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 00:00:21,540 --> 00:00:23,350 ROBERT FIELD: So in the previous lecture, 9 00:00:23,350 --> 00:00:26,430 I told you about many electron atoms. 10 00:00:26,430 --> 00:00:30,870 And in order to describe their structure, 11 00:00:30,870 --> 00:00:33,930 we have to build a Slater determinant, 12 00:00:33,930 --> 00:00:37,410 because that builds in the antisymmetry 13 00:00:37,410 --> 00:00:39,910 between all pairs of orbitals. 14 00:00:39,910 --> 00:00:43,480 Now, Slater determinants are expanded 15 00:00:43,480 --> 00:00:46,980 into a very large number of additive terms. 16 00:00:46,980 --> 00:00:50,340 And there are algebraic methods, which 17 00:00:50,340 --> 00:00:52,740 I describe in a supplement, which 18 00:00:52,740 --> 00:00:57,450 enable you to reduce these algebraic terms, the sum 19 00:00:57,450 --> 00:00:59,490 over an enormous number of terms, 20 00:00:59,490 --> 00:01:03,720 to something almost identical to what you would get 21 00:01:03,720 --> 00:01:05,790 if you didn't antisymmetrize. 22 00:01:05,790 --> 00:01:08,160 They're just a few extra things. 23 00:01:08,160 --> 00:01:12,940 And so once you know the mechanics, it's not a big deal. 24 00:01:12,940 --> 00:01:16,830 It's a little bit more work than just calculating 25 00:01:16,830 --> 00:01:21,840 the individual orbital matrix elements, 26 00:01:21,840 --> 00:01:24,780 but it's nothing special. 27 00:01:24,780 --> 00:01:27,630 And Henry has produced a beautiful thing, 28 00:01:27,630 --> 00:01:32,080 a beautiful handout on Slater-Condon parameters, 29 00:01:32,080 --> 00:01:36,050 which is what you get when you evaluate these Slater 30 00:01:36,050 --> 00:01:39,090 determinal matrix elements. 31 00:01:39,090 --> 00:01:42,510 And they are the things that you use to describe 32 00:01:42,510 --> 00:01:44,340 electronic structure. 33 00:01:44,340 --> 00:01:48,570 So electronic structure is a really complicated 34 00:01:48,570 --> 00:01:52,470 many electron descriptive thing. 35 00:01:52,470 --> 00:01:58,300 But if you can reduce for every atom to a small number 36 00:01:58,300 --> 00:02:02,230 of parameters, most of which you understand intuitively 37 00:02:02,230 --> 00:02:06,460 how they depend on properties of the atom, 38 00:02:06,460 --> 00:02:11,230 like the ionization energy of the individual orbitals, 39 00:02:11,230 --> 00:02:19,220 you can say, yes, I understand all of the periodic table. 40 00:02:19,220 --> 00:02:21,060 That's where it comes from. 41 00:02:21,060 --> 00:02:26,880 The rules enable you to identify the lowest lying states 42 00:02:26,880 --> 00:02:29,310 and to calculate their properties 43 00:02:29,310 --> 00:02:34,140 and to be able to intuit their properties 44 00:02:34,140 --> 00:02:37,644 and to be aware of when something unexpected happens, 45 00:02:37,644 --> 00:02:39,060 which is what we're always looking 46 00:02:39,060 --> 00:02:40,590 for as physical chemists. 47 00:02:40,590 --> 00:02:43,510 We're always looking for a simple minded picture 48 00:02:43,510 --> 00:02:45,940 that breaks. 49 00:02:45,940 --> 00:02:48,460 Because then we can go deeper into our understanding 50 00:02:48,460 --> 00:02:49,600 of all the pieces. 51 00:02:49,600 --> 00:02:54,900 But we're always parameterizing. 52 00:02:54,900 --> 00:02:57,170 OK. 53 00:02:57,170 --> 00:03:00,630 So there's a lot of subtleties that I didn't have time 54 00:03:00,630 --> 00:03:04,950 to talk about in the evaluation of these Slater determinal 55 00:03:04,950 --> 00:03:06,900 matrix elements. 56 00:03:06,900 --> 00:03:10,240 And since whatever problems I give 57 00:03:10,240 --> 00:03:13,420 you connected with Slater determinal matrix elements 58 00:03:13,420 --> 00:03:16,690 or Slater determinants are going to be really simple. 59 00:03:16,690 --> 00:03:19,660 They're probably going to be 2 by 2 or 3 by 3. 60 00:03:19,660 --> 00:03:22,270 You don't really need to know and understand 61 00:03:22,270 --> 00:03:24,370 all the rules and all the shortcuts. 62 00:03:24,370 --> 00:03:29,110 But you've got the basic mechanism. 63 00:03:29,110 --> 00:03:34,190 So a Hamiltonian doesn't depend on spins. 64 00:03:34,190 --> 00:03:37,880 But because of antisymmetry, the spatial part 65 00:03:37,880 --> 00:03:43,190 of the wave function does depend on which spin state you've got. 66 00:03:43,190 --> 00:03:46,670 And so the Hamiltonian knows about spins 67 00:03:46,670 --> 00:03:51,020 in a peculiar, indirect way, which is beautiful. 68 00:03:51,020 --> 00:03:54,260 You've got to admire how clever this Hamiltonian is 69 00:03:54,260 --> 00:03:58,450 to be able to capture these spin effects. 70 00:03:58,450 --> 00:04:00,950 And of course, you have to capture them too. 71 00:04:00,950 --> 00:04:03,740 All right. 72 00:04:03,740 --> 00:04:09,320 So today we're finally getting to molecules. 73 00:04:09,320 --> 00:04:18,860 And the simplest molecule is the homonuclear molecule. 74 00:04:18,860 --> 00:04:21,740 Two hydrogens with one electron. 75 00:04:21,740 --> 00:04:23,570 That's the simplest problem. 76 00:04:23,570 --> 00:04:28,280 And because it is representative of all 77 00:04:28,280 --> 00:04:30,650 of the qualitative effects and most 78 00:04:30,650 --> 00:04:34,040 of the quantitative calculations you do, 79 00:04:34,040 --> 00:04:36,200 it's a really great introduction, 80 00:04:36,200 --> 00:04:40,660 because it captures most of the important stuff 81 00:04:40,660 --> 00:04:42,740 that molecules or small molecules 82 00:04:42,740 --> 00:04:45,960 do, that the electronic structure of small molecule. 83 00:04:45,960 --> 00:04:52,310 So the key to being able to describe small molecules using 84 00:04:52,310 --> 00:04:56,900 this LCAO, Linear Combination of Atomic Orbitals method, 85 00:04:56,900 --> 00:04:59,180 is the variational method. 86 00:04:59,180 --> 00:05:02,280 And this variational method says, 87 00:05:02,280 --> 00:05:07,070 we're going to throw a whole bunch of basis functions 88 00:05:07,070 --> 00:05:08,270 at the problem. 89 00:05:08,270 --> 00:05:12,150 A whole bunch of orbitals, which we choose because we like them 90 00:05:12,150 --> 00:05:15,870 or because the computer likes them. 91 00:05:15,870 --> 00:05:22,100 And we find that the linear combination 92 00:05:22,100 --> 00:05:24,670 of coefficients of these orbitals 93 00:05:24,670 --> 00:05:26,120 that minimizes the energy. 94 00:05:29,550 --> 00:05:34,410 And you can do this in wave mechanics or matrix mechanics. 95 00:05:34,410 --> 00:05:37,110 It's more elegant in matrix mechanics, 96 00:05:37,110 --> 00:05:40,270 and you should be comfortable going from one to the other. 97 00:05:40,270 --> 00:05:42,760 Now, I'm only going to be talking about a 2 by 2 problem. 98 00:05:42,760 --> 00:05:44,820 So you can go easily from one to the other 99 00:05:44,820 --> 00:05:49,740 without even taking a breath. 