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# CHEM 125a: Freshman Organic Chemistry I

## Lecture 7

## - Quantum Mechanical Kinetic Energy

### Overview

After pointing out several discrepancies between electron difference density results and Lewis bonding theory, the course proceeds to quantum mechanics in search of a fundamental understanding of chemical bonding. The wave function ψ, which beginning students find confusing, was equally confusing to the physicists who created quantum mechanics. The Schrödinger equation reckons kinetic energy through the shape of ψ. When ψ curves toward zero, kinetic energy is positive; but when it curves away, kinetic energy is negative!

Professor McBride’s web resources for CHEM 125 (Fall 2008)

http://webspace.yale.edu/chem125_oyc/#L07

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html## Freshman Organic Chemistry I## CHEM 125a - Lecture 7 - Quantum Mechanical Kinetic Energy## Chapter 1. Limits of the Lewis Bonding Theory [00:00:00]
Now how big is that? Here, if you look at that yellow thing, when it shrinks down, that’s how big it is, that’s how big the vibration is. It’s very small. But these are very precise measurements. Right? Now why did they do so precise measurements? Did they really care to know bond distances to that accuracy? Maybe for some purposes they did, but that wasn’t the main reason they did the work very carefully. They did it carefully in order to get really precise positions for the average atom, so they could subtract spherical atoms and see the difference accurately. Okay? Because if you have the wrong position for the atom that you’re subtracting, you get nonsense. Okay, and what this is going to reveal is some pathologies of bonding, from the point of view of Lewis concept of shared electrons. Okay, so here’s a picture of this molecule. And remember, we had — rubofusarin, which we looked at last time, had the great virtue that it was planar. So you could cut a slice that went through all the atoms. This molecule’s definitely not planar, so you have to cut various slices to see different things. So first we’ll cut a slice that goes through those ten atoms. Okay? And here is the difference electron density. What does the difference density show? Somebody? Yes Alex?
Now the colored atoms, on the right, are the positions of the atoms through which the plane slices, but the atoms are subtracted out; so what you see is the bonding in that plane. So you see those bonds, because both ends of the bond are in the plane, so the bonds are in the plane, and you see just what you expect to see. But there are other things you see as well. You see the C-H bonds, although they don’t have nearly as much electron density as the C-C bonds did. Right? You also see that lump, which is the unshared pair on nitrogen. Right? But you see these two things, which are bonds, but they’re cross-sections of bonds, because this particular plane cuts through the middle of those bonds. Everybody see that? Okay, so again that’s nothing surprising. But here is something surprising. There’s another bond through which that plane cuts, which is the one on the right, through those three-membered rings, right? And what do you notice about that bond?
## Chapter 2. Introduction to Quantum Mechanics [00:08:35]Now this raises the question, is there a better bond theory than Lewis theory, maybe even one that’s quantitative, that would give you numbers for these things, rather than just say there are pairs here and there. Right? And the answer, thank goodness, is yes, there is a great theory for this, and what it is, is chemical quantum mechanics. Now you can study quantum mechanics in this department, you can study quantum mechanics in physics, you can probably study it other places, right? And different people use the same quantum mechanics but apply it to different problems. Right? So what we’re going to discuss in this course is quantum mechanics as applied to bonding. So it’ll be somewhat different in its flavor — in fact, a lot different in its flavor — from what you do in physics, or even what you do in physical chemistry, because we’re more interested — we’re not so interested in getting numbers or solving mathematical problems, we’re interested in getting insight to what’s really involved in forming bonds. We want it to be rigorous but we don’t need it to be numerical. Okay? So it’ll be much more pictorial than numerical. Okay, so it came with the Schrödinger wave equation that was discovered in, or invented perhaps we should say, in — I don’t know whether — it’s hard to know whether to say discovered or invented; I think invented is probably better — in 1926. And here is Schrödinger the previous year, the sort of ninety-seven-pound weakling on the beach. Right? He’s this guy back here with the glasses on. Okay? He was actually a well-known physicist but he hadn’t done anything really earthshaking at all. He was at the University of Zurich. And Felix Bloch, who was a student then — two years before he had come as an undergraduate to the University of Zurich to study engineering, and after a year and a half he decided he would do physics, which was completely impractical and not to the taste of his parents. But anyhow, as an undergraduate he went to these colloquia that the Physics Department had, and he wrote fifty years later — see this was 1976, so it’s the 50th anniversary of the discovery of, or invention, of quantum mechanics. So he said:
And we write it now: So, for example, this is five years later in Leipzig, and it’s the research group of Werner Heisenberg, who’s sitting there in the front, the guy that — this was about the time he was being nominated or selected for the Nobel Prize. Right? So he’s there with his research group, and right behind him is seated Felix Bloch, who himself got the Nobel Prize for discovering NMR in 1952. So he’s quite a young guy here, and he’s with these other — there’s a guy who became famous at Oxford and another one who became the head of the Physics Department at MIT. Bloch was at Stanford. So these guys know they’re pretty hot stuff, so they’re looking right into the camera, to record themselves for posterity, as part of this distinguished group; except for Bloch. What’s he thinking about? [Laughter] What in the world is And that summer these smart guys, who were hanging around Zurich at that time, theoretical physicists, the young guys went out on an excursion, on the lake of Zurich, and they made up doggerel rhymes for fun about different things that were going on, and the one that was made up by Bloch and Erich Hückel, whom we’ll talk about next semester, was about [Short film clip is played]
