 So, we understand an awful lot about the hydrogen atom at this point. In atomic coordinates, we can write down the Schrodinger equation in this way, and it looks a lot simpler than it did the first time we wrote it down. That works for hydrogen as well as hydrogen like ions if the atomic charge is greater than one. We're not going to make a whole lot of progress just talking about hydrogen atoms. We won't learn very much chemistry with just hydrogen, however, so we need to be able to extend this to understand atoms larger than hydrogen. So let's start by thinking about what the Schrodinger equation would look like for the next step up, a helium atom, one with just two protons in the nucleus and two electrons. So if I draw the nucleus of the helium atom with a charge of plus two, and then I can place electrons, so here's an electron at some distance r1 away from the nucleus, it has a charge of negative one minus E in SI units or minus one in atomic units, and then I can put another electron at another distance away from the nucleus. So our job is to understand first what the Schrodinger equation looks like and then how to solve the Schrodinger equation for this system whose main difference is that it has two electrons now instead of one. So we can still write down Schrodinger's equation for the helium nucleus with these two electrons relatively easily. I guess the first thing to understand is the wave function needs to describe the probability of finding electrons at two different positions. So it's not only going to depend on the position of a single electron, it's going to depend on the position of two different electrons. And those positions are three-dimensional vectors, and so if we again think of those in terms of r, theta, and phi, I've got an r1, theta, and one phi one describing the location of electron one, I've got an r2, theta, two, and phi two describing the location of electron number two. So rather than three coordinates in our wave function, r, theta, phi like we had previously, now we've got a total of six coordinates to describe our two different electrons. So what that means is this Schrodinger equation we write down, the wave function is going to be a function of six coordinates, and the derivatives hiding inside of the del squared need to involve derivatives with each of these six coordinates. But in atomic units and in the simplified term, it's still not that difficult to write down. There's still kinetic energy terms. So electron one has some kinetic energy, and when I'm talking about the kinetic energy of just this electron, I don't need to worry about electron two at all. So it's just the del squared with respect to the r1, theta one, phi one terms. So del squared for coordinates of electron one. Likewise, there's a kinetic energy term that looks very similar for electron two. It's also just minus one-half del squared of the wave function, but these derivatives hiding inside that del squared are for electron two. So their derivatives will respect to r2, or theta two, or phi two. Schrodinger's equation has kinetic energy terms. It also has potential energy terms. So this for the hydrogen atom, this potential energy term where this came from originally was the Coulomb interaction between the electron and the nucleus. This electron still interacts with the nucleus in exactly the same way. So I can say that the potential energy, so that's potential energy for electron one, and this is now r1, the location, the distance of electron one away from the nucleus. There's a similar term for the Coulomb interaction of electron two with the nucleus. That's the potential energy of electron two. So far so good, and I've just converted each one of these terms into a term for electron one and a term for electron two for the potential energy as well. I can't stop there, however, if I'm thinking about potential energy. There's Coulomb interaction energy between this negative charge and this positive charge, between electron number one's negative charge and this positive charge. But there's also a Coulomb interaction between the two electrons themselves. So those two electrons are at some distance away from each other. If I know the values of r1 and r2 and theta and phi, I could compute r12 as a function of these other variables. But for now, I'll just write it as some distance r12, which hides all these other variables. And then the Coulomb interaction between those two electrons, that's a negative, interacting with a negative. So it's a positive energy of E squared, which is one in atomic units divided by the distance between them multiplied by the wave function. So that is also a potential energy term. So that's the electron potential energy term, or we could call that, I suppose, the one interacting with two potential energy term. So that's the left hand, the Schrodinger equation, all of my kinetic energy terms, all of my potential energy terms. All that must equal energy times the wave function. And that's Schrodinger's equation for the helium atom. So let's pause a minute and ask ourselves what it would take to solve that equation. So if all we had were these first four terms, kinetic energy one and two, potential energy one and two. If I hadn't written down the Coulomb energy term for the two electrons interacting, then everything would be fine. It would be no more difficult than solving the hydrogen atom Schrodinger equation. Because what I could do is separate the variables. Notice that this term only involves r1, theta1, phi1. This term also only involves r1. I can break the equation into some pieces that only involve electron one, some other pieces that only involve electron two. And by separating the equation that way, I could write the wave function as a product of two wave functions, a wave function for electron one, another wave function for electron two. And I could proceed exactly as I did for the hydrogen atom. Unfortunately, we have this fifth term which we can't separate. If I were to write down how r12 depends on r1 and its angle and phi and how it depends on r2 and theta two and phi two, turns out that's a big complicated tangle of variables that I can't separate into a piece that looks just like electron one added to a piece that looks just like electron two. So this equation is not separable. I can't write down the answer as a product of two solutions. If I could, if this term didn't exist, that would mean if the two electrons didn't interact at all, then we would solve for the wave functions as if I had two non-interacting electrons around a charge two nucleus. But that's not the way real electrons behave. So to solve for the properties of actual real helium, we need to deal with this interaction term. So it turns out even it's not that difficult to go beyond helium. If I start thinking about lithium, the only difference would be I'm gonna have kinetic energy for three electrons instead of two, potential energy for three electrons and then the potential of interaction among those electrons themselves. So again, it's not hard to write down the Schrodinger equation for any atom we like with n electrons. If it has n electrons, then the kinetic energy terms are gonna look like minus one-half del squared summed over each of my electrons. So there's a del squared term for each electron. Likewise, there's a potential energy term that looks like minus z. One over r i summed from one to n. The potential energy of interaction between each one of these electrons, I need to have. I need to sum over every pair of electrons i and j and then have one over r i j multiplying psi. Multiplying psi as the potential energy term for the electron-electron interaction. All that has to equal the energy times psi. So just as for the helium atom, kinetic energy terms and potential energy terms only involve single electrons variables. It's these Coulomb interaction terms, interaction of electron i with electron j, that tangle the variables together and make this a non separable equation. So we're stuck, at least in the sense that we can't solve this shorting equation on paper the same way we have for hydrogen. But it won't surprise you to hear that since we're in physical chemistry, our approach is gonna be to take an approximate view. Let's make some approximations and see if we can still understand the fair amount about the properties of a helium atom or other atoms, even though we can't solve the expression exactly.