100 00:05:49,740 --> 00:05:51,730 But anyway, OK. 101 00:05:51,730 --> 00:05:54,570 In approaching this, you're going 102 00:05:54,570 --> 00:05:59,160 to see what's called the general eigenvalue problem where 103 00:05:59,160 --> 00:06:02,610 instead of diagonalizing a simple matrix, 104 00:06:02,610 --> 00:06:06,540 you have to deal with the Hamiltonian matrix 105 00:06:06,540 --> 00:06:09,750 and an overlap matrix, which makes 106 00:06:09,750 --> 00:06:13,650 the solution of the secular equation, 107 00:06:13,650 --> 00:06:18,600 finding the eigenvalues and eigenstates a little bit more 108 00:06:18,600 --> 00:06:20,020 complicated. 109 00:06:20,020 --> 00:06:22,650 So you're going to see it first appear here. 110 00:06:22,650 --> 00:06:26,100 And it can be solved by a simple matrix 111 00:06:26,100 --> 00:06:29,430 equation, which is in a supplement to a later set 112 00:06:29,430 --> 00:06:30,670 of lecture notes. 113 00:06:30,670 --> 00:06:36,480 But this is the thing that consumes the vast majority 114 00:06:36,480 --> 00:06:40,140 of computer time in maybe in the world, 115 00:06:40,140 --> 00:06:43,330 but certainly among chemists. 116 00:06:43,330 --> 00:06:48,820 So the crucial thing, and I wrote it all in caps, 117 00:06:48,820 --> 00:06:56,830 in dealing with this complicated many body problem, 118 00:06:56,830 --> 00:06:59,050 we make a crucial assumption. 119 00:06:59,050 --> 00:07:02,760 And that's called the Born-Oppenheimer approximation. 120 00:07:02,760 --> 00:07:08,120 It's based on the fact that electrons move really fast 121 00:07:08,120 --> 00:07:12,930 and nuclei move really slowly relative to electrons. 122 00:07:12,930 --> 00:07:16,620 And so what we do is we solve the problem 123 00:07:16,620 --> 00:07:19,640 with the nuclei clamped. 124 00:07:19,640 --> 00:07:22,430 Because that's a good approximation. 125 00:07:22,430 --> 00:07:25,870 And then after we solve the electronic problem, 126 00:07:25,870 --> 00:07:29,950 we have what we call a potential energy surface, which 127 00:07:29,950 --> 00:07:35,070 is the electronic energy as a function of nuclear coordinate. 128 00:07:35,070 --> 00:07:37,260 And it's parametric. 129 00:07:37,260 --> 00:07:40,620 But then once we have the potential energy surface, 130 00:07:40,620 --> 00:07:43,530 we can calculate the rotation and vibration. 131 00:07:43,530 --> 00:07:46,050 So basically, what we're doing in making 132 00:07:46,050 --> 00:07:48,120 the Born-Oppenheimer approximation is 133 00:07:48,120 --> 00:07:51,630 we're setting up to treat vibration and rotation 134 00:07:51,630 --> 00:07:54,220 by perturbation theory. 135 00:07:54,220 --> 00:07:56,950 Now, you never see it mentioned in that way, 136 00:07:56,950 --> 00:07:59,410 but that is, in fact, what we're doing. 137 00:07:59,410 --> 00:08:01,270 And the reason we can get away with it 138 00:08:01,270 --> 00:08:04,915 is the difference in velocity of electrons versus nuclei. 139 00:08:07,450 --> 00:08:11,050 But it's a fundamental philosophical point of view 140 00:08:11,050 --> 00:08:13,690 that we take about molecular structure 141 00:08:13,690 --> 00:08:18,400 that we can clamp the nuclei, get most of the insight, 142 00:08:18,400 --> 00:08:22,120 and then allow the nuclei to move and the electrons 143 00:08:22,120 --> 00:08:25,460 with them and calculate everything else. 144 00:08:25,460 --> 00:08:29,110 So this is central to chemistry, the Born-Oppenheimer 145 00:08:29,110 --> 00:08:29,950 approximation. 146 00:08:29,950 --> 00:08:32,335 Without that, we would have a bag of atoms. 147 00:08:35,200 --> 00:08:38,650 And that would be a very bad starting point, 148 00:08:38,650 --> 00:08:41,409 because we believe in bonds. 149 00:08:41,409 --> 00:08:43,730 And a bag of atoms is saying, well, yeah, 150 00:08:43,730 --> 00:08:44,900 we could be statistical. 151 00:08:44,900 --> 00:08:48,710 We could say every atom is capable of being anywhere 152 00:08:48,710 --> 00:08:51,470 and they don't interact in any rational way, 153 00:08:51,470 --> 00:08:54,040 and we would have no insight. 154 00:08:54,040 --> 00:08:58,210 So this is where most of the ideas that you learn 155 00:08:58,210 --> 00:09:03,880 to use as physical chemists and probably any kind of chemist 156 00:09:03,880 --> 00:09:06,670 comes from the Born-Oppenheimer approximation. 157 00:09:06,670 --> 00:09:11,005 And it's widely misunderstood and misrepresented. 158 00:09:11,005 --> 00:09:13,250 But we won't go into that. 159 00:09:13,250 --> 00:09:21,040 OK, we're going to derive two orbitals for hydrogen. H2 plus. 160 00:09:21,040 --> 00:09:25,570 The sigma bonding orbital and the sigma star 161 00:09:25,570 --> 00:09:29,030 anti-bonding orbital. 162 00:09:29,030 --> 00:09:32,770 This is enough to basically understand 163 00:09:32,770 --> 00:09:36,070 how you get bonding and anti-bonding in any molecule. 164 00:09:39,000 --> 00:09:42,930 And I like to say that we understand 165 00:09:42,930 --> 00:09:46,840 bonding in a very simple way. 166 00:09:46,840 --> 00:09:51,850 If you have wave functions for two adjacent atoms 167 00:09:51,850 --> 00:09:57,040 and they overlap in the region between the two nuclei, 168 00:09:57,040 --> 00:09:59,460 you get constructive interference, 169 00:09:59,460 --> 00:10:03,810 you get twice as much amplitude in the region 170 00:10:03,810 --> 00:10:09,570 where the electrons interact with both nuclei, which means 171 00:10:09,570 --> 00:10:11,860 four times the probability. 172 00:10:11,860 --> 00:10:14,730 So a little bit of overlap leads to a big energy 173 00:10:14,730 --> 00:10:19,990 effect because of this constructive interference. 174 00:10:19,990 --> 00:10:22,340 And that's really, really important. 175 00:10:22,340 --> 00:10:25,190 And all sorts of hand waving arguments 176 00:10:25,190 --> 00:10:28,370 about what makes a chemical bond have a lot of currency, 177 00:10:28,370 --> 00:10:31,640 even here at MIT, but that's what a bond is. 178 00:10:31,640 --> 00:10:33,416 It's constructive interference. 179 00:10:39,740 --> 00:10:43,580 All right, so let's get started. 180 00:10:43,580 --> 00:10:48,350 And the lecture notes are extremely long and extremely 181 00:10:48,350 --> 00:10:49,730 beautiful and clear. 182 00:10:49,730 --> 00:10:51,800 They're written by Troy. 183 00:10:51,800 --> 00:10:56,340 I have reduced them to a minimal picture, 184 00:10:56,340 --> 00:11:02,090 but I don't want to detract from them. 185 00:11:02,090 --> 00:11:05,270 The insight is very rich. 186 00:11:05,270 --> 00:11:08,030 OK, so what we have to do first is 187 00:11:08,030 --> 00:11:10,430 to look at a coordinate system. 188 00:11:10,430 --> 00:11:13,880 So we have one hydrogen nucleus. 189 00:11:16,460 --> 00:11:17,180 Another one. 190 00:11:22,190 --> 00:11:23,900 There's a center of mass. 191 00:11:23,900 --> 00:11:30,590 And so we say we have RA, this vector from the center mass 192 00:11:30,590 --> 00:11:35,630 to this RB like that. 193 00:11:35,630 --> 00:11:39,350 And up here is the electron. 