6 - Order of Atomic Orbitals
Abstract (TL;DR):
The arc on orbital chemistry ends with an explanation of two very important rules: the Aufbau Principle, which states that electrons must fill lower energy orbitals before filling higher energy ones, and Hund’s Rules, which state that (1) all subshells must be filled with one electron before being completed by an electron with an opposite spin and (2) the electrons that enter the subshells and half-fill them must have the same spin. Finally, we cover the specific order that electrons fill orbitals according to the Madelung Rule (or diagonal rule), which allows you to add both the principle and orbital quantum numbers (n + l) to determine what the lowest energy orbitals are (to satisfy the Aufbau Principle), and we explain that Noble Gases are unreactive because all of their orbitals are filled.
—
Image by jimmymm-ilustra
Updated 6/2026
Did you notice that the scale of what we discuss is getting bigger? However, no matter how large things get, there’s always going to be a ribbon to tie things up with. That’s the beauty of science having its gentle fingers on every pulse.
The ribbon around everything that we’ve discussed in the last 5 chapters is the Aufbau Principle, which describes the order of how electrons organize. Specifically, electrons seek stability according to the rules of the Pauli Principle, which states that there can only be two electrons in any one orbital.
“Aufbau,” a German word meaning “construction" or “build-up”, is an apt name. The principle explains that electrons must occupy the orbitals with the lowest energy before going into higher energy orbitals. This follows what we’ve learned from the very beginning: Electrons have no need nor desire to be at a high energy level; if they can exist in a lower energy state, they will be.
The Building-Up
Before we begin, you might want to keep a periodic table handy.
Image via ScienceNotes
Building from chapter 4 and 5, the organization of each orbital conventionally takes the form of Principle-Orbital-Spin (n-l-s). The Aufbau Principle dictates that we start from lowest to highest energy level. As such, it’s surprisingly trivial to label these orbitals to see just how electrons organize themselves along the periodic table.
Image by William Reusch via Michigan State University
The first number, the Principle Quantum Number is what shows the energy level, and, thus, the electron shell of an element. If we are trying to occupy the lowest energy states first, we start with 1 – ground state energy. Next, the Orbital Quantum Number, the subshell number. Its lowest value is 0, corresponding to its label - s. Last, the Spin Quantum Number gives you whether one or both electron spins have been filled. The two spins - negative and positive - of each subshell must be filled before moving to the next subshell, which we designate as either 1 or 2 (since we can’t tell which is “spinning” negatively or positively).
The unnamed Magnetic Quantum Number, the subshell orientation number, is necessary for telling how a subshell is filled by electrons. As a reminder, telling how many orientations there are is as simple as looking at the orbital quantum number and counting the spectrum from negative to positive. For example, if the orbital number is 0, there’s only 1 subshell. If it’s 1, there are 3 (-1, 0, 1).
The n-l-s system dictates that the lowest possible energy state is 1s1. This value is the numerical representation of an orbital. This particular orbital corresponds to hydrogen.
But remember, there are two electrons, "spinning" oppositely, per subshell. With the addition of the second electron, we hit 1s2. This state corresponds to helium.
Once all subshells, and, therefore, the entire shell’s electrons have been filled, according to the Aufbau Principle, the next shell can be filled. We know that the next principle number, 2, marks an increase in energy states, which also means that the distance that electrons are from the nucleus increases. Ergo, the more energy electrons have, the more capacity they have to break away from the attractive forces of the nucleus.
With the increase in distance and the addition of even more electrons comes increases in possible electron configurations, resulting in not only new subshells, but also new orientations. Each orbital, however, always follows the Aufbau Principle.
This is an orbital diagram. Each arrow represents an electron and its spin.
Remember, p represents an orbital quantum number of 2. That means there’s a magnetic quantum number of 5, representing 5 subshells. Since the p orbital has three different orientations, each able to have two electrons, it allows for 6 electrons in total.
Image via PrintableDiagram
The Stability Rules
So why does the principle work like this? Fundamentally, because of a set of rules that must be true to explain the behavior of electrons in an atom.
Friedrich Hund, a German physicist that frequently worked with Schrödinger and other quantum physicists during this era, put together three rules that guides the logic behind the Aufbau principle. These are known as Hund’s Rules.
The Rule of Maximum Multiplicity
As we know, electrons seek to keep their individual energies low. However, electrons also have to make some compromises in how they organize themselves in an orbital so that the atom remains stable.
Two electrons in a single orbital “feel” each other’s negative charge and naturally repel each other, driving their energies up. But when a particular shell has multiple subshells, like in the case of the 2p orbitals, electrons have an option to get around that. Since all orbitals with subshells have the same energy, also known as degeneracy, electrons can avoid each other while filling the orbital of the lowest energy state.
