 In principle, there are many ways atoms interact in molecules, but we're gonna look primarily at large molecules, proteins, and we're gonna try to group this way we normally do it in our programs. It's not necessarily the only way of doing it, but I would argue it's the most common way. The first and simplest interaction, if we look at a almost, I was about to say mess, but this is a core of a protein. It's a beautiful structure, but let's look at just one interaction in the middle here. If these two atoms are moving relative to each other, that's gonna correspond to a stretching or compression of that bond. That's actually quite expensive to do because remember those electron orbital pairing I told you. If we're stretching that, we're gonna move the electrons away from its equilibrium state. But that will definitely happen a bit at room temperature. There are many ways we can model this and if we're gonna do it with quantum chemistry, it becomes really complicated. So with quantum chemistry, this would be a so-called quantum chemical oscillator. And I'm not sure if all of you have studied quantum chemistry or quantum mechanics, but in quantum mechanics, what that would mean is that you would have a ground state that actually corresponds to a fixed length. And if you increase the energy, you're gonna start to have some of the molecules populate a higher state with also a fixed length that's slightly higher and then et cetera, increasingly higher energies. But each length is discrete and that's what we see in these horizontal bars. That would be very complicated to do for us. So we're not gonna do it. But maybe we can cheat instead. So if you look at that blue curve, that blue curve is, again, very simple approximation. It's called a Morse bond potential. And what that describes is roughly at very small distances, atoms will repel each other exponentially. That's the Pauli exclusion principle that I already introduced you to and you might know since before. If you start stretching atoms, you're also gonna have bad things happening, right? That you're taking something away from its equilibrium condition. But if you do stretch things far enough, at some point, the bond will break and bonds can break. And when bonds break, the atoms will eventually be happy. They won't be quite as happy as when they were together. But you can certainly say heat and gas until the point where the two hydrogen atoms and the H2 molecules go away from each other and form a plasma. So the blue potential does describe that. But then we think a little bit more about that. Wait a second. If we're gonna study proteins around room temperature, how frequently is it gonna be that we start having things? Well, we're never gonna have plasma phase. That would be 100,000s of Kelvin's. But how frequent is it that we actually break molecules inside the normal proteins in ourselves? Well, it does happen once in a blue moon for very specific chemical reactions. But typically when a molecule is stable in an equilibrium state, it's neither gonna form nor break bonds. And I would argue that accounts for 99% of what we do. And if that's 99%, we should optimize for that. So I'm gonna take the approximation and make an even worse approximation. So I said, let's just assume that it's not a quantum harmonica oscillator, but a classical harmonica oscillator. So there's just a second-order potential here that the potential of the energy is simply some sort of force constant multiplied by the square of the deviation from the equilibrium bond length. Is that bad? Well, in a way it is bad, but physics has been bad in that way for centuries. You might know that under another name, Hooke's Law. Hooke's Law has to do with this, the force when you're extending a spring, right? I hate to break it to you, that's wrong too. If you don't know that, it's kind of fun because I think every three-year-old kid knows that. If you have one of these springs that you're using to walk down a stair or something, if you're a three-year-old, you can't help but trying to see what happens if you tear it apart, right? And at some point, you've distorted it so much that it won't go back. That has to do with memory effects and metals and everything, so Hooke's Law is not too. The reason why we use Laws like that in physics that you can imagine having absolutely any type of function, a potential describing a system. We're gonna talk more about potentials in the next lecture, but this far, we can just say, if I have a potential energy here that is some sort of arbitrary shape, there is something around the x-axis here. We don't know what it is for now. This might be the deviation from the equilibrium or the length of a spring, whatever. The y-axis here is an energy and in physics and life, low energy is good. That corresponds to weight being on the floor rather than on the table. So this would be the best possible state here. Now, describing this entire equation is difficult, but what if I'm only interested in deviations around this part here? Well, I could start by saying the first approximation here would be the actual value of the function here, but it turns out that's not very important because when it comes to energies, I can talk about how far I've lifted the weight from the floor, but is this my floor or a floor in another building? So I can just say that the first approximation that is, I'll just adjust my y-axis here to say that this is zero. The second part I would need to account for if I do a series expansion here, that would be the derivative, but here's the cool thing. What is the first derivative around a local minimum or maximum for that matter? Well, it's zero, it disappears. So I don't have to think of the first derivative. And the next higher part I then need to consider is gonna be the second derivative. So if I now take the second derivative and fit that around this local minimum, that's what I get exactly a potential like this one. And that's why they're so common in physics. So physicists apply this not because it's right, but if I don't know anything around a local minimum, I can always describe some things with the type of function. How good it is, that depends on how far we go away, but for bonds in our case, it's gonna turn out to be an excellent approximation. There is only one approximation that's even better. We're gonna come back to that when we do actual simulations. Remember that quantum mechanical oscillator? It turns out that if we stick to room temperature, 99% of the bonds would be in the ground state. So an even simpler way of fixing this would be to give all the bond lengths a fixed distance and say that they should not deviate from that distance. Slightly harder to implement on a computer, but much easier to understand in terms of physics.