194 00:11:39,350 --> 00:11:47,520 And it's at position R. This is at position RB. 195 00:11:50,550 --> 00:11:52,410 Position RA. 196 00:11:55,130 --> 00:12:00,310 Now, the center of mass can be anywhere in space, 197 00:12:00,310 --> 00:12:02,930 and we don't really care. 198 00:12:02,930 --> 00:12:06,290 So we tend to not talk about where the center of mass is. 199 00:12:06,290 --> 00:12:09,200 But there is another set of coordinates for it. 200 00:12:09,200 --> 00:12:09,800 OK. 201 00:12:09,800 --> 00:12:15,420 And now we have this, which is called R sub A. 202 00:12:15,420 --> 00:12:16,490 And it's a vector. 203 00:12:16,490 --> 00:12:23,000 And this is R sub B. So we have uppercase symbols 204 00:12:23,000 --> 00:12:26,520 for nuclear positions and lowercase for electrons, 205 00:12:26,520 --> 00:12:30,340 except they're between electrons and nuclei. 206 00:12:30,340 --> 00:12:32,250 OK. 207 00:12:32,250 --> 00:12:34,580 And once we have this picture, we 208 00:12:34,580 --> 00:12:38,570 can write what the Hamiltonian is, because we're basically 209 00:12:38,570 --> 00:12:42,990 looking at the interactions between these three particles. 210 00:12:42,990 --> 00:12:46,520 And so we have kinetic energy and we have potential energy. 211 00:12:46,520 --> 00:12:54,350 And the Hamiltonian-- excuse me. 212 00:12:54,350 --> 00:13:04,810 --in atomic units is this del squared, 213 00:13:04,810 --> 00:13:10,140 which is the second derivatives with respect to whatever we 214 00:13:10,140 --> 00:13:12,270 have here, which is little r. 215 00:13:12,270 --> 00:13:14,590 And we have 1/2. 216 00:13:14,590 --> 00:13:17,940 It's 1/2 because of atomic units and because the mass 217 00:13:17,940 --> 00:13:21,980 of the electron is treated as one. 218 00:13:21,980 --> 00:13:30,340 And then we have a similar sort of term for del for nucleus A. 219 00:13:30,340 --> 00:13:33,500 And we have a two, because 1/2 in atomic units. 220 00:13:33,500 --> 00:13:39,067 But we have the mass of A. Now, the minus signs, 221 00:13:39,067 --> 00:13:40,400 why do we have minus signs here? 222 00:13:45,830 --> 00:13:48,450 You know this. 223 00:13:48,450 --> 00:13:50,520 AUDIENCE: This is the Schrodinger. 224 00:13:50,520 --> 00:13:52,130 ROBERT FIELD: Yeah, right. 225 00:13:52,130 --> 00:13:54,920 Or we say it's p squared over 2m. 226 00:13:54,920 --> 00:13:57,850 And p is-- right. 227 00:13:57,850 --> 00:14:01,640 OK, but what we're trying to understand stuff, 228 00:14:01,640 --> 00:14:02,640 this is negative. 229 00:14:02,640 --> 00:14:06,860 That means it sort of leads to a stabilization. 230 00:14:06,860 --> 00:14:12,881 And many people talk about bonding in terms of velocity 231 00:14:12,881 --> 00:14:16,100 or momentum, but I don't. 232 00:14:16,100 --> 00:14:21,860 And then we have del squared B over 2 233 00:14:21,860 --> 00:14:25,190 and B and then a repulsion term. 234 00:14:28,250 --> 00:14:30,240 A whole bunch of terms that we have. 235 00:14:30,240 --> 00:14:32,860 So these are the kinetic energy terms. 236 00:14:32,860 --> 00:14:35,960 And then we have the potential energy terms. 237 00:14:35,960 --> 00:14:48,920 1 over RA vector minus little r vector minus 1 over RB 238 00:14:48,920 --> 00:14:51,800 minus little r. 239 00:14:51,800 --> 00:14:57,000 And then the repulsion between RA minus RB. 240 00:15:01,430 --> 00:15:03,110 OK, that's it. 241 00:15:03,110 --> 00:15:05,330 Now, that's a lot of stuff. 242 00:15:05,330 --> 00:15:08,780 And we'd like to make a lot of it go away. 243 00:15:08,780 --> 00:15:10,910 But again, we have potential. 244 00:15:10,910 --> 00:15:13,240 We have kinetic energy and we have potential energy. 245 00:15:18,480 --> 00:15:21,200 So as I've written it, the Hamiltonian, 246 00:15:21,200 --> 00:15:24,200 which is our big problem, is a function 247 00:15:24,200 --> 00:15:33,920 of ra rb r big RA big RB. 248 00:15:37,690 --> 00:15:41,860 That's just ridiculous. 249 00:15:41,860 --> 00:15:44,770 We don't want to look at a function of all 250 00:15:44,770 --> 00:15:46,210 these variables. 251 00:15:46,210 --> 00:15:48,070 We would like to be able to simplify it 252 00:15:48,070 --> 00:15:52,900 so that we can draw pictures on a two dimensional surface 253 00:15:52,900 --> 00:15:55,060 or just to reduce it now. 254 00:15:57,930 --> 00:16:02,220 We like to think in terms of orders of magnitude. 255 00:16:02,220 --> 00:16:06,320 And so it is a very simple matter to say, 256 00:16:06,320 --> 00:16:08,690 well, let's simplify this Hamiltonian 257 00:16:08,690 --> 00:16:11,120 by clamping the nuclei. 258 00:16:11,120 --> 00:16:12,530 They don't move. 259 00:16:12,530 --> 00:16:17,950 If they don't move, then the big Rs are no longer operators. 260 00:16:17,950 --> 00:16:20,060 They're just parameters. 261 00:16:20,060 --> 00:16:28,820 And the contributions to the kinetic energy 262 00:16:28,820 --> 00:16:32,360 of the heavy particles gone. 263 00:16:32,360 --> 00:16:35,240 So there's no rotation and no vibration. 264 00:16:35,240 --> 00:16:36,360 Now, it doesn't go away. 265 00:16:36,360 --> 00:16:37,830 We have to come back with it. 266 00:16:37,830 --> 00:16:40,220 But we come back with it when we have 267 00:16:40,220 --> 00:16:44,330 a good representation of the electronic structure 268 00:16:44,330 --> 00:16:46,340 for the nuclei clamped. 269 00:16:46,340 --> 00:16:50,540 And if we can do it for one internuclear separation, 270 00:16:50,540 --> 00:16:53,550 we can do it for all internuclear separations. 271 00:16:53,550 --> 00:16:59,150 And so if you have a grid of say 1,000 points, 272 00:16:59,150 --> 00:17:03,250 you do the same calculation 1,000 times, except you don't. 273 00:17:03,250 --> 00:17:06,970 Because you know how to extrapolate from one 274 00:17:06,970 --> 00:17:08,200 point to another. 275 00:17:08,200 --> 00:17:10,510 But basically, you are thinking about this 276 00:17:10,510 --> 00:17:14,500 as the problem of finding the electronic wave 277 00:17:14,500 --> 00:17:18,220 function at a grid of points. 278 00:17:18,220 --> 00:17:24,849 And so that turns all of the big R's into parameters, not 279 00:17:24,849 --> 00:17:32,110 operators, and simplifies the problem enormously. 280 00:17:32,110 --> 00:17:37,010 And so the reduced Hamiltonian becomes-- 281 00:17:37,010 --> 00:17:39,270 we'll call it the electronic Hamiltonian. 282 00:17:39,270 --> 00:17:44,280 R semicolon ra rb. 283 00:17:44,280 --> 00:17:46,860 So r is the variable, and these are just parameters 284 00:17:46,860 --> 00:17:48,680 that we sometimes include. 285 00:17:51,960 --> 00:17:58,560 And then the Hamiltonian is just del squared little r over 2 286 00:17:58,560 --> 00:18:00,480 plus V effective. 