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5.111 Principles of Chemical Science, Fall 2005 Transcript – Lecture 14

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5.111 Principles of Chemical Science, Fall 2005 Transcript – Lecture 14 MIT OpenCourseWare http://ocw.mit.edu 5.111 Principles of Chemical Science, Fall 2005 Please use the following citation format: Sylvia Ceyer and Catherine Drennan, 5.111 Principles of Chemical Science, Fall 2005. (Massachusetts Institute of Technology: MIT OpenCourseWare). http://ocw.mit.edu (accessed MM DD, YYYY). License: Creative Commons Attribution-Noncommercial-Share Alike. Note: Please use the actual date you accessed this material in your citation. For more information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms MIT OpenCourseWare http://ocw.mit.edu 5.111 Principles of Chemical Science, Fall 2005 Transcript – Lecture 14 All right. Today we're going to start on a topic, one of the harder topics that we cover this semester. We're going to start talking about molecular orbital theory. So this is a quantum mechanical description of the wave functions in a molecule. And we're going to use molecular orbital theory to look at the bonding in diatomic molecules. That is what we will do for the rest of today. We will look solely at diatomic molecules, and we will look mostly at homonuclear diatomic molecules. At the end we will look at one heteronuclear diatomic molecule. And then on Monday we're actually going to use a different kind of approach to look at the chemical bonding in molecules that are larger than diatomic. So we're going to use molecular orbital theory for these diatomic molecules, and then on Monday we will go a little bit further. We will look at bigger molecules. And we are going to use something else. We're going to use valence bond theory and hybridization ideas. But today we're going to deal with these diatomic molecules with the molecular orbital approach. All right? So here it goes. We're going to start talking about hydrogen, as we usually do when we start talking about chemical bonding. I am going to represent my hydrogen atom here by this nucleus here, a. This is going to be hydrogen atom a. And I am going to represent the wave function of that hydrogen atom by a circle because, after all, the 1S wave function in hydrogen is spherically symmetric. I am going to represent it by a circle around the nucleus. So this is going to be 1Sa. And then I am going to, of course, have a second hydrogen atom represented here by this nucleus b and a circle around it because that wave function is spherically symmetric. And what I am going to do in this molecular orbital theory approach is that I am going to take these two wave functions and I am going to bring them a distance of the bond length apart. In other words, I am going to take those two hydrogen atoms and I am going to put the two nuclei here, a and b, at a distance of a bond length apart. And I am going to let these two wave functions overlap. And these are wave functions. And what we learned a few weeks ago is that waves constructively interfere or destructively interfere. What we're going to do first here is we're going to let these two waves constrictively interfere where they overlap. When we do that, this part of the wave function here is going to constructively interfere with this part of this wave function. And then the sum of those two is going to look something like this where now there is this constructive interference between these two nuclei. There is a lot of wave function right here because we superimposed those two wave functions, each from one hydrogen atom. And so there is constructive interference here, there's a lot of wave function right there. And in so doing, what we're going to call this resultant wave function here, we are going to call that the molecular wave function or the molecular orbital. And how we got it was simply a linear combination of atom orbitals. This is a linear combination of atomic orbitals. We just brought those two atoms close enough so that their wave functions overlapped, constructive interference, and the result is a molecular wave function. Now, we are going to give that molecular wave function a name. We're going to call it a sigma wave function. The sigma designates the symmetry of that wave function with respect to the bond axis. This is the bond axis right here. The sigma represents the fact that this is cylindrically symmetric about the molecular bond axis. That's what that sigma means. And we are going to call this a sigma 1S orbital. It's a sigma 1S orbital because this was a linear combination of two 1S atomic orbitals. But now what I want to do is I want to remind you that the physically significant part of this wave function was actually the wave function squared. Remember that? We talked about taking the wave function, squaring it. And that interpretation is a probability density. That was what was critical to our understanding of where the electrons are in a molecule. So that is what we're going to do here. We're going to take this sigma 1S molecular wave function now and we're going to square it. And that is then proportional to, or interpreted as a probability density. I will just abbreviate that by P, but this is a probability density, remember, not a probability. Well, if that is the case then I can multiply this out. Because sigma 1S right here, that sigma 1S is really equal to this linear combination of 1Sa, 1Sb. That is sigma 1S. So let me take 1Sa plus 1Sb, multiply it by 1Sa plus 1Sb. That's what sigma 1S is. I'm just going to multiply this out. And when I do that I am going to get 1Sa squared plus 1Sb squared plus 2 times 1Sa, 1Sb. That is my probability density. But now, to make this a little bit more real for you, let's plot it. I am going to plot the probability density as a function of the position along the bond axis. In this case here the nucleus a is here and the nucleus b is there. So we have a molecule now. I am just going to plot the probability density as a function of the position along the internuclear axis. And what is it going to look like? Well, it's going to look like this. It is going to look something like that where the probability density is the highest right at nucleus a and the probability density is the highest right at nucleus b. Because, after all, any wave function, an S wave function, the highest value of that wave function is right at r is equal to zero or right at the nucleus. So that is what that probability density looks like. And I have just been informed here that the Xeroxing on that particular figure did not turn out so well, but that is what it should read, if you cannot read it very well. But the reason I plot this here is to contrast it with the following. Suppose I take just one hydrogen atom and the other hydrogen atom, but I take those wave functions and I just square them and then I add them up. Suppose I take a hydrogen atom with its wave function 1Sa. I square it. And then I take another wave function for another hydrogen atom 1Sb. I square it. And I add those two up. In this case, what that is doing is taking two hydrogen atoms and just taking their wave function and squaring them. In this particular case, what I haven't done is I haven't allowed any interference to happen. I haven't allowed the wave functions to constructively interfere. I did not add up these wave functions first before I squared them. I squared them and then I added them up. That is different than taking those wave functions, letting them constructively interfere and then squaring them. And you can see the difference is when I plot this sum and compare it to that sum. So I am going to plot now this sum. When I do that it's going to look, well, pretty much like this. But then right at the center, right here, the probability density for this case, for two hydrogen atoms which are now nonbonded, they're nonbonded because I didn't let those two wave functions constructively interfere. The result is that this probability density right in the center here, that probability density is less than what it would be if I let those two hydrogen atom wave functions constructively interfere. And that is the key in this chemical bond is this term right here. It's this term that makes all of the difference. This is our constructive interference. This is the extra density that we get because those two wave functions have constructively interfered. That is what gives me this pile up of electron density right in the center here. This is the glue in this chemical bond. Make sense? OK. All right. We have formed now a molecular wave function by this linear combination of atomic wave function. Now, what happens to the energy when we do that? Well, you can imagine what happens to the energy. The energy goes down. We are going to form a new molecular state. Let's draw an energy level diagram.
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