To do this, electrons with a single spin fill each orbital within a subshell before pairing up with opposite spin electrons. This is the first of Hund’s Rules, also known as the rule of maximum multiplicity.
Pauli’s Exclusion Principle helps to explain why the electrons must all have the same spin. We know that two electrons of the same type can’t occupy the same orbital. By extension, two electrons of the same spin tend to avoid each other to keep the total energy low. Electrons with opposite spins tend to be closer to each other within an atom because this configuration doesn’t violate the Pauli exclusion, but it also means that the charges have more repulsion energy. Therefore, the best organization of electrons is to, first, fill orbitals with those that have the same spin within a subshell.
In the above picture, carbon (C) and oxygen (O) represent this best. Carbon’s first 2p orbital is filled with an electron in one spin and then its second 2p orbital is filled with an electron with the same spin. It is only when you get to oxygen that the first subshell is filled with an oppositely-spinning electron.
The Rule of Total Orbital Angular Momentum
We must recall that electrons exist within a cloud of possible locations and don’t want to be near each other. Even though, from the first rule, electrons have already spread out as much as possible, they can still end up near each other as they move around their nuclei. This second rule, therefore, focuses on what’s needed to minimize interactions between electrons for the stability of the entire atom (instead of just the individual electron).
First, let’s take a look at nitrogen, which represents a case where the electron cloud distribution is balanced across the atom. There is one electron with parallel spins in each of the 2p orbitals, thanks to Hund’s first rule. These electrons are, therefore, evenly distribution in all directions around the atom; no particular orbital in the 2p subshell is overemphasized and there is no crowding of electrons. Said another way, the orbitals cancel out each other’s contribution of energy to the atom as a whole. Nitrogen is, thus, a symmetrical and energetically balanced atom.
We can also understand this using our orbital quantum number (l), which explains an electron’s orbital angular momentum, and magnetic quantum number, m. The p orbitals correspond to an l of 1, as we covered in Part 4. Knowing l gives us the magnetic quantum numbers, which are the integer values in the range of -l to l, that any of our three electrons can fall into. In this case, that’s -1, 0, and 1, each of which correspond to a subshell. Hund’s first rule tells us that electrons must fill each orbital first. In nitrogen, there happens to only be three electrons left to fill each of the three orbitals. Therefore one electron has a magnetic quantum number of -1, a second has a number of 0, and the last has a number of 1 — each electron has its own space.
Hund’s second rule, however, concerns itself with the stability of the entire atom, not just that of the individual electrons. In other words, we are interested in the total magnetic quantum number and the total orbital angular momentum (which is where the rule’s nickname comes from). We can use the connection between the magnetic and orbital quantum numbers to get the total orbital angular momentum using the total magnetic quantum number.
First, we add the individual magnetic quantum numbers together for the total magnetic quantum number, which we will call M.
Along with nitrogen’s electron configuration, we have enough information to tell what the total orbital angular momentum, which we will call L, is. This is because there is only one possible way the electrons can be organized, thanks to Hund’s first rule; that is, an even distribution of electrons, in each orbital that surrounds the nucleus. This even distribution of electrons around the nucleus of an atom…sounds familiar, doesn’t it? We discussed such a case in Part 5! There is only one orbital angular momentum value that corresponds to a symmetrical distribution of electrons around the nucleus, l = 0. This is also true of the total orbital angular momentum. In this case, the total orbital angular momentum of nitrogen is also 0.
Nitrogen is a good way to understand Hund’s second rule, due to how straightforward its electrons are organized. But what about in a more interesting case, like that of oxygen? Unlike nitrogen, which had three electrons to place in three p orbitals, oxygen has four electrons to place in those same three orbitals.
Image via LibreTexts Chemistry
So what do the magnetic quantum numbers look like for the electrons in oxygen? Well, we still only have three magnetic quantum numbers, but two electrons now occupy one of them. But this opposite spin electron can go in any of the three boxes (remember, these are degenerate orbitals, they are all energetically the same). One possibility is like the above image, where two electrons exist in the subshell that corresponds to a m = -1, the third exists at m = 0, and the fourth exists at m = 1. Adding them together to get the total magnetic quantum number gives us an M of -1. But another possibility is that the opposite spin electron occupies the m = 0 subshell, which would change the M to 0. The last possibility, where the opposite spin electron occupies the m = 1 subshell, gives an M of 1.