287 00:18:03,240 --> 00:18:10,230 And this would be as a function of RA and RB. 288 00:18:10,230 --> 00:18:12,750 But it's because of R approximation, 289 00:18:12,750 --> 00:18:14,850 it's just a function of R. So these we pick. 290 00:18:14,850 --> 00:18:19,130 And this is the thing that we're worrying about. 291 00:18:19,130 --> 00:18:25,970 And then there's the 1 over RA minus RB, 292 00:18:25,970 --> 00:18:27,170 which is just a constant. 293 00:18:27,170 --> 00:18:31,420 This is the repulsion between two charged particles. 294 00:18:31,420 --> 00:18:33,150 So this is a lot nicer. 295 00:18:33,150 --> 00:18:35,520 We can also sort of forget about this. 296 00:18:35,520 --> 00:18:37,800 I mean, it's there, but it's a trivial thing 297 00:18:37,800 --> 00:18:39,000 that we can add in. 298 00:18:39,000 --> 00:18:41,940 So we have basically the kind of problem we like. 299 00:18:41,940 --> 00:18:45,050 A kinetic energy problem and a potential energy problem. 300 00:18:50,200 --> 00:18:52,964 So we're going to take this approach 301 00:18:52,964 --> 00:18:54,630 and we're going to learn how to apply it 302 00:18:54,630 --> 00:18:56,340 to the simplest system. 303 00:18:56,340 --> 00:18:59,010 And the simplest system is homonuclear. 304 00:19:03,000 --> 00:19:04,710 That's better than heteronuclear, 305 00:19:04,710 --> 00:19:08,370 because if we had two different atoms, 306 00:19:08,370 --> 00:19:13,500 we'd have to somehow represent their difference in properties. 307 00:19:13,500 --> 00:19:16,050 But because it's homonuclear, we just basically 308 00:19:16,050 --> 00:19:22,150 have an atom, the same atom, repeated, 309 00:19:22,150 --> 00:19:24,820 and there's a lot of symmetry that results. 310 00:19:24,820 --> 00:19:29,596 And the other is only two electrons. 311 00:19:29,596 --> 00:19:30,970 I mean, sorry, only one electron. 312 00:19:35,560 --> 00:19:39,620 Remember, we did hydrogen first, and that was wonderful. 313 00:19:39,620 --> 00:19:42,100 We got everything that was-- 314 00:19:42,100 --> 00:19:45,970 basically everything is related to anything else. 315 00:19:45,970 --> 00:19:49,120 Any property is related to any other property of the hydrogen 316 00:19:49,120 --> 00:19:52,280 atom through this orbital approximation. 317 00:19:52,280 --> 00:19:56,920 And so one electron makes us feel really comfortable, 318 00:19:56,920 --> 00:20:01,570 because we are going to expect that there 319 00:20:01,570 --> 00:20:05,590 is some structure, which means not just a random number 320 00:20:05,590 --> 00:20:08,950 or random collection of descriptive observations 321 00:20:08,950 --> 00:20:11,230 but some relationships. 322 00:20:11,230 --> 00:20:13,600 And we don't have to even antisymmetrize. 323 00:20:13,600 --> 00:20:16,510 So it's really something we can hit out of a park 324 00:20:16,510 --> 00:20:20,340 without getting too stressed. 325 00:20:20,340 --> 00:20:24,030 But there is a problem, and that is, 326 00:20:24,030 --> 00:20:29,970 how do we find the wave functions if we 327 00:20:29,970 --> 00:20:34,710 take a simple basis set? 328 00:20:34,710 --> 00:20:39,480 1s a plus 1s b. 329 00:20:39,480 --> 00:20:43,330 So these are hydrogenic wave functions. 330 00:20:43,330 --> 00:20:45,570 We know them. 331 00:20:45,570 --> 00:20:48,057 Now, they may not be the most convenient thing 332 00:20:48,057 --> 00:20:49,890 when you're doing a large scale calculation. 333 00:20:49,890 --> 00:20:52,590 In fact, nobody uses hydrogenic functions, 334 00:20:52,590 --> 00:20:56,140 even though they make a lot of sense, 335 00:20:56,140 --> 00:20:57,790 because they're not computationally as 336 00:20:57,790 --> 00:20:59,440 easy as Gaussian functions. 337 00:21:02,160 --> 00:21:09,080 So if you need to describe a problem with, say, 10 338 00:21:09,080 --> 00:21:15,290 hydrogenic functions and 1,000 Gaussians, 339 00:21:15,290 --> 00:21:17,730 you're way ahead of the game when you use 1,000 Gaussians, 340 00:21:17,730 --> 00:21:20,660 because the integrals are trivial. 341 00:21:20,660 --> 00:21:22,790 But in terms of understanding, it's 342 00:21:22,790 --> 00:21:27,050 really nice to take things we know and do everything 343 00:21:27,050 --> 00:21:27,880 in terms of them. 344 00:21:30,530 --> 00:21:32,140 OK. 345 00:21:32,140 --> 00:21:34,436 And so this is where we get the variational. 346 00:21:39,300 --> 00:21:42,020 So what we do is we say we're going 347 00:21:42,020 --> 00:21:46,250 to express the wave function as a linear combination 348 00:21:46,250 --> 00:21:48,800 of atomic orbitals. 349 00:21:48,800 --> 00:21:52,280 And that means we can adjust the coefficient 350 00:21:52,280 --> 00:21:57,510 of each atomic orbital to minimize the energy. 351 00:21:57,510 --> 00:22:00,370 And the variational principle or the variational theorem 352 00:22:00,370 --> 00:22:06,050 says, no matter what you do, you can't 353 00:22:06,050 --> 00:22:08,900 calculate an energy lower than the lowest 354 00:22:08,900 --> 00:22:13,290 energy of the system. 355 00:22:13,290 --> 00:22:15,290 Even if you don't know what it is, 356 00:22:15,290 --> 00:22:17,510 you know that anything you calculate 357 00:22:17,510 --> 00:22:19,110 is a little bit higher. 358 00:22:19,110 --> 00:22:20,690 And what you want to do, then, is 359 00:22:20,690 --> 00:22:25,460 to make the energy as small as possible by doing a min max 360 00:22:25,460 --> 00:22:29,010 problem on the coefficients. 361 00:22:29,010 --> 00:22:31,190 And so the variational method is basically that. 362 00:22:36,190 --> 00:22:38,220 OK. 363 00:22:38,220 --> 00:22:42,545 So basis set. 364 00:22:46,150 --> 00:22:49,590 I'm probably not going to finish my notes, even though these 365 00:22:49,590 --> 00:22:54,386 are a massively reduced version of what Troy had written, 366 00:22:54,386 --> 00:22:56,010 because I have some other things to say 367 00:22:56,010 --> 00:22:59,930 that aren't in the notes, and they're really important. 368 00:22:59,930 --> 00:23:04,000 When we do perturbation theory, we implicitly 369 00:23:04,000 --> 00:23:08,170 are dealing with an infinite basis set and an infinite set 370 00:23:08,170 --> 00:23:12,820 of known matrix elements. 371 00:23:12,820 --> 00:23:21,440 But we can't ask a computer to diagonalize an infinite matrix. 372 00:23:21,440 --> 00:23:24,220 So we have a tricky way of doing this 373 00:23:24,220 --> 00:23:27,850 so that we can use perturbation theory 374 00:23:27,850 --> 00:23:33,750 to get a good approximation of almost anything you want. 375 00:23:33,750 --> 00:23:35,940 But if your basis set is really terrible, 376 00:23:35,940 --> 00:23:39,500 then the perturbation theory is going to be terrible too. 377 00:23:39,500 --> 00:23:41,340 I mean, it's going to give correct answers. 378 00:23:41,340 --> 00:23:43,810 You're just going to have to do more calculations. 