1 00:00:01,970 --> 00:00:04,710 Welcome back everyone. I'm Charles Clark. 2 00:00:04,710 --> 00:00:08,654 And in this part, we're going to look at the application of the Gaussian functions 3 00:00:08,654 --> 00:00:12,540 that you derived at the end of the last segment and their value for a wide range 4 00:00:12,540 --> 00:00:19,320 of practical calculations using something called the Variational Theorem. 5 00:00:19,320 --> 00:00:26,442 Now, I must say that I always carry a Gaussian with me wherever I go. 6 00:00:26,442 --> 00:00:31,806 this one, as very simple to remember. it's the ground state of a harmonic 7 00:00:31,806 --> 00:00:36,702 oscillator, and you can use it in a variety of applications for estimating 8 00:00:36,702 --> 00:00:43,558 ground state energies of other systems. That's an exercise that you conduct in 9 00:00:43,558 --> 00:00:47,410 the assignment this week. So, let's proceed. 10 00:00:49,850 --> 00:00:54,818 [SOUND] So, here is the three-dimensional version of the way function that you 11 00:00:54,818 --> 00:01:01,175 derived the end of last segment. And here's the one I prefer to remember 12 00:01:01,175 --> 00:01:06,139 that has the, it, it expresses the the length scale, the harmonic oscillator, 13 00:01:06,139 --> 00:01:13,110 through this parameter d which is the square root of h bar over m omega. 14 00:01:13,110 --> 00:01:18,840 so d has to have the units of length. And now, have a quick InVideo quiz just 15 00:01:18,840 --> 00:01:27,589 to refresh your appreciation of that. Well, the answer to that inline quiz was 16 00:01:27,589 --> 00:01:32,261 a of course sort of apparent right here and the messages, the way function has 17 00:01:32,261 --> 00:01:37,655 dimensional units. it's the units of the inverse square root 18 00:01:37,655 --> 00:01:41,790 of length times the number of dimensions. So, in one dimension, it's 1 over the 19 00:01:41,790 --> 00:01:45,714 root length. in three dimensions, it's 1 over length 20 00:01:45,714 --> 00:01:50,698 to the three halves power. [SOUND] Gaussians have become a major 21 00:01:50,698 --> 00:01:55,967 workhorse of actually of chemistry rather than physics. 22 00:01:55,967 --> 00:02:01,863 And I think it's it seems plausible to think that much more work using quantum 23 00:02:01,863 --> 00:02:08,270 mechanics is done by chemists rather than by physicists. 24 00:02:08,270 --> 00:02:13,311 So, for example the Noble Prize in Chemistry in 1998 was awarded to this 25 00:02:13,311 --> 00:02:17,855 man, John Pople for his development of computational methods in quantum 26 00:02:17,855 --> 00:02:23,161 chemistry. And what he did in particular, was lead 27 00:02:23,161 --> 00:02:27,717 the development, the very large scale program development effort for using 28 00:02:27,717 --> 00:02:35,848 Gaussians to compute molecular structure. And if you if you look at Google Scholar 29 00:02:35,848 --> 00:02:41,320 and look at the just search on Gaussian orbitals, seems that the top two entries 30 00:02:41,320 --> 00:02:47,108 that I got. this one here is co-authored by Pople, 31 00:02:47,108 --> 00:02:54,448 7307 citations and next to it, a paper that reports Gaussian basis. 32 00:02:54,448 --> 00:02:59,780 And so there's choices of particular values of Gaussian orbitals that are 33 00:02:59,780 --> 00:03:07,800 optimized for doing calculations of of molecular structure and light atoms. 34 00:03:07,800 --> 00:03:13,160 And this it's been cited 14,000, over 14,000 times. 35 00:03:13,160 --> 00:03:17,210 Now, I don't know about you, but where I live that's a large number. 36 00:03:17,210 --> 00:03:22,212 For example, if you look at the most cited paper in Physics, using the same 37 00:03:22,212 --> 00:03:28,152 Google search engine, it's this paper by Steven Weinberg. 38 00:03:28,152 --> 00:03:34,240 Also Nobel laureate in Physics cited a little bit over 10,000 times. 39 00:03:34,240 --> 00:03:37,610 That's a very large number, largest number in Physics. 