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Due to Hund's rule, electrons start filling up the orbitals without pairing up. When this is happening, do the electrons all fill up the 'up' spin? Could they fill in the 'down' spin? Why do they prioritize the 'up' spin if they do?

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No it does not matter whether you start with 'spin up' or 'spin down' electrons as long as you are consistent: If you start filling some orbitals with 'spin up' electrons you will have to keep doing this for all the orbitals you want to fill.

But: The convention is that you start with 'spin up' as this usually is taken to represent the $\frac{1}{2} \! \hbar$ eigenvalue of the $z$ component of the spin angular momentum operator and you should adapt to that in order to avoid confusion.

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    $\begingroup$ awesome. thank you! Is it then, fair to say that up/down of the first filling will be random? $\endgroup$ – IlIIlllIIIlIIlllIlIllIIIlIlI Sep 6 '14 at 2:55
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    $\begingroup$ @jkim Well, there are a few subtleties and I don't know how much you actually know about quantum mechanics. But if you only want to use the Aufbau principle and put some up/down arrows into boxes to describe the electron configuration of atoms and molecules then the choice between up and down should be random. But: The convention is that you start with 'spin up' as this usually is taken to represent the $+ \frac{1}{2} \! \hbar$ eigenvalue of the $z$ component of the spin angular momentum operator and you should adapt to that in order to avoid confusion. $\endgroup$ – Philipp Sep 6 '14 at 3:43

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