379 00:23:43,810 --> 00:23:46,590 The variational method says, OK, let's 380 00:23:46,590 --> 00:23:51,720 choose not an infinite basis set but a physically appropriate 381 00:23:51,720 --> 00:23:56,340 basis set or a computationally minimal basis set. 382 00:23:56,340 --> 00:24:03,280 And so we are making decisions about form and size. 383 00:24:03,280 --> 00:24:06,450 So we can use hydrogenic orbitals, 384 00:24:06,450 --> 00:24:07,770 because we know what they are. 385 00:24:07,770 --> 00:24:10,560 We know how to evaluate lots of integrals with them 386 00:24:10,560 --> 00:24:13,890 without too much trouble. 387 00:24:13,890 --> 00:24:15,700 And how many? 388 00:24:15,700 --> 00:24:20,260 Well, we've got two atoms. 389 00:24:20,260 --> 00:24:27,000 And so you sort of need one orbital on one atom 390 00:24:27,000 --> 00:24:29,880 and one orbital on another to even begin 391 00:24:29,880 --> 00:24:33,430 to capture the molecularness of the problem. 392 00:24:33,430 --> 00:24:38,550 So we've chosen the simplest possible form but not 393 00:24:38,550 --> 00:24:40,290 necessarily the most convenient form. 394 00:24:43,010 --> 00:24:46,950 And the size is two. 395 00:24:46,950 --> 00:24:48,250 Now, we could use four. 396 00:24:48,250 --> 00:24:49,840 We could use 100. 397 00:24:49,840 --> 00:24:52,000 And you could use a million. 398 00:24:52,000 --> 00:24:55,330 Most quantum chemical calculations, 399 00:24:55,330 --> 00:24:57,310 which you're going to be doing soon, 400 00:24:57,310 --> 00:25:01,720 involve not just millions but often 401 00:25:01,720 --> 00:25:04,240 billions of basis functions. 402 00:25:04,240 --> 00:25:05,710 And the computer does all the work 403 00:25:05,710 --> 00:25:07,720 and doesn't complain to you. 404 00:25:07,720 --> 00:25:12,260 It doesn't require a trip to the beach or anything like that. 405 00:25:12,260 --> 00:25:15,230 It just says, OK, here is what you wanted me to do. 406 00:25:15,230 --> 00:25:17,770 I don't know whether you-- it's a stupid question 407 00:25:17,770 --> 00:25:23,770 or a smart question, but I'm answering not what you thought 408 00:25:23,770 --> 00:25:27,640 you asked but what you did ask. 409 00:25:27,640 --> 00:25:28,390 OK. 410 00:25:28,390 --> 00:25:31,990 But we're going to show how it works with the simplest 411 00:25:31,990 --> 00:25:35,420 possible basis set. 412 00:25:35,420 --> 00:25:39,350 So our electronic wave function will 413 00:25:39,350 --> 00:25:43,190 be a function of electron position 414 00:25:43,190 --> 00:25:46,650 and parametrically nuclear position. 415 00:25:46,650 --> 00:25:56,290 C1, 1S A R R. I'm going to stop including 416 00:25:56,290 --> 00:26:02,310 this parametrically dependent on R. But eventually I'll do that. 417 00:26:02,310 --> 00:26:03,010 But for now. 418 00:26:08,810 --> 00:26:12,120 OK, so that's the wave function. 419 00:26:12,120 --> 00:26:13,820 The variational function. 420 00:26:13,820 --> 00:26:17,910 And we need to be able to calculate the average energy, 421 00:26:17,910 --> 00:26:21,710 because we know the average energy is going to be larger 422 00:26:21,710 --> 00:26:23,850 than the true energy. 423 00:26:23,850 --> 00:26:27,530 But if we do a good calculation, the energy 424 00:26:27,530 --> 00:26:30,530 difference between the truth and what you get 425 00:26:30,530 --> 00:26:32,970 is small as possible. 426 00:26:32,970 --> 00:26:35,660 And so you can say for a particular basis that we have, 427 00:26:35,660 --> 00:26:38,690 the best possible energies, if you want to do better, 428 00:26:38,690 --> 00:26:41,570 you have to use a bigger or smarter basis set. 429 00:26:45,720 --> 00:26:48,180 And so how do we calculate the average energy? 430 00:26:48,180 --> 00:26:50,600 Well, we do the usual thing where 431 00:26:50,600 --> 00:26:59,280 you say psi H psi d tau over psi star psi d tau. 432 00:26:59,280 --> 00:27:03,670 Now, I'm including this because we need it 433 00:27:03,670 --> 00:27:07,470 when we don't have orthonormal basis functions. 434 00:27:07,470 --> 00:27:11,730 And for this basis set, they're not orthonormal. 435 00:27:11,730 --> 00:27:13,930 They can't be. 436 00:27:13,930 --> 00:27:20,070 The wave functions for one atom are all mutually orthogonal. 437 00:27:20,070 --> 00:27:22,080 But because you have two atoms and there's 438 00:27:22,080 --> 00:27:25,560 overlap between them, they are not orthogonal, 439 00:27:25,560 --> 00:27:28,200 and they're not normalized. 440 00:27:28,200 --> 00:27:30,270 And so we have to do this. 441 00:27:34,030 --> 00:27:43,240 And so the problem is we want to minimize 442 00:27:43,240 --> 00:27:55,960 the energy with respect to C1 or C2 or C1 star or C2 star. 443 00:27:58,900 --> 00:28:01,870 Now, this is a subtle point. 444 00:28:01,870 --> 00:28:08,840 The complex conjugate of C is linearly independent of C. 445 00:28:08,840 --> 00:28:11,650 But they're related to each other. 446 00:28:11,650 --> 00:28:18,240 And so you can say that these derivatives are 0, 447 00:28:18,240 --> 00:28:22,640 and you can choose C1 and C2 or C1 star and C2 star, 448 00:28:22,640 --> 00:28:27,880 and you get the same result. And so it's 449 00:28:27,880 --> 00:28:30,940 important to understand this. 450 00:28:30,940 --> 00:28:33,430 So this is all by means of introduction. 451 00:28:33,430 --> 00:28:36,167 I'm going to start repeating myself. 452 00:28:36,167 --> 00:28:37,375 But let's draw some pictures. 453 00:28:42,930 --> 00:28:50,270 So the complicated thing is the potential. 454 00:28:50,270 --> 00:28:53,610 And so to illustrate the R dependence, 455 00:28:53,610 --> 00:28:57,900 I'm going to try to draw things. 456 00:28:57,900 --> 00:29:00,990 So we have r small and R large. 457 00:29:00,990 --> 00:29:03,390 This is the positions of the nuclei. 458 00:29:03,390 --> 00:29:06,525 And the potential will look like this. 459 00:29:21,280 --> 00:29:25,370 So the potential goes to minus infinity 460 00:29:25,370 --> 00:29:28,790 at the position of the nucleus, because the electron 461 00:29:28,790 --> 00:29:31,830 is attracted to the nucleus. 462 00:29:31,830 --> 00:29:34,420 And when the distance is 0, we have an-- 463 00:29:34,420 --> 00:29:38,410 it's a forgivable infinity because it-- 464 00:29:38,410 --> 00:29:39,730 anyway. 465 00:29:39,730 --> 00:29:44,600 So what I should really have done is exaggerated here. 466 00:29:44,600 --> 00:29:47,210 So when the particles are close together, 467 00:29:47,210 --> 00:29:52,340 the potential in the region between the nuclei 468 00:29:52,340 --> 00:29:55,430 doesn't go as close to 0 as it does when we do this. 469 00:30:00,300 --> 00:30:02,820 And so that's an important fact. 470 00:30:05,930 --> 00:30:10,640 So as the particles are brought together, 471 00:30:10,640 --> 00:30:15,450 something happens in this intermediate region. 472 00:30:15,450 --> 00:30:17,750 And when they're completely separated, 473 00:30:17,750 --> 00:30:19,520 it's as if they are completely separated. 