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# State the Pauli exclusion principle in your own words. Explain why the Pauli exclusion principle is important to the understanding of the periodic table?

**The Pauli Exclusion Principle is in fact the main reason why we have the idea of an "" and why some transition metals like Chromium have access to** ##\mathbf(12)## ** electrons, and sometimes up to even** ##\mathbf(18)##.

The **Pauli Exclusion Principle** essentially states:

No two electrons may have entirely identical quantum states; at least one quantum number must be different.

I've given a formal explanation of the octet rule . Please read that before proceeding, as I will be furthering that discussion.

**Following that, we then realize that the octet rule is centered around the Pauli Exclusion Principle.**

**EXCEPTIONS TO THE OCTET RULE**

We can then determine how Chromium, for example, can use ##12## , sometimes, for the same reason, and not just ##8##. Here's what I mean.

Chromium's is:

(the original diagram is different, but it was wrong, because it had electrons in the ##4p##, which would mean ##30## electrons, not ##24##. It also has the ##3d## higher in energy than the ##4s##, but it's actually not, for Chromium, according to Eric Scerri, replying to David Talaga.)

##1s^2 2s^2 2p^6 3s^2 3p^6 color(blue)(3d^5 4s^1)##

where the blue atomic orbitals are the valence orbitals.

**QUINTUPLE BONDS?!**

The energy levels are so close together, however, that Chromium actually sometimes has access to ##12## , rather than just ##6##. That's why Chromium can sometimes make **SIX bonds**. Just take a look at this!

One **single** bond and one **quintuple** bond, and one interaction (dashed bond)! Okay, so how in the world?!

**QUANTUM NUMBER CONSIDERATIONS**

We can realize that Chromium sometimes has access to its ##3p## orbitals as well, as that would give it ##12## electrons, and the ##3p## is closest in energy to the ##3d## orbital (when moving downwards in energy).

So, we can consider the following :

##n = 3##:

##l = 1, 2## ##m_l = -2, -1, 0, +1, +2## ##m_s = pm"1/2"##

(covering the ##3p## and ##3d## orbitals)

##n = 4##:

##l = 0## ##m_l = 0## ##m_s = pm"1/2"##

(covering the ##4s## orbital)

In the same **type** of atomic orbital (examining only the ##3d##, for instance, or examining only the ##4s##, etc), all quantum numbers are the same, except for ##m_l## and ##m_s##, which CAN be different.

**UNIQUE QUANTUM STATES**

As a result, for the ##3d## orbital, we have **five unique quantum states** for the five available **spin-up** electrons (##m_l = -2, -1, 0, +1, +2## with ##m_s = +"1/2"##). ##m_l## just ultimately tells us that there are five different ##3d## orbitals (##3d_(z^2)##, ##3d_(x^2 - y^2)##, ##3d_(xy)##, ##3d_(xz)##, and ##3d_(yz)##).

This, however, doesn't include the spin-down electrons due to Hund's rule of favoring the maximum spin state, which, for Chromium's ##3d## orbitals, is ##+"5/2"##, and due to how there are exactly ##5## electrons here.

Next, for the ##3p## orbitals, we have three pairs of electrons. We have ##m_l = -1, 0, +1## with ##m_s = -"1/2"##, as well as ##m_l = -1, 0, +1## with ##m_s = +"1/2"##. That makes for a total of **six unique quantum states**, and thus six possible electrons that can exist in the ##3p## orbitals.

Finally, the ##4s## orbital has only one valence electron, which obviously can only exist in one possible way at a time. The quantum numbers corresponding to it are ##l = 0##, ##m_l = 0## with ##m_s = +"1/2"##. Whether you believe that a single electron can flip its spin or not, either way, that means **one unique quantum state**.

**TAKE-HOME MESSAGE**

Hence, Chromium could sometimes have ##5 + 6 + 1 = \mathbf(12)## **unique quantum states** for each of the ##12## electrons available for bonding, following the Pauli Exclusion Principle.

Each electron can only occupy one state at a time (like how one twin can only be that twin for all time), so with ##12## electrons, ##12## states are implicitly possible. That's how I would rationalize why in the world Chromium can make **SIX bonds** sometimes. :)