40 00:03:37,610 --> 00:03:45,935 but as you can see the the work in chemistry is it generates a lot of effort 41 00:03:45,935 --> 00:03:54,116 in terms of number citations. That's not a statement of its absolute 42 00:03:54,116 --> 00:03:59,576 importance, but it does indicate the the effect that this development has had in 43 00:03:59,576 --> 00:04:06,834 the field of science as a whole. Why has there been such a great attention 44 00:04:06,834 --> 00:04:12,432 paid to these Gaussian orbitals. Well, it has to do with the use of very 45 00:04:12,432 --> 00:04:17,192 important theorem of quantum mechanics, the Variational Theorem, which states 46 00:04:17,192 --> 00:04:21,532 that for any Hamiltonian and for any function that we choose, so this is not 47 00:04:21,532 --> 00:04:28,450 necessary an eigenfunction, it's just some arbitrary function. 48 00:04:28,450 --> 00:04:32,006 I should say, this is a course in physics, not in mathematics. 49 00:04:32,006 --> 00:04:34,644 So, there's some qualifications, so I say here. 50 00:04:34,644 --> 00:04:39,400 But, let's just let those be debated in mathematician study group. 51 00:04:39,400 --> 00:04:45,386 for any Hamiltonian and for any function that's normalized, the expectation value 52 00:04:45,386 --> 00:04:51,044 of the Hamiltonian over that function is greater than or equal to the ground-state 53 00:04:51,044 --> 00:05:00,829 energy. [SOUND]. 54 00:05:00,829 --> 00:05:05,380 That is the exact, it's greater than or equal to the exact minimum energy. 55 00:05:05,380 --> 00:05:11,035 And so this means that if you have computational capabilities and a rich an 56 00:05:11,035 --> 00:05:17,212 intelligent way of choosing lots and lots of trial functions you can, you can make 57 00:05:17,212 --> 00:05:22,084 start to make estimates, of the ground-state energy by continuing 58 00:05:22,084 --> 00:05:31,390 refinement of approximations, to get you to the lower value. 59 00:05:31,390 --> 00:05:37,066 Now, for those of you who are, sitting peacefully and undisturbed right now, if 60 00:05:37,066 --> 00:05:42,570 you haven't yet proven any theorem in quantum mechanics, this would be a good 61 00:05:42,570 --> 00:05:48,562 one to prove. So, for such people, you might consider 62 00:05:48,562 --> 00:05:53,010 pausing and thinking about how you would prove this. 63 00:05:53,010 --> 00:05:57,880 I'm going to go, run through the proof. It's just a few lines on the next slide. 64 00:05:57,880 --> 00:06:03,187 And I'll just if you want a suggestion this is a proof which is quite 65 00:06:03,187 --> 00:06:09,277 straightforward, if you expand the, the trial wave function psi in a sequence of 66 00:06:09,277 --> 00:06:19,205 eigenfunctions of the Hamiltonian Age. So, I'll pause while you think and then 67 00:06:19,205 --> 00:06:24,210 well, if you pause, I'm going to continue but you might pause and, and think about 68 00:06:24,210 --> 00:06:29,640 how that, how to prove like that might look. 69 00:06:29,640 --> 00:06:33,632 [SOUND] Well, here is the, here is the proof of the Variation of Theorem. 70 00:06:33,632 --> 00:06:38,294 So, this is the statement of theorem up to this point and now what we do is just 71 00:06:38,294 --> 00:06:43,326 use a familiar expansion of the arbitrary weigh function psi, I wish I should have 72 00:06:43,326 --> 00:06:50,216 put a, made this a cat form. In terms of the I, in terms of a, well it 73 00:06:50,216 --> 00:06:54,494 could be any orthonormal basis, but in particular we'll choose as our basis the 74 00:06:54,494 --> 00:06:59,440 functions that are eigenfunctions, the Hamiltonian. 75 00:06:59,440 --> 00:07:03,296 Okay. So we expand psi in the eigenfunctions of 76 00:07:03,296 --> 00:07:08,813 Hamiltonian H. And now, we take the expectation value of 77 00:07:08,813 --> 00:07:15,684 psi H psi. That's the thing that we're supposed to 78 00:07:15,684 --> 00:07:21,472 be computing. And so now, we just, we just put this 79 00:07:21,472 --> 00:07:27,706 form of psi into the the the Ha-, the Hamiltonian. 80 00:07:27,706 --> 00:07:32,152 So, we're just, this part is the, the cat side of that and it's the, the 81 00:07:32,152 --> 00:07:38,496 [INAUDIBLE] side here and then we just sandwich the Hamiltonian. 82 00:07:38,496 --> 00:07:44,000 And between are these are all very routine mechanical procedures. 