474 00:30:19,520 --> 00:30:20,936 They're not talking to each other. 475 00:30:23,770 --> 00:30:27,360 And the wave functions look like this. 476 00:30:43,810 --> 00:30:46,012 I missed. 477 00:30:46,012 --> 00:30:47,470 And this is what I was telling you. 478 00:30:47,470 --> 00:30:48,326 This is a bond. 479 00:30:48,326 --> 00:30:49,450 This is why we have a bond. 480 00:30:52,430 --> 00:30:54,920 Because we have constructive interference 481 00:30:54,920 --> 00:30:57,890 in the spatial region where the electron can 482 00:30:57,890 --> 00:31:00,020 be attracted to both nuclei. 483 00:31:00,020 --> 00:31:05,240 Whereas here, there is no amplitude in between them 484 00:31:05,240 --> 00:31:08,060 or negligible amplitude, and so there 485 00:31:08,060 --> 00:31:11,030 isn't much of a contribution from the attraction 486 00:31:11,030 --> 00:31:12,780 to both nuclei. 487 00:31:12,780 --> 00:31:15,920 And so this is the separated atom or approaching separated 488 00:31:15,920 --> 00:31:16,920 atom limit. 489 00:31:16,920 --> 00:31:18,530 And this is the molecular limit. 490 00:31:18,530 --> 00:31:19,970 And now it's possible that you can 491 00:31:19,970 --> 00:31:24,610 squeeze the atoms too close together 492 00:31:24,610 --> 00:31:27,640 and bad things start to happen. 493 00:31:27,640 --> 00:31:29,710 Because then what you're doing is you're 494 00:31:29,710 --> 00:31:33,260 putting two electrons into the same spin orbital. 495 00:31:33,260 --> 00:31:37,030 And we know that that's overlap repulsion 496 00:31:37,030 --> 00:31:39,490 or it violates the exclusion principle 497 00:31:39,490 --> 00:31:42,520 and there is an energy penalty that you 498 00:31:42,520 --> 00:31:45,190 pay in order to get too close. 499 00:31:45,190 --> 00:31:48,840 But basically, this is what we're going to be explaining. 500 00:31:48,840 --> 00:31:51,220 And we're going to recover this from a real calculation. 501 00:31:59,450 --> 00:32:01,370 OK. 502 00:32:01,370 --> 00:32:07,940 So again, we want to minimize the average energy. 503 00:32:07,940 --> 00:32:11,810 And the average energy, as I've written somewhere, 504 00:32:11,810 --> 00:32:25,240 is given by psi star H psi d tau psi star psi d tau. 505 00:32:25,240 --> 00:32:27,510 And if we are writing these things 506 00:32:27,510 --> 00:32:34,440 as C1 1S A times C2 1S B, then we 507 00:32:34,440 --> 00:32:39,600 end up getting stuff that we know how to calculate now. 508 00:32:44,720 --> 00:33:03,320 1S A. This orbital is not orthogonal to 1S B. 509 00:33:03,320 --> 00:33:04,100 It can't be. 510 00:33:07,210 --> 00:33:09,180 And so this is something that we had 511 00:33:09,180 --> 00:33:12,510 been able to avoid in all of our one dimensional problems. 512 00:33:12,510 --> 00:33:17,250 We could always think about a orthonormalized basis set. 513 00:33:17,250 --> 00:33:20,870 But the effect that we're really interested in 514 00:33:20,870 --> 00:33:25,320 is based on the fact that these guys aren't orthogonal. 515 00:33:25,320 --> 00:33:31,650 You can't just say we're going to write an orthonormal basis 516 00:33:31,650 --> 00:33:32,490 set. 517 00:33:32,490 --> 00:33:34,170 It's not possible. 518 00:33:34,170 --> 00:33:38,250 If you want to construct an orthonormal basis set, 519 00:33:38,250 --> 00:33:41,010 there is a transformation you can do. 520 00:33:41,010 --> 00:33:43,770 And it's a simple transformation, 521 00:33:43,770 --> 00:33:45,820 and it's done all the time. 522 00:33:45,820 --> 00:33:50,280 And you'll see it, but not today. 523 00:33:50,280 --> 00:33:52,230 But you still do the calculations 524 00:33:52,230 --> 00:33:56,610 on the non-orthogonal basis functions. 525 00:33:56,610 --> 00:33:59,850 And then you do a transformation that takes into account 526 00:33:59,850 --> 00:34:03,150 the lack of orthonormality. 527 00:34:03,150 --> 00:34:06,310 For now just accept it. 528 00:34:06,310 --> 00:34:10,380 And so we're going to have to worry about an integral 529 00:34:10,380 --> 00:34:11,864 like this one. 530 00:34:11,864 --> 00:34:15,310 SA 1 SB. 531 00:34:18,300 --> 00:34:20,340 And we call this the overlap integral. 532 00:34:27,760 --> 00:34:32,280 And this is something that we have to not ignore. 533 00:34:35,179 --> 00:34:40,010 So now we can approach this problem using wave function 534 00:34:40,010 --> 00:34:41,239 notation. 535 00:34:41,239 --> 00:34:45,170 And it's most familiar. 536 00:34:45,170 --> 00:34:49,580 But we can also use these things saying, OK, this 537 00:34:49,580 --> 00:34:54,320 is the complete set of the variational coefficients. 538 00:34:54,320 --> 00:34:58,770 And this is a step towards the matrix picture. 539 00:34:58,770 --> 00:35:07,410 Because it's basically a linear array of mixing coefficients. 540 00:35:07,410 --> 00:35:11,240 And you could think of that as a vector. 541 00:35:11,240 --> 00:35:14,210 And in fact, that's how you do the vector picture, 542 00:35:14,210 --> 00:35:18,880 but we'll use sort of a hybrid notation for the time being. 543 00:35:18,880 --> 00:35:28,970 But we can also say, OK, this means C1 C2 Cn. 544 00:35:28,970 --> 00:35:38,420 And this means C1 star Cn star. 545 00:35:38,420 --> 00:35:42,380 And here we are in the matrix picture. 546 00:35:42,380 --> 00:35:45,100 OK, but I will use sort of a hybrid picture 547 00:35:45,100 --> 00:35:48,470 until you're ready to take the complete jump. 548 00:35:48,470 --> 00:35:53,050 And so the average energy expressed 549 00:35:53,050 --> 00:35:55,900 as a function of this set of coefficients 550 00:35:55,900 --> 00:35:57,740 is going to be integral. 551 00:35:57,740 --> 00:36:26,180 Psi C star H psi C d tau over integral psi C star psi. 552 00:36:30,120 --> 00:36:31,830 The reason I'm being hesitant here 553 00:36:31,830 --> 00:36:35,400 is because I work in the matrix picture, 554 00:36:35,400 --> 00:36:41,720 and this hybrid picture is away from both comfort zones. 555 00:36:41,720 --> 00:36:44,850 But it is leading you towards what you need to see, 556 00:36:44,850 --> 00:36:46,910 and so we're going to do it. 557 00:36:46,910 --> 00:36:47,520 OK. 558 00:36:47,520 --> 00:36:50,400 So in the wave function picture, this thing 559 00:36:50,400 --> 00:37:02,635 becomes just C1 1S A plus C2 1S B, 560 00:37:02,635 --> 00:37:15,250 the whole thing conjugated, H C1 1S A C2 psi 1S B d tau. 561 00:37:19,200 --> 00:37:24,780 And the same sort of thing down here, except no Hamiltonian. 562 00:37:24,780 --> 00:37:29,140 And so we can now slavishly write down 563 00:37:29,140 --> 00:37:33,190 all of the terms that result. And what 564 00:37:33,190 --> 00:37:40,100 we get is from the Hamiltonian matrix element. 565 00:37:40,100 --> 00:37:52,900 So for H, we get C1 star C1 H11 plus C1 star C2 566 00:37:52,900 --> 00:38:09,130 H1 H12 plus C2 star C1 H21 plus C2 star C2 H22. 