One of the most important rules in quantum chemistry is the Pauli Exclusion Principle. It states that every electron in an atom must have a unique set of quantum numbers: no two electrons in the same atom can have the same quantum numbers.

This turns out to be the keystone of modern chemistry and explains the shape of the periodic table. Why? Electrons want to reside in orbitals with the lowest energy, thus if we think about adding electrons 1 by 1 to an atom we should put the electrons in the lowest energy orbitals first

The lowest energy shell has n=1. It has only one subshell, l=0 (l runs from 0 to n-1), which is an s subshell, and thus only one orbital, m_l = 0 (m_l runs from -l to +l). An electron may have either spin up or spin down: let's place a spin up (m_s = +1/2) electron in this orbital.

If we have