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# CHEM 125a: Freshman Organic Chemistry I

## Lecture 12

## - Overlap and Atom-Pair Bonds

### Overview

This lecture begins by applying the united-atom “plum-pudding” view of molecular orbitals, introduced in the previous lecture, to more complex molecules. It then introduces the more utilitarian concept of localized pairwise bonding between atoms. Formulating an atom-pair molecular orbital as the sum of atomic orbitals creates an electron difference density through the cross product that enters upon squaring a sum. This “overlap” term is the key to bonding. The hydrogen molecule is used to illustrate how close a simple sum of atomic orbitals comes to matching reality, especially when the atomic orbitals are allowed to hybridize.

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http://webspace.yale.edu/chem125_oyc/#L12

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html## Freshman Organic Chemistry I## CHEM 125a - Lecture 12 - Overlap and Atom-Pair Bonds## Chapter 1. The United-Atom “Plum-Pudding” View for Ethane and Methanol [00:00:00]
Now, as you split the nucleus up and pull pieces in different directions, it doesn’t have the same symmetry it had when it was all together in the nucleus, a spherical kind of symmetry. So the nodes get distorted. But still you can see them there. And we saw them last time and went through all the occupied and vacant valance orbitals of ammonia and methane, and saw how they looked like atomic orbitals. This, not surprisingly, because it’s so fundamental, the potential energy and the kinetic energy, applies to every system. It applies to you, viewed as a single atom, right? With a zillion electrons. Okay? But pieces have moved around, so the orbitals change. We’ll look at two more complicated cases and then we’ll get on to a different way of looking at bonding. So we’ll look at ethane and methanol. And we use — I didn’t tell you last time, explicitly, where I got the molecular orbitals from. I got them from my laptop. There’s a program — the particular one is called Spartan, that I use — and it calculates what molecular orbitals look like, using approximate molecular orbital (that is, Schrödinger equation kind of) theory. Okay, so let’s just look at the ones of ethane and methanol. Now both of these have seven pairs of valence electrons. There are also core electrons, and if we were looking at the orbitals for all the electrons, we’d include those. And exactly how we’re going to count those — you could do it one way or the other — whether you consider the core electrons, the Now we have two heavy atoms, carbon and carbon in methane, carbon and oxygen in methanol. So there are two heavy atoms and therefore two boring core orbitals. So for purposes of making analogies, we’ll use the atomic Now we can — as I do this, on the left side of the pictures I’m going to show one view, and then, because it’s more complicated than methane and ammonia were, I’m going to also show a picture rotated by ninety degrees. So I’m going to rotate around that axis, and on the right I’ll show a different view of the same orbital. Okay, so that’s the lowest valence level molecular orbital of ethane. And it’s not very exciting; it’s just a distorted sphere. And you can see the way in which it’s distorted. It’s distorted vertically, because the two carbons are pulled apart; so it’s got sort of a narrow waist to it. And then it’s pulled out, where each of the protons left the middle atom to come out and be hydrogens. Okay? Now if we look at methanol, it’ll be a little different. Can you anticipate how it might be different if, on the bottom we’ll have CH
s, then 2p, now _{z}2p, if we define the horizontal axis here on the left as the x axis. So again, it’s what you expect, and it’s pulled out, stretched vertically from being a dumbbell by where the nuclei went. And here’s _{x}2p, which we can see more clearly on the right, perpendicular to the _{y}2p._{x}Okay? And notice that it’s obviously so, that the methanol orbital will be less symmetric, in all cases, than the one for ethane. But still we can recognize the nodes, because they must go that way. It must be no nodes, one node of three different kinds, and so on. Okay, now this is
