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Electron Configurations and Periodicity
Electron Spin and the Pauli Exclusion Principle
Understanding Electron Spin [Page 1 of 2]
We’ve come a long way. With a little help from Schrödinger, we now understand perfectly the hydrogen atom, and we
know not to ask, “Where is the electron, and what momentum does it have?” but rather, “Where are we going to have
the highest probability of finding the electron?” Remember, the solutions to the Schrödinger equation give rise to a
shape of orbitals, whether it’s an s orbital or a p orbital or a d orbital. We know what those look like. We know that they
describe probability of finding the electron but they don’t promise an absolute position. We also know from the
Schrödinger equation the energies of the electron, and we know that the energies of the electron in the hydrogen
atom are quantized, meaning the electron in hydrogen can be this value or this value or this value, but nothing in
between. And we know the general shape associated with these different states. So we’ve really done a lot to master
this. The next logical step is to try to understand helium, lithium, beryllium, get into the rest of the periodic table. And
the reason this is so important and why we’re dwelling on this so much is that in knowing where the electrons are,
we’re going to be able to predict and understand how the atoms come together to form molecules.
So, going from hydrogen to helium, what’s different? Well, we know that in the case of helium, we have one more
electron. If we just show a little cartoon of this here, we have a 2+ nucleus instead of a 1+ nucleus now, and we have
two electrons. Again, I draw this little cartoon, but remember, these are not electrons in orbit around the nucleus.
They’re clouds of electron density, if you will, probability of finding the electron at different positions around the
nucleus. But in my cartoon, we have two electrons that we have to worry about.
In considering the energy of this system, we have to worry about the kinetic energy of the electrons, the potential
energy (meaning their attraction to the nucleus), and a new term that we haven’t seen before, and that is the repulsion
between two particles that have the same charge. That is going to be the problem. We don’t know where the electrons
are. How do we begin to write an equation that describes the repulsion when we don’t know what the distance is
between the two electrons? In fact, although we can write down an equation (still the Schrödinger equation) that, if we
could solve it, would answer the question of, “Where are the electrons in helium?” there is no exact mathematical
solution to it. To this day, in fact, there is no absolute solution to that calculation. We have to, using the most
sophisticated computer technology we have, make approximations about how to treat these two electrons.
Now, here’s the assumption, in essence, that we’re going to make. We’re going to assume that each one of these
electrons is still in a hydrogen-like orbital, and, to a first approximation at least, the two electrons don’t see each other
at all. Now, we’re going to remember that that’s our crucial assumption, and that it’s wrong. I mean, these electrons do
see each other. But we’ll come back and kind of modify that in a few minutes. We’ll come back and modify our theory
just a little bit—a fudge factor, if you will—to take into account the fact that we’re ignoring those two electrons seeing
each other. For the moment, humor me here.
Let’s suppose we just had this electron, let’s say, in a 1s orbital, the lowest energy state of the hydrogen wave
functions, and so we know the shape of where that electron is, the orbital that it’s in. And this other electron we’ll also
put in a 1s orbital, just in its own set of orbitals. Right away we have a problem. We’ve violated what’s called the Pauli
exclusion principle. The Pauli exclusion principle states that no two electrons can have the same set of quantum
numbers. And each of these would be described uniquely as a principle quantum number of 1, an angular quantum
number of 0, and a magnetic quantum number of 0. So we’ve got a problem.
It turns out that the answer to this dilemma lies in a key observation made by Stern and Gerlach, who took an oven,
heated up a sample of silver, generated silver atoms, and passed a beam of those silver atoms through an
inhomogeneous magnetic field. To their astonishment, the beam split into two different beams that were detected at
two different positions. So what this told them was that there were in fact two different kinds of silver atoms, one
corresponding to the beam that is deflected in this direction in the magnetic field, and one type of silver that was
deflected exactly the opposite direction in the same magnetic field.
What’s going on here? What this is telling us—if you did the same experiment for hydrogen, you’d have the same
outcome. What it has to do with is the fact that the electron, in essence, can interact with a magnetic field. It has its
own property of magnetization. It’s as if—this is a loose analogy; it’s not exactly correct but it helps us picture it—it’s
as if the electron were a little ball spinning, and it could spin clockwise or counterclockwise. Now, depending on which
way it was spinning, it would interact with a magnetic field in an opposite sense. And though that is imposing our
classical ideas on an electron, it helps us at least have a higher comfort level with what ultimately will be called a “spin
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Electron Configurations and Periodicity
Electron Spin and the Pauli Exclusion Principle
Understanding Electron Spin [Page 2 of 2]
quantum number.” Once again, this is just a property of the electron, describing how it interacts in a magnetic field, but
it gives us a fourth quantum number—a fourth property, if you will—of the electron, that allows us to differentiate two
electrons simply by the fact that one could have the opposite spin of the other.
So our picture then becomes a little different. We say that for helium—once again, two electrons—the first electron
could be in a 1s orbital with its spin, let’s say, down, in one direction. Down, up, it doesn’t mean anything other than
it’s going to be the opposite spin. So one electron is down, let’s say, and the second electron also could go in the 1s
orbital and not violate the Pauli exclusion principle if its spin was up.
So we now need a fourth quantum number to describe this. We’d say that this red electron here is (1, 0, 0, –1/2), and
the green is (1, 0, 0, 1/2). That gives us a different set of quantum numbers, a different address, or almost a different
ZIP code that each one of these electrons has. And kind of the physical meaning of the Pauli exclusion principle is
just: you don’t want to have two electrons in the same space at the same time. Again, that’s a very loose analogy, but
that’s more or less what it’s saying physically: each electron must have its own set of quantum numbers that uniquely
describes where it is. This allows us to understand the helium atom with two electrons, both in a 1s orbital.
Let’s take this one step further now, and in the periodic table go to lithium. Lithium has a charge of 3+ instead of 2+ in
the nucleus. No problem. We already learned from previous tutorials that as we increase nuclear charge, it just pulls
the electrons in tighter, so we can understand that. The same sets of solutions would come out of the Schrödinger
equation. But where are we going to put it? We can’t put it in the 1s orbital anymore. In other words, we can’t assign it
a set of quantum numbers (1, 0, 0, anything) because we only have two choices for the spin and we’ve used them
both in the first two electrons.
So where do we put the third electron in lithium? We have to go to a higher energy, because there’s nothing else
down at the n = 1 level. We have then a choice of the 2s or the 2p orbital. Where is it going to go? They’re the same
energy. Does that mean that it’s just random chance that it’s going to go in a p orbital or an s orbital? It turns out that
that’s not the case, that although these orbitals are exactly the same energy in hydrogen, they aren’t going to be the
same energy in lithium. We’ll understand why when we understand the idea of electronic shielding, and that’s next.
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