## Chapter 3. Understanding Psi as a Function of Position [00:16:36]
N particles, how many positions do you have to specify to know where they all are? How many numbers do you need? You need x, y, z for every particle. Right? So you need 3N arguments for Ψ. So Ψ is a function that when you tell it where all these positions are, it gives you a number. Now curiously enough, the number can be positive, it can be zero, it can be negative, it can even be complex, right, although we won’t talk about cases where it’s complex. The physicists will tell you about those, or physical chemists. Okay? And sometimes it can be as many as 4N+1 arguments. How could it be 4N+1?
So here’s the Schrödinger equation, Now I already told you the right side of this equation is the total energy. So when you see a system, what does the total energy consist of? Potential energy and kinetic energy. So somehow this part on the left,
Okay, but it turned out that although this was fine for our great-grandparents, it’s not right when you start dealing with tiny things. Right? Here’s what kinetic energy really is. It’s a constant. This is the thing that gets it in the right units: mv, but one over the mass of each particle — and here’s where we get it — [Students react] — times second derivatives of a wave function. That’s weird. I mean, at least it has twos in it, like ^{2}v, right? [Laughter] That’s something. And in fact it’s not completely coincidental that it has twos in it. There was an analogy that was being followed that allowed them to formulate this. And you divide it by the number ^{2}Ψ. So that’s a pretty complicated thing. So if we want to get our heads around it, we’d better simplify it. And oh also there’s a minus sign; it’s minus, the constant is negative that you use. Okay, now let’s simplify it by using just one particle, so we don’t have to sum over a bunch of particles, and we’ll use just one dimension, x; forget y and z. Okay? So now we see something simpler. So it’s a negative constant times one over the mass of the particle, times the second derivative of the function, the wave function, divided by Ψ. That’s kinetic energy really, not ½mv. Okay? Or here it is, written just a little differently. So there’s a constant, ^{2}C, over the mass, right? And then we have the important part, is the second derivative. Does everybody know that the second derivative is a curvature of a function. Right? What’s the first derivative?
Ψ such that the changes in kinetic energy compensate changes in potential energy.## Chapter 4. Understanding Negative Kinetic Energy and Finding Potential Energy [00:33:24]Now what’s coming? Let’s just rehearse what we did before. So first there’ll be one particle in one dimension; then it’ll be one-electron atoms, so one particle in three dimensions; then it will be many electrons and the idea of what orbitals are; and then it’ll be molecules and bonds; and finally functional groups and reactivity. Okay, but you’ll be happy to hear that by a week from Friday we’ll only get through one-electron atoms. So don’t worry about the rest of the stuff now. But do read the parts on the webpage that have to do with what’s going to be on the exam. Okay, so normally you’re given a problem, the mass and the charges — that is, that potential energy as a function of position — and you need to find
[Students speak over one another]
sin(ax). Right? The a comes out, that constant, each time you take a derivative. So now what does the kinetic energy look like? It’s a^{2 }times the same thing. Okay? So again, the potential energy is constant. Right? It doesn’t change with position. But what is different? It has higher kinetic energy if it’s a shorter wavelength. And notice that the kinetic energy is proportional to one over the wavelength squared, right?; a; ^{2}a shortens the wave, it’s proportional to a, one over the wavelength squared. Okay. Now let’s take another function, exponential, so ^{2}e. What’s the second derivative of ^{x}e? Pardon me?^{x}
e?^{x}
-C/m, and again it would be a constant potential energy greater than the total energy. This is not just a mathematical curiosity, it actually happens for every atom in you, or in me. Every atom has the electrons spend some of their time in regions where they have negative kinetic energy. It’s not just something weird that never happens. And it happens at large distance from the nuclei where 1/r — that’s Couloumb’s Law — where it stops changing very much. When you get far enough, 1/r gets really tiny and it’s essentially zero, it doesn’t change anymore. Right? Then you have this situation in any real atom. So let’s look at getting the potential energy from the shape of Ψ via the kinetic energy. Okay, so here’s a map of Ψ, or a plot of Ψ, it could be positive, negative, zero — as a function of the one-dimension x, wherever the particle is. Okay? Now let’s suppose that that is our wave function, sometimes positive, sometimes zero, sometimes negative. Okay? And let’s look at different positions and see what the kinetic energy is, and then we’ll be able to figure out, since the total will be constant, what the potential energy is. Okay? So we’ll try to find out what was the potential energy that gave this as a solution? This is again the Jeopardy approach. Okay? Okay, so the curvature minus — remember it’s a negative constant — minus the curvature over the amplitude could be positive — that’s going to be the kinetic energy; it could be positive, it could be zero, it could be negative, or it could be that we can’t tell by looking at the graph. So let’s look at different positions on the graph and see what it says. First look at that position. What is the kinetic energy there? Positive, negative, zero? Ryan, why don’t you help me out?
[Laughter]
[Students speak over one another]
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