xz’ means that the product of x and z appears in the wave function. Right? So when both x and z are positive, then the product is positive; on the top right, red. Right? When x is negative, and z is positive, the top left, it’s negative, the product of them. So that’s why it has the name xz.Okay, and there’s d, which is that thing that has a doughnut that goes around the middle. Right? It’s hard to see the doughnut. Can you see? It’s blue on top. You can easily see the red on the top left, which is what’s blue in the middle; the sign has changed. Right? But the doughnut is highly distorted, because as you go around, first a proton on the top pulls it up, then a proton on the bottom pulls it down, then up, down. So it’s like a crown, the doughnut has been made into a crown around the end, by the protons pulling in that way. Okay. Then here’s the _{z}^{2}3p. So it has the horizontal nodal plane, but also it has a spherical node, which you can see in either picture really. Okay?_{z}Then here’s the 3. Or here’s the _{px}3d; which you don’t see so well here; well, but if you turned it down you would. And here’s the _{xy}3d; which again, to see it well, you’d have to turn it. And here’s the _{x^2-y^2}4f orbital. So you can see, especially say in the top ones, compared with the atomic orbital, that it’s exactly the same general pattern of nodes, slightly distorted.And incidentally, remember all the ## Chapter 2. The Orbital Shape of 1-Flouroethanol [00:13:25]Okay, now just finally and very quickly, I want to look at 1-flouoroethanol, which is a molecule that would have, I think, no stability at all, but you can put it into the — as a practical matter, you’re never going to put it in a bottle, but you can easily put it into the computer and calculate what its molecular orbitals would look like. And I did it to show you something that’s very, very unsymmetrical, and has atoms of very different nuclear charge. Okay? So what will the very, very, very lowest orbital look like, do you think, for this thing? It has a fluorine, an oxygen, two carbons and five hydrogens. So what do you think? If you were an electron, and you had that set of nuclei, arranged this way, where would you want to go? Elizabeth?
p kind of orbital. Elizabeth?_{y}
## Chapter 3. Localized Pairwise Bonding between Atoms and the Idea of Overlap [00:19:38]Okay now, now we’re getting into really — we just looked at this view, at the single united atom view. But the other view is the one that’s going to be more generalizable, and that’s the one where we looked at bonding. Right? So you have to probe a little harder to get a qualitative understanding of what chemical bonds are. And that’s what we’re going to do now by choosing a higher contour with which to look at a molecule. Now, true molecular orbitals, to the extent that orbitals are true all together — why aren’t they true all together; why aren’t orbitals true all together? Yes, Alex?
1/√2 times one atomic orbital plus another atomic orbital. Right? Now, have you ever seen adding orbitals like that before? That’s what hybridization is; we added s and p. But this is different, because when we added s and p before, they were on the same nucleus, and we did it to get a new orbital for that particular nucleus for that atom; to distort it one way or the other, for example, or to rotate a p orbital. Right? But this is very different, because we’re adding orbitals that are on different nuclei: A, nucleus A, and nucleus B. See the difference? Adding is just — wave functions are numbers, we just add the numbers. But in the first case, hybridization, those two functions were on the same nucleus. Now they’re on different nuclei, what we’re adding together. Okay, now why is it sensible to think that you might get pairwise molecular orbitals that can be expressed like this? How do you interpret an orbital? Corey? What good is an orbital? What do you use it for?
[Students speak over one another]
[Laughter]
Okay, but notice that here we’re multiplying A times B. But this is a completely different instance of multiplying from what we had before. Right? Before we multiplied two orbitals to try to get a two-electron wave function. This has nothing to do with this, because both of these are functions of the same electron; it’s like one electron that we’re squaring here. So this A times B, this product, this overlap, comes from the squaring. It was when we squared it that we got that. Okay? Now, because we have this extra term, we have not only ^{2 }summed over all space, or integrated, if it’s normalized?
So from the point of view of the electron distribution, that was the glue holding the atoms together. So it’s held together, both because the energy goes down and because you put this glue in the middle, which is what causes the energy to go down. Okay? So that’s bonding. And remember we also had this. So as the energy went up in the middle one, the energy is lower [correction: higher] here than it was in the atoms apart. So the nuclei push one another apart now, without the glue in the middle, and that was anti-bonding. So we’ve seen it before in one-dimension, but it’s true in three-dimensions as well. Now let’s think about this again. So here’s atom A. Now where is the square of that function significant? Is it significant there? No. Is it significant there? Yes, it’s a little bit significant at least. How about there? A little bit. Right? Okay. Now suppose we have another atomic orbital there. Now, where is the product significant, of A times B? Okay? So is the product A times B significant there? No. Is it significant there? Nick, what do you say?