83 00:07:44,000 --> 00:07:47,944 Now the critical point here is that the Hamiltonian is diagonal in the 84 00:07:47,944 --> 00:07:54,760 eigenfunc-, in its owneigenfunctions. So, this matrix element here reduces just 85 00:07:54,760 --> 00:08:00,280 to the energy. And we just, we just take its a chronicle 86 00:08:00,280 --> 00:08:08,658 delta MN but just replace it by En. And so, now we have a we have oh, you 87 00:08:08,658 --> 00:08:21,285 know, a weighted sum of the energies and Well the, so the product, the product of 88 00:08:21,285 --> 00:08:28,140 these two terms here is a positive number because they're complex conjugates. 89 00:08:28,140 --> 00:08:33,526 I guess I should have emphasized that. So, this in equality implies, that is if 90 00:08:33,526 --> 00:08:38,774 we replace E, this sum here is greater than the sum where instead of En, we have 91 00:08:38,774 --> 00:08:46,119 the smallest value of all the [UNKNOWN] values, which is E0. 92 00:08:46,119 --> 00:08:51,189 And then, then this, then, then there's this identity here, this, this series 93 00:08:51,189 --> 00:08:55,844 collapses. So, I hope this, you can think about this 94 00:08:55,844 --> 00:09:00,191 intuitively. You're taking your, you're projecting the 95 00:09:00,191 --> 00:09:05,356 arbitrary trial function onto all of these, the eigenstates of the system. 96 00:09:05,356 --> 00:09:09,778 And you're taking the weighted average energy and that's gotta be greater than 97 00:09:09,778 --> 00:09:12,940 the least energy. That's a basic idea. 98 00:09:12,940 --> 00:09:19,030 I hope this was clear. Well, why should anyone really care about 99 00:09:19,030 --> 00:09:22,111 this? And the reason has to do with a 100 00:09:22,111 --> 00:09:27,720 remarkable result derived by Max Born and Robert Oppenheimer, which sort of laid 101 00:09:27,720 --> 00:09:35,804 the grounds for practical and accurate calculation of materials properties. 102 00:09:35,804 --> 00:09:41,393 And basically the Born-Oppenheimer approach shows that the electrons are 103 00:09:41,393 --> 00:09:46,209 really the glue that holds everything together. 104 00:09:46,209 --> 00:09:51,423 And if you want to find the [INAUDIBLE] equilibrium configuration of atoms in a 105 00:09:51,423 --> 00:09:56,242 solid or the positions of nuclei in a molecule, what you're going to do is 106 00:09:56,242 --> 00:10:01,219 minimize the total energy of the system that will be, that will be the ground 107 00:10:01,219 --> 00:10:09,010 state of you know, a very complicated material system. 108 00:10:09,010 --> 00:10:13,564 And in, in most of the cases the cases of most of the materials that you're 109 00:10:13,564 --> 00:10:17,980 familiar with, this approach can be implemented by solving the Schrodinger 110 00:10:17,980 --> 00:10:22,603 equation for the ground-state of the system at some fixed value of the, of the 111 00:10:22,603 --> 00:10:29,654 nuclear coordinate. So, for a molecule, we'd say, you take 112 00:10:29,654 --> 00:10:35,030 the nucle-, nuclei of some arbitrary distance R, solve the total energy, and 113 00:10:35,030 --> 00:10:39,986 then, as a function of R, you'll get an, you'll get an E of R, which, which 114 00:10:39,986 --> 00:10:49,439 displays a minimum typically. And that, that minimum then, defines the 115 00:10:49,439 --> 00:10:56,550 equilibrium, the equilibrium bond length in the molecule. 116 00:10:56,550 --> 00:11:00,798 So, generalizing this to a complicated system [INAUDIBLE] it's computationally 117 00:11:00,798 --> 00:11:06,220 intensive, of course, but it's an approach that's had great success. 118 00:11:06,220 --> 00:11:12,468 And indeed in 1998, the Nobel Prize in chemistry was shared by Walter Kohn, who 119 00:11:12,468 --> 00:11:18,364 developed a so, density functional techniques that, that greatly improved 120 00:11:18,364 --> 00:11:25,120 the computational speed of of such methods. 121 00:11:25,120 --> 00:11:31,784 So, this the the ability of quantum chemistry and condensed theoretical, 122 00:11:31,784 --> 00:11:37,762 condensed matter Physics to understand and discover new materials, was 123 00:11:37,762 --> 00:11:44,186 recognized by the Nobel Prize in the 1998.