In other words, oxygen has a range of M values, from -1 to 1. We’ve seen this before too, even more recently. To get a range of integers from -1 to 1 in our individual magnetic quantum number calculation, m, our orbital angular momentum, l, had to be 1. Likewise, an M from -1 to 1, corresponds to a total angular momentum, L, of 1. Unlike nitrogen, this means that the electrons are distributed along a particular direction in the atom to keep the energy low; it is no longer symmetrical.
Much like the orbital quantum number (l) determines the shape of an individual electron’s cloud, the total orbital quantum number (L), explained by Hund’s second rule, determines the shape of the atom’s electron cloud.
The Rule of Combined Angular Momenta
Hund’s rules have now resolved the two momenta that an electron would have, it’s spin and its motion around the atom. The last rule is yet another bow that ties the two together to ensure the atom is at its lowest energy state.
In the last part, I noted that electrons don’t behave like typical magnetic dipoles, due to their lack of a true orbit around the atomic nucleus. However, the intrinsic movement of electrons does generate a magnetic moment.
For clarity, here is another orbital diagram with 36 elements. Notice Hund’s Rule being applied from boron (B) to neon (Ne) and from aluminum (Al) to argon (Ar).
Notice how here, the 2p orbital is divided into its three dimensional configurations, 2px, 2py and 2pz. Both ways of writing the 2p orbital are acceptable.
Image via Meta-Synthesis
Diagonal Rule
There’s one more curve ball that you should be aware of.
Electrons tend to fill orbitals in a unique, but organized way after the 2p subshell in most cases. When you get to the third principle number, you gain yet another subshell – the d subshell.
As we’ve said 1s1 is hydrogen and 1s2 is helium. With each element in the periodic table, the number of electrons within the system grows by one. But when you get to Argon (Ar) and move to Potassium (K), it goes from 3p to 4s. Why would we skip the 3d shell? After all, we went from 1s, to 2s-2p, then 3s-3p. Shouldn’t 3d come next?
There was a little factor that I purposely did not mention until this point. You can actually define the energy of orbitals by adding the principle quantum number and the orbital quantum number – n + l. This is called the Madelung Rule, which gives the order that orbitals are filled in. So, if we take hydrogen, for example, with an n of 1 and an l of 0, or s, we see that it has an energy level of 1, which is the lowest. Hydrogen and helium, technically, both fit into this 1s category, but we know that helium has two electrons, while hydrogen has only one. Therefore, helium has more energy in the system.
To belabor the point, it would be as if you had a solitary magnet on the table. It won’t move on its own until you put something of a similar pole near it, in which case it will move away. Electrons experience a similar reaction to each other, and therefore, the element itself is more energetic the more electrons there are.
But, returning to potassium (K), which has an n of 4 and an l of 0 (4s), it has an energy level lower than Scandium (Sc), which has an n of 3 and an l of 2 (3d). Potassium, which occupies the 4s orbital, has an energy level of 4, while scandium has an energy level of 5. Therefore, the 4s level is filled first.
This staggering creates a well-known diagonal. In fact, we can easily tell how to fill orbitals by following this Diagonal Rule.
Image via HyperPhysics
It’s simply amazing how we manage to organize something as chaotic as an electron. I suppose, however, that this is how they organize; we are just observers in this game, of which we are a sentient and thinking part.
There’s one more amazing thing that these orbitals reveal about the periodic table. It explains the Noble Gases.
The Noble Gases
Time for another callback. Remember in third article, when I mentioned noble gases and valence electrons? I said that noble gases were almost completely nonreactive.
Look at this periodic table.
Image via Chemistry LibreTexts
This puts a little more attention on the electron configuration. Do me a favor: take a look at the noble gas group, or the column most to the right.
Their orbitals are completely filled! There is not one spin arrow that is unaccompanied. You might have noticed that beryllium (Be) and magnesium (Mg) also fit this criterion. However, recall that every principle quantum number over 2 has a p or higher orbital. Those two elements, and all other elements in Group 2, don’t fill the p orbital, like neon and argon do in their respective rows.
Noble gases are nonreactive because there are no electrons required to complete their electron shells. Therefore, an element paired with a noble gas will lead to no result; there is no response to the phone call, as it were. Without space free for more electrons, there’s almost no way for elements to communicate.
If you look at your complete periodic table and the image of the orbital diagram used in the Diagonal Rule section, you can see this is the case for Argon (Ar) and Krypton (Kr) as well. Yes, Superman is from a very noble planet indeed.
Conclusion
With orbitals fully explained, you now have a basic understanding of modern atomic theory and, consequently, the Elemental Flow arc has come to its end. Now that you know what electrons are, in terms of how they move and organize themselves in an element, you are well equipped to understand how elements combine. In fact, I’ve already given you the answer to that. But, I suppose we will see soon, as we move with the flow eternal.