Ψ assigns are the same number. So _{B}2(Ψ is as large as _{A} Ψ_{B})(Ψ; because _{A})^{2}+(Ψ_{B})^{2}Ψ times _{A}Ψ is the same as (_{B}Ψ. So the electron density is nearly doubled in the middle from what it would’ve been if it had just been two atoms. So that’s the region of significant overlap, and that’s what we care about. So the overlap integral, summing this product — or integrating it — over all that space, that’s a certain density. Right? We squared in order to get that. Right? That’s part of the density. So we sum that, the density that comes from that product, over all space, and that’s called the overlap integral. If the atoms are very far apart, the overlap integral is essentially zero. If the atoms are close together, the overlap orbital will be finite, and the better the — the more the orbitals overlap, the bigger the overlap integral, obviously. And that measures the net change that arises on bonding, the difference density, as we’ve just seen._{A})^{2}## Chapter 4. Hydrogen at Bonding Distance: A Case for Overlap [00:36:37]Now let’s look at some theoretical examples here. So let’s look at the total electron density as calculated for two — adding two
Okay, so high accuracy is required to calculate a correct value of the bond energy. This simple thing won’t do it. Well it’s in the right direction, you’re halfway there, so it’s a pretty good start. Right? But to do the difference, as in the same way you needed high precision to do X-ray difference maps, you need better orbitals than this, if you want to calculate good bond energies. So you need to make the orbitals better. Okay? Now — so but already we can take heart that the very crudest model shows most, 52%, of the energy of the bond, and it shows the electron density building up by 0.02 electrons per cubic bohr radius. And what we saw qualitatively was there was a shift from the atom to the bond, of electron density. Okay, now we can adjust the molecular orbital to get a better approximation of the true thing. How will we know when we’ve adjusted it and it’s gotten better? If we adjust it and get a lower, calculate a lower average energy — I should’ve said a lower average energy, because if we don’t have a true wave function we get different values for the total energy; the kinetic won’t exactly offset the potential as you move from place to place. But if you get the lowest average energy, then that is, by definition almost, more realistic, because you can easily prove that the true energy is the lowest possible energy; that makes a certain amount of sense. The lowest possible calculated energy is the true energy. So if you change your wave function and get a lower average energy, you’re closer to the truth. Okay? That’s called the Variational Principle. Okay, so here we’ve changed the form of the molecular orbital. And how did we change it? What does a
[Students speak over one another]
1/√2(A+B). But those A+B are no longer true atomic functions, they’re a little fatter, a little skinnier. Okay? And we can change how fat or skinny it is, until we get the lowest molecular energy. See, that’s a way you can vary it and find the best value. And that’s what was done here to optimize the exponent. And now you get a total electron density that looks essentially the same as it did before. And if you look at the difference density, how has it changed, if you do this? First, notice that the energy got lower. We’re now to 73% of the lowering of the bond energy. So the total energy’s gotten lower, it’s better.And how has the electron density changed? It got higher in the middle. Because what we did was spread the exponent out a little bit, so you had more overlap in the middle. Okay? So this wouldn’t have been good for the single atom, to spread it out, but it gives a better function for the molecule. And it’s still very, very simple. And what you see it did is it increases the bonding density and the bonding strength. You get a larger shift from the atoms to the bond. Now, how else could you change the shape of the atomic orbital in order to increase the overlap; some way other than making a single exponential and having it get fatter or thinner? Can you think of some other way? Here you have an atomic orbital, a sphere, and you want to change its shape so that it overlaps better over here. Right? How could you change the shape of an atomic orbital, without doing really gross damage to it, making it a cube or something like that, or a line? How could you change it so it looked pretty much still like an atom did, has a lot of the virtues of the atom, but is shifted over here? Sam? I can’t hear.
Okay, so here’s a pairwise atomic orbital: < Okay, but we’re going to keep it simple, use only [end of transcript] Back to Top |
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