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Since orbitals are just regions of electron density, they allow electrons to occupy the same space. I feel like in some sense this contradicts the Pauli exclusion principle limiting two fermions, or at least make it a fairly weak claim since an orbital isn't really separate from any other orbital. So what is physically going on here to restrict the electrons to have separate quantum numbers?

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2 Answers 2

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Two points relevant here.

First, of course, we can only put two electrons in any orbital - one with spin up and one with spin down because of the Pauli exclusion principle.

Secondly, When all other things are equal the state with highest multiplicity gives the lowest energy.... For example, in the carbon atom there are two electrons in the 3 2p orbitals. We can put these two electons in with spins parallel or with spins opposite. If we have spins parallel the total spin of the two electrons is 1 and the 'multiplicity' ($2S+1$) is 3 or 'triplet'. With spins opposite the total spin of the two electrons is zero and the multiplicity is 1 or 'singlet'. Now when the two electrons have parallel spins and are both in the 2p orbials the Pauli exclusion principle forces them to occupy different regions of space and reduces the repulsion between the two electrons by keeping them more apart than two electrons in a singlet state. I think this effect is called Pauli repulsion. The effect of Pauli repulsion is that, if everythng else is equal, higher multiplicity states have lower energy than higher multiplicity states. So for he ground electronc configuration of the carbon atom with 2 electrons in the 2p orbitals the triplet state is lower in energy than the singlet sate.

Hope this helps.

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The Pauli Exclusion principle does not tell you anything intuitive at first. It states only that two Fermions are not allowed to occupy the same quantum state. If there were two quantum states that have the same probability distribution in space, then those two Fermions could occupy the same spatial probability distribution. The principle does not explicitly tell us that two Fermions can occupy the same position, it's only an implication from the real statement of the principle. Thus two orbitals that overlap in space do not in any way violate the exclusion principle.

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