Advanced Electron Configurations

The Pauli exclusion principle means that no two electrons in one atom have identical sets of all four quantum numbers. In one spatial orbital, the two possible electrons have opposite spins. Electrons in different degenerate orbitals can have equal energy and parallel spins. That last arrangement is compatible with Pauli and is important when we apply Hund’s rule.

Using this principle, you can illustrate electron configurations using something called the box method. This method provides a box (or a spot) for every orbital, and puts in up to two arrows, one up, one down, representing the electrons that would exist in that orbital.

When drawing the boxes, you group together orbitals contained in each block, so the p-block would have 3 boxes representing the 3 p-orbitals, and the d block would have 5 boxes. Let’s take a look at the configuration for Arsenic:

Aufbau is the lower-energy-first filling guide. Hund’s rule describes a different step: within a set of degenerate orbitals, occupy them singly with parallel spins before pairing. Paired electrons in the same orbital have opposite spins. An orbital-box drawing needs all three ideas, so do not attribute single occupancy of equal-energy p or d orbitals to Aufbau alone.

Here are a few more examples of this process:

Fluorine:

$\begin{aligned} \underline{\uparrow\downarrow} && \underline{\uparrow\downarrow}\ && \underline{\uparrow\downarrow}\ \underline{\uparrow\downarrow}\ \underline{\uparrow\phantom{\downarrow}} \\ 1s^2 \phantom{1}&& 2s^2 \phantom{1}&& 2p^5 \phantom{\uparrow\downarrow1} \end{aligned}$

Calcium:

$\begin{aligned} \underline{\uparrow\downarrow} && \underline{\uparrow\downarrow}\ && \underline{\uparrow\downarrow}\ \underline{\uparrow\downarrow}\ \underline{\uparrow\downarrow} && \underline{\uparrow\downarrow}\ && \underline{\uparrow\downarrow}\ \underline{\uparrow\downarrow}\ \underline{\uparrow\downarrow} && \underline{\uparrow\downarrow} \\ 1s^2 \phantom{1}&& 2s^2 \phantom{1}&& 2p^6 \phantom{\uparrow\downarrow1} &&3s^2 \phantom{1}&&3p^6 \phantom{11111}&&4s^2 \phantom{1} \end{aligned}$

Manganese:

$\begin{aligned} \underline{\uparrow\downarrow} && \underline{\uparrow\downarrow}\ && \underline{\uparrow\downarrow}\ \underline{\uparrow\downarrow}\ \underline{\uparrow\downarrow} && \underline{\uparrow\downarrow}\ && \underline{\uparrow\downarrow}\ \underline{\uparrow\downarrow}\ \underline{\uparrow\downarrow} && \underline{\uparrow\downarrow}\ && \underline{\uparrow\phantom{\downarrow}}\ \underline{\uparrow\phantom{\downarrow}}\ \underline{\uparrow\phantom{\downarrow}}\ \underline{\uparrow\phantom{\downarrow}}\ \underline{\uparrow\phantom{\downarrow}} \\ 1s^2 \phantom{1}&& 2s^2 \phantom{1}&& 2p^6 \phantom{\uparrow\downarrow1} &&3s^2 \phantom{1}&&3p^6 \phantom{11111}&&4s^2 \phantom{1}&&3d^{5} \phantom{445244241} \end{aligned}$

Zinc:

$\begin{aligned} \underline{\uparrow\downarrow} && \underline{\uparrow\downarrow}\ && \underline{\uparrow\downarrow}\ \underline{\uparrow\downarrow}\ \underline{\uparrow\downarrow} && \underline{\uparrow\downarrow}\ && \underline{\uparrow\downarrow}\ \underline{\uparrow\downarrow}\ \underline{\uparrow\downarrow} && \underline{\uparrow\downarrow}\ && \underline{\uparrow\downarrow}\ \underline{\uparrow\downarrow}\ \underline{\uparrow\downarrow}\ \underline{\uparrow\downarrow}\ \underline{\uparrow\downarrow} \\ 1s^2 \phantom{1}&& 2s^2 \phantom{1}&& 2p^6 \phantom{\uparrow\downarrow1} &&3s^2 \phantom{1}&&3p^6 \phantom{11111}&&4s^2 \phantom{1}&&3d^{10} \phantom{445244241} \end{aligned}$

Copper:

$\begin{aligned} \underline{\uparrow\downarrow} && \underline{\uparrow\downarrow}\ && \underline{\uparrow\downarrow}\ \underline{\uparrow\downarrow}\ \underline{\uparrow\downarrow} && \underline{\uparrow\downarrow}\ && \underline{\uparrow\downarrow}\ \underline{\uparrow\downarrow}\ \underline{\uparrow\downarrow} && \underline{\uparrow\phantom{\downarrow}}\ && \underline{\uparrow\downarrow}\ \underline{\uparrow\downarrow}\ \underline{\uparrow\downarrow}\ \underline{\uparrow\downarrow}\ \underline{\uparrow\downarrow} \\ 1s^2 \phantom{1}&& 2s^2 \phantom{1}&& 2p^6 \phantom{\uparrow\downarrow1} &&3s^2 \phantom{1}&&3p^6 \phantom{11111}&&4s^1 \phantom{1}&&3d^{10} \phantom{445244241} \end{aligned}$