567 00:38:09,130 --> 00:38:11,470 OK, we think we know some stuff. 568 00:38:11,470 --> 00:38:15,590 We think this is going to be the energy of the hydrogen atom. 569 00:38:15,590 --> 00:38:17,140 But it isn't. 570 00:38:17,140 --> 00:38:23,990 Because the orbital one sees both nuclei. 571 00:38:23,990 --> 00:38:28,150 And so this is something which we just call this guy. 572 00:38:28,150 --> 00:38:30,100 We're calling this epsilon. 573 00:38:30,100 --> 00:38:32,635 And that is a function of internuclear distance. 574 00:38:35,400 --> 00:38:41,275 But this guy is the same, because it's homonuclear. 575 00:38:47,220 --> 00:38:52,200 And this, well, we can call it V12 or V. 576 00:38:52,200 --> 00:38:55,570 And these two guys are the same thing. 577 00:38:55,570 --> 00:39:02,900 So we've reduced the problem to a simple equation. 578 00:39:02,900 --> 00:39:07,260 But I'm not going to go all the way there. 579 00:39:07,260 --> 00:39:10,330 But you know how to do this. 580 00:39:10,330 --> 00:39:12,390 You know how to do the overlap integral. 581 00:39:12,390 --> 00:39:14,610 You're gonna have a similar sort of thing. 582 00:39:14,610 --> 00:39:19,950 S11 is the overlap of the orbital 1 with itself. 583 00:39:19,950 --> 00:39:21,330 And that one is 1. 584 00:39:25,390 --> 00:39:27,460 But S12 is S. 585 00:39:27,460 --> 00:39:37,240 So anyway, what we end up doing is a lot of algebra. 586 00:39:37,240 --> 00:39:41,250 And what we end up getting when we impose the condition 587 00:39:41,250 --> 00:39:47,350 that the partial of everything with respect to C1 star 588 00:39:47,350 --> 00:39:50,230 is equal to 0, we get an equation 589 00:39:50,230 --> 00:39:51,820 for the average energy. 590 00:39:51,820 --> 00:39:56,170 And the average energy is-- 591 00:39:56,170 --> 00:39:57,530 oh, I'm sorry. 592 00:39:57,530 --> 00:39:58,590 C1. 593 00:39:58,590 --> 00:40:02,390 When we take the derivative with respect to C1, what we get 594 00:40:02,390 --> 00:40:04,040 is this equation. 595 00:40:04,040 --> 00:40:21,793 C1 star H11 plus C2 star H22 over C1 star S11 plus C2 star 596 00:40:21,793 --> 00:40:22,293 S21. 597 00:40:25,920 --> 00:40:29,380 And so we could also do the same thing, 598 00:40:29,380 --> 00:40:33,150 taking the partial derivative with respect to C2 599 00:40:33,150 --> 00:40:35,144 and setting that equal to 0. 600 00:40:35,144 --> 00:40:36,310 But we get another equation. 601 00:40:39,890 --> 00:40:52,350 And so that other equation would be E average is equal to-- 602 00:40:59,380 --> 00:41:01,010 no, I'm not even going to write it, 603 00:41:01,010 --> 00:41:02,593 because I didn't write it in my notes, 604 00:41:02,593 --> 00:41:06,860 and I'm operating on batteries. 605 00:41:06,860 --> 00:41:08,390 You can write that. 606 00:41:08,390 --> 00:41:11,900 And so we have two equations for the average energy. 607 00:41:11,900 --> 00:41:16,580 And well, that's kind of good, because we have two unknowns. 608 00:41:16,580 --> 00:41:19,940 So we're going to be able to solve for them. 609 00:41:19,940 --> 00:41:21,130 OK. 610 00:41:21,130 --> 00:41:31,480 Now, we could also approach this problem using matrix methods. 611 00:41:31,480 --> 00:41:33,910 And since that's the way computers think 612 00:41:33,910 --> 00:41:37,600 about the problem and that's the way most people think 613 00:41:37,600 --> 00:41:40,900 about the problem, you really want to understand the matrix 614 00:41:40,900 --> 00:41:41,890 picture. 615 00:41:41,890 --> 00:41:45,040 And at risk of repeating myself, we 616 00:41:45,040 --> 00:41:53,750 know for this two level problem, we have H11 H12 H21 H22. 617 00:41:53,750 --> 00:42:01,840 And we know that the S matrix S11 S12 S21 S22. 618 00:42:01,840 --> 00:42:10,180 So we have an equation C1 star C2 star 619 00:42:10,180 --> 00:42:26,620 times the H matrix is equal to E average times the C1 star C2 620 00:42:26,620 --> 00:42:31,420 star times the S matrix. 621 00:42:31,420 --> 00:42:34,860 So this is the H matrix and this is the S matrix. 622 00:42:34,860 --> 00:42:37,410 And you can write this down. 623 00:42:37,410 --> 00:42:43,650 And when you take the derivatives with respect to C, 624 00:42:43,650 --> 00:42:57,670 you get C dagger H is equal to E average times C dagger S. 625 00:42:57,670 --> 00:43:01,780 Now, if instead of taking derivatives with respect 626 00:43:01,780 --> 00:43:04,780 to the Cs we took the derivatives with respect 627 00:43:04,780 --> 00:43:10,130 to the C stars, we'd get a different equation. 628 00:43:10,130 --> 00:43:22,870 And that equation would be HC is equal to E average times S then 629 00:43:22,870 --> 00:43:26,970 C. This guy looks familiar. 630 00:43:29,510 --> 00:43:33,610 It looks like Hamiltonian times something 631 00:43:33,610 --> 00:43:38,444 is equal to the energy times-- 632 00:43:38,444 --> 00:43:40,040 uh oh. 633 00:43:40,040 --> 00:43:43,730 We have something else here. 634 00:43:43,730 --> 00:43:45,042 It's not just the energy. 635 00:43:45,042 --> 00:43:46,250 We have the overlap integral. 636 00:43:49,040 --> 00:43:54,050 So these two things are examples of the general eigenvalue 637 00:43:54,050 --> 00:43:56,210 equation. 638 00:43:56,210 --> 00:44:01,060 And so in almost all calculations, 639 00:44:01,060 --> 00:44:07,480 the fact that S is present leads to problems of algebra. 640 00:44:11,490 --> 00:44:18,810 But in this particular problem, we know S12 is equal to S21. 641 00:44:18,810 --> 00:44:26,330 And this epsilon parameter can be-- 642 00:44:26,330 --> 00:44:32,060 so we have H11 is equal to H22. 643 00:44:32,060 --> 00:44:35,490 They're both parametrically dependent on internuclear 644 00:44:35,490 --> 00:44:39,950 distance, and they're equal to epsilon of R. 645 00:44:39,950 --> 00:44:43,760 So what we end up with is an equation that 646 00:44:43,760 --> 00:44:44,865 looks really familiar. 647 00:44:50,780 --> 00:44:53,450 We have a 2 by 2 matrix, which we can diagonalize. 648 00:44:53,450 --> 00:44:56,060 We can find the energies. 649 00:44:56,060 --> 00:44:59,420 And so what you end up getting when you do that. 650 00:44:59,420 --> 00:45:03,470 And I want to spare you my attempting 651 00:45:03,470 --> 00:45:06,226 to do a derivation in real time. 652 00:45:06,226 --> 00:45:07,600 You can look at the notes and you 653 00:45:07,600 --> 00:45:11,950 can see that the results are what you care about 654 00:45:11,950 --> 00:45:13,910 and what we're going to interpret. 655 00:45:13,910 --> 00:45:14,870 So here we go. 656 00:45:18,940 --> 00:45:20,760 So what you find. 657 00:45:20,760 --> 00:45:25,740 The average energy for one of the orbitals that we get. 658 00:45:25,740 --> 00:45:27,000 Remember, we got a 2 by 2. 659 00:45:27,000 --> 00:45:31,740 We're going to have two eigenvalues and eigenstates. 660 00:45:31,740 --> 00:45:41,880 And so for one of those, we get epsilon plus V12 1 plus S1. 