00:48:20,750 And this is a singlet. 696 00:48:24,330 --> 00:48:28,810 And for helium, we call the singlet state 697 00:48:28,810 --> 00:48:35,550 singlet s because total angular momentum is 0. 698 00:48:35,550 --> 00:48:39,570 And spin, it has 0. 699 00:48:39,570 --> 00:48:42,180 And the triplet we call a triplet s. 700 00:48:42,180 --> 00:48:49,300 Now, if we did 1s 2p, we would get singlet and triplet p 701 00:48:49,300 --> 00:48:49,800 states. 702 00:48:53,780 --> 00:48:56,420 But the important thing is, how big is K? 703 00:48:56,420 --> 00:48:59,810 And remember, K is an approximation. 704 00:48:59,810 --> 00:49:04,100 It's a real quantity, but for hydrogenic orbitals, 705 00:49:04,100 --> 00:49:05,420 it's going to be incorrect. 706 00:49:11,490 --> 00:49:22,820 And so 2K 12 is 2.4 electron volts. 707 00:49:22,820 --> 00:49:24,770 And that's a perfectly doable calculation 708 00:49:24,770 --> 00:49:26,900 because you're dealing with simple functions. 709 00:49:26,900 --> 00:49:28,400 And you can do this integral. 710 00:49:28,400 --> 00:49:30,710 And you can actually do it in closed form. 711 00:49:30,710 --> 00:49:32,960 But it's 2.4 electron volts. 712 00:49:32,960 --> 00:49:45,110 And experiment, it's 0.8. 713 00:49:45,110 --> 00:49:48,630 And this tells you something really important. 714 00:49:48,630 --> 00:49:52,320 The repulsion between two electrons, 715 00:49:52,320 --> 00:49:55,440 when you have the correct wave function, 716 00:49:55,440 --> 00:50:01,350 is less than it would be if you're using the wrong wave 717 00:50:01,350 --> 00:50:02,120 functions. 718 00:50:04,940 --> 00:50:08,900 And basically, whenever you do anything using the wrong wave 719 00:50:08,900 --> 00:50:11,570 functions, your energy levels are too large 720 00:50:11,570 --> 00:50:14,480 or the energy differences are too large. 721 00:50:14,480 --> 00:50:16,990 And in this case, the electrons. 722 00:50:16,990 --> 00:50:19,490 And when you solve the exact Schrodinger equation, 723 00:50:19,490 --> 00:50:24,220 the electrons know about how to absorb it to avoid each other. 724 00:50:24,220 --> 00:50:26,200 It's called correlation energy. 725 00:50:26,200 --> 00:50:29,960 And this is what makes all the calculations hard because there 726 00:50:29,960 --> 00:50:33,830 is no closed form analytic expression for the correlation 727 00:50:33,830 --> 00:50:34,670 energy. 728 00:50:34,670 --> 00:50:40,280 And the way you optimize the correlation energy 729 00:50:40,280 --> 00:50:43,850 is throwing a huge basis set at the problem. 730 00:50:43,850 --> 00:50:46,610 And this is why it takes a long. 731 00:50:46,610 --> 00:50:49,450 And you never know that you've converged until you fill 732 00:50:49,450 --> 00:50:52,990 in another million functions. 733 00:50:52,990 --> 00:50:55,150 OK, but so there are several lessons. 734 00:50:55,150 --> 00:51:01,510 And that is we have something that's completely classical, 735 00:51:01,510 --> 00:51:03,970 and we understand what it is. 736 00:51:03,970 --> 00:51:06,100 And then, we have this extra thing 737 00:51:06,100 --> 00:51:08,905 that is a consequence of the permutation symmetry. 738 00:51:11,430 --> 00:51:15,660 And so this extra thing we calculated too big. 739 00:51:15,660 --> 00:51:19,320 But it says that if we have states 740 00:51:19,320 --> 00:51:22,540 that belong to the same electronic configuration-- 741 00:51:22,540 --> 00:51:26,670 in other words, if we specify the orbitals that are occupied, 742 00:51:26,670 --> 00:51:29,160 there's going to be more than one electronic state. 743 00:51:31,820 --> 00:51:35,090 And so configurations split into many different electronic 744 00:51:35,090 --> 00:51:35,900 states. 745 00:51:35,900 --> 00:51:38,840 And we know the machinery that enables us to calculate 746 00:51:38,840 --> 00:51:41,090 their relative energies. 747 00:51:41,090 --> 00:51:44,620 And some of it is simple Coulomb's law. 748 00:51:44,620 --> 00:51:46,930 And some of it is something that's 749 00:51:46,930 --> 00:51:51,210 caused by the required permutation symmetry. 750 00:51:51,210 --> 00:51:52,290 So I'm over time. 751 00:51:52,290 --> 00:51:55,590 But I can't believe that I got to this point 752 00:51:55,590 --> 00:51:59,250 because I delivered everything that was really important. 753 00:51:59,250 --> 00:52:02,520 When we have more than two electrons, everything I've said 754 00:52:02,520 --> 00:52:05,340 is true, it's just more complicated. 755 00:52:05,340 --> 00:52:09,690 And we have to evaluate matrix elements 756 00:52:09,690 --> 00:52:13,650 between determinantal wave functions, which 757 00:52:13,650 --> 00:52:16,620 looks like, if you have n electrons, 758 00:52:16,620 --> 00:52:20,580 there's n factorial terms in a determinate. 759 00:52:20,580 --> 00:52:23,290 And we have n factor l squared matrix elements. 760 00:52:23,290 --> 00:52:24,540 And you don't want to do that. 761 00:52:24,540 --> 00:52:26,820 But there turns out to be a simple algebra that 762 00:52:26,820 --> 00:52:31,740 makes it all pop out, almost with the same level of effort 763 00:52:31,740 --> 00:52:32,940 as this. 764 00:52:32,940 --> 00:52:35,190 And so we'll talk about many electron atoms, 765 00:52:35,190 --> 00:52:36,660 and then we'll go on to molecules 766 00:52:36,660 --> 00:52:39,600 after the many electron atoms lecture. 767 00:52:39,600 --> 00:52:41,150 OK.