Whoops! What happened to copper? Shouldn’t the 4s orbital have 2 electrons, and the 3-d have only 9? The Aufbau Principle would agree with you, but there are a few exceptions to that principle. In the case of copper, by taking one electron from the 4s orbital and using it to completely fill the 3-d orbital, the atom achieves a lower energy, meaning that it is more stable. In fact, there are many exceptions to the Aufbau principle amongst the transition metals. You should remember two main ones: Copper and Chromium. Chromium’s box configuration looks like:

$\begin{aligned} \underline{\uparrow\downarrow} && \underline{\uparrow\downarrow}\ && \underline{\uparrow\downarrow}\ \underline{\uparrow\downarrow}\ \underline{\uparrow\downarrow} && \underline{\uparrow\downarrow}\ && \underline{\uparrow\downarrow}\ \underline{\uparrow\downarrow}\ \underline{\uparrow\downarrow} && \underline{\uparrow\phantom{\downarrow}}\ && \underline{\uparrow\phantom{\downarrow}}\ \underline{\uparrow\phantom{\downarrow}}\ \underline{\uparrow\phantom{\downarrow}}\ \underline{\uparrow\phantom{\downarrow}}\ \underline{\uparrow\phantom{\downarrow}} \\ 1s^2 \phantom{1}&& 2s^2 \phantom{1}&& 2p^6 \phantom{\uparrow\downarrow1} &&3s^2 \phantom{1}&&3p^6 \phantom{11111}&&4s^1 \phantom{1}&&3d^{5} \phantom{445244241} \end{aligned}$

Notice that it is similar to Copper’s except instead of every orbital in the d-block being full, it all has one electron. That link will make these two exceptions easier to remember.
Now, as to the other exceptions, you won’t typically encounter them, but two others that are easier to remember are Molybdenum and Silver. These are directly below Chromium and Copper, respectively, and have upper s and d configurations very similar:

Molybdenum, enrichment: its observed neutral ground-state shorthand is [Kr]4d⁵5s¹, consistent with five singly occupied 4d orbitals and one 5s electron in this introductory drawing. The original 4d¹⁰ label was inconsistent with its five arrows. Memorizing exception configurations is excluded from the required AP targets; keep this example as optional enrichment rather than an assessment requirement.

Silver:
$\begin{aligned} \underline{\uparrow\phantom{\downarrow}}\ && \underline{\uparrow\downarrow}\ \underline{\uparrow\downarrow}\ \underline{\uparrow\downarrow}\ \underline{\uparrow\downarrow}\ \underline{\uparrow\downarrow} \\ 5s^1 \phantom{1}&&4d^{10} \phantom{445244241} \end{aligned}$

For ions, keep the element’s proton count fixed and change the electron count. Main-group ions often match a nearby noble-gas configuration: Na⁺ is [Ne], while Cl⁻ is [Ar]. That shortcut is not a universal left/right shift for every element. Transition-metal cations lose ns electrons before (n−1)d electrons; for example, Fe²⁺ is [Ar]3d⁶. Check the actual occupied states instead of treating the ion as a new element.

Example Problem 5.1.1

What are the electron configuration for Al+3:

Now, when you’ve been drawing those electron diagrams, you simply go and draw them all based on jumping down the periodic table, from Hydrogen to whichever element you want the configuration for.

Isn’t there an easier way? Well, yes. We call it Noble Gas Configuration. This entails starting at the previous noble gas, and instead of writing all the orbital information, you go straight from that gas. For example, let’s compare Lead’s Configuration.

Going by the standard configuration, we have:

For Pb, a consistent expanded ground-state configuration is 1s² 2s² 2p⁶ 3s² 3p⁶ 4s² 3d¹⁰ 4p⁶ 5s² 4d¹⁰ 5p⁶ 6s² 4f¹⁴ 5d¹⁰ 6p², totaling 82 electrons. Its shorthand is [Xe]4f¹⁴5d¹⁰6s²6p². The expanded sequence includes the 3p⁶ occupancy that the original omitted.

That’s one long configuration. To use the noble gas configuration, we start at the first noble gas above lead. In this case, it is Xenon. Then we start adding electrons from there:

[Xe]6s24f145d106p2

That is quite a bit nicer to look at, isn’t it?

Diamagnetic atoms are repelled by any magnetic field. This is typically a very low repulsion, so you won’t see any magnets being held up by diamagnetic materials. However, if the diamagnetic material is cooled to a very low temperature, it sometimes is powerful enough to support the mass of a magnet.

Paramagnetic materials are attracted to magnetic fields. Again, there isn’t that strong of an attraction in your typical environment, but it is there.

To determine whether a material is diamagnetic or paramagnetic, you need only look at the electron configuration. If there are any unpaired electrons, then that atom will be paramagnetic. Otherwise, if the atom’s electrons are all paired, then it will be diamagnetic.