Why Electrons Pair in Orbitals Around Atoms

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trollcast
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When electrons pair in orbitals around an atom what's the best description of why it happens, there's a bit of variation of the answers on google.

The lowest energy state is the most favourable, therefore electrons will inhabit the orbitals of the lowest energy first ie. 1s, 2s, 2d and so on.

However due to the Pauli exclusion principle no 2 fermions can occupy the same quantum state, however electrons with opposite spin numbers do not have the same quantum state therefore 2 electrons of opposite spins can occupy the same orbital.

It just the explanation in my chemistry books starts talking about repulsion and stuff but the above sounds more concise?
 
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Difficult to know what your chemistry books are discussing without more information.

However the rest of your post is accurate.

Another way to state the Pauli principle is that no two electrons in one atom can possesses the same set of quantum numbers.

Also note that there are only two options for the spin quantum number which implies two per orbital.