661 00:45:46,730 --> 00:45:49,430 So these are symbols that we've seen. 662 00:45:49,430 --> 00:45:53,430 And this tells us now this, this, 663 00:45:53,430 --> 00:45:56,350 and this are functions of R. So we're 664 00:45:56,350 --> 00:45:58,960 going to have an energy for the sigma orbital, which 665 00:45:58,960 --> 00:46:01,150 is a function of R. 666 00:46:01,150 --> 00:46:07,120 And we have another solution, which we're calling sigma star. 667 00:46:09,910 --> 00:46:19,650 And that one is epsilon minus V12 over 1 minus S12. 668 00:46:19,650 --> 00:46:24,840 So we'd like to be able to know which of these is 669 00:46:24,840 --> 00:46:27,210 the absolute-- one of these is-- 670 00:46:30,620 --> 00:46:38,850 so we have sigma and sigma star. 671 00:46:38,850 --> 00:46:46,100 Both as a function of R. And so if we 672 00:46:46,100 --> 00:46:49,430 draw a molecular orbital diagram and we say, OK, well here we 673 00:46:49,430 --> 00:46:59,770 have the energy of 1SA, which is epsilon as a function of R. 674 00:46:59,770 --> 00:47:09,650 And this is epsilon energy 1SB, which is equal to epsilon of R. 675 00:47:09,650 --> 00:47:13,310 Normally we choose R as equal to infinity 676 00:47:13,310 --> 00:47:15,770 for the separated atom. 677 00:47:15,770 --> 00:47:18,410 So these are actually just the energies 678 00:47:18,410 --> 00:47:20,810 of the individual orbitals. 679 00:47:20,810 --> 00:47:24,320 And what we end up getting is a molecular orbital diagram 680 00:47:24,320 --> 00:47:30,790 which, in an exaggerated form, is like this. 681 00:47:30,790 --> 00:47:33,130 We get bonding, because it's more 682 00:47:33,130 --> 00:47:36,140 stable than the separate atoms, and anti-bonding, 683 00:47:36,140 --> 00:47:37,630 which is less stable. 684 00:47:37,630 --> 00:47:40,600 And so one question we'd like to ask is, 685 00:47:40,600 --> 00:47:43,630 well, one question is why is it bonding? 686 00:47:43,630 --> 00:47:44,680 Why is this more stable? 687 00:47:44,680 --> 00:47:47,710 And the answer is constructive interference 688 00:47:47,710 --> 00:47:51,070 in the important energy region. 689 00:47:51,070 --> 00:47:53,980 But another question is, is this bigger? 690 00:47:53,980 --> 00:47:59,480 Is this a larger destabilization than this stabilization? 691 00:47:59,480 --> 00:48:02,500 And it's kind of hard to scope that out, 692 00:48:02,500 --> 00:48:07,790 because here we have two parameters, which are added, 693 00:48:07,790 --> 00:48:09,820 and here we have a difference of two parameters. 694 00:48:12,780 --> 00:48:14,620 And here we have a denominator, which 695 00:48:14,620 --> 00:48:19,210 is going to be larger for the sigma orbital 696 00:48:19,210 --> 00:48:21,580 than the sigma star. 697 00:48:21,580 --> 00:48:26,160 And so we don't know whether the plus versus minus V 698 00:48:26,160 --> 00:48:28,030 is dominant. 699 00:48:28,030 --> 00:48:30,310 But the energy denominator is saying, 700 00:48:30,310 --> 00:48:33,330 yeah, this one, the anti-bonding orbital 701 00:48:33,330 --> 00:48:36,300 is more anti-bonding than the bonding one. 702 00:48:36,300 --> 00:48:38,520 Now, you can actually do some algebra 703 00:48:38,520 --> 00:48:41,700 to calculate what is the difference in energy 704 00:48:41,700 --> 00:48:49,380 between the sigma star as a function of R. 705 00:48:49,380 --> 00:48:54,250 But evaluated at R sub E, which is the minimum-- 706 00:48:54,250 --> 00:49:00,000 the energy where the bonding orbital has its minimum energy. 707 00:49:00,000 --> 00:49:04,830 Then if you evaluate at that particular value of R 708 00:49:04,830 --> 00:49:13,960 and you make that comparison B 1S evaluated at R sub B, 709 00:49:13,960 --> 00:49:25,360 and you get epsilon times S12 minus-- 710 00:49:30,170 --> 00:49:32,300 so I'm doing sigma star, so that's 711 00:49:32,300 --> 00:49:44,120 this one minus B12 over 1 plus 1 minus S12. 712 00:49:44,120 --> 00:49:48,590 And for the sigma orbital, I have a similar equation, 713 00:49:48,590 --> 00:49:59,830 but I get epsilon S12 minus B12 over 1 minus S12. 714 00:49:59,830 --> 00:50:03,520 So we have the same numerator for these two cases 715 00:50:03,520 --> 00:50:05,330 but a different denominator. 716 00:50:05,330 --> 00:50:07,990 And so this is saying this simple variational calculation 717 00:50:07,990 --> 00:50:13,110 says you're going to get a pair of orbitals, one bonding, 718 00:50:13,110 --> 00:50:14,650 one anti-bonding. 719 00:50:14,650 --> 00:50:17,460 And the anti-bonding one is more destabilized 720 00:50:17,460 --> 00:50:19,540 relative to the separate atoms then 721 00:50:19,540 --> 00:50:22,050 the bonding one is stabilized. 722 00:50:22,050 --> 00:50:25,930 And this is a kind of general expectation 723 00:50:25,930 --> 00:50:27,370 that the anti-bonding orbitals are 724 00:50:27,370 --> 00:50:31,090 more destabilized than the bonding ones. 725 00:50:31,090 --> 00:50:33,970 But it's not a proof, because its just a proof 726 00:50:33,970 --> 00:50:36,490 for the simplest possible case. 727 00:50:36,490 --> 00:50:38,570 So I'm done. 728 00:50:38,570 --> 00:50:43,000 And in the next lecture, we're going 729 00:50:43,000 --> 00:50:46,840 to use molecular orbital diagrams like this 730 00:50:46,840 --> 00:50:50,405 to describe a series of examples. 731 00:50:53,210 --> 00:51:01,241 Homonuclear diatomic molecules, heteronuclear AH, and AB. 732 00:51:01,241 --> 00:51:04,300 H is special, and this is more general. 733 00:51:04,300 --> 00:51:06,120 And what we'd like to be able to do 734 00:51:06,120 --> 00:51:09,180 is without a great deal of thought 735 00:51:09,180 --> 00:51:11,590 construct the molecular orbital diagram 736 00:51:11,590 --> 00:51:14,995 for basically any diatomic molecule using some insight. 737 00:51:18,790 --> 00:51:23,190 Now, this also transfers to polyatomic molecules. 738 00:51:23,190 --> 00:51:25,020 And it's especially beautiful the way 739 00:51:25,020 --> 00:51:29,130 it transfers, because electronic structure is usually 740 00:51:29,130 --> 00:51:32,850 centered at a particular part of the molecule. 741 00:51:32,850 --> 00:51:35,670 And so, basically, you're reducing polyatomic molecules 742 00:51:35,670 --> 00:51:38,580 to a diatomic that you understanding the bonding 743 00:51:38,580 --> 00:51:40,050 and anti-bonding. 744 00:51:40,050 --> 00:51:44,910 OK, so I'll continue on Monday. 745 00:51:44,910 --> 00:51:48,660 And your problem set for this week is due on Monday. 746 00:51:48,660 --> 00:51:50,490 And the following set for the next week 747 00:51:50,490 --> 00:51:54,410 will also be due on Monday. 748 00:51:54,410 --> 00:51:55,260 Not that Monday. 749 00:51:55,260 --> 00:51:56,135 The following Monday. 750 00:51:59,547 --> 00:52:01,380 And that's the next to the last problem set. 751 00:52:01,380 --> 00:52:06,040 So you're inside of the home base. 752 00:52:06,040 --> 00:52:09,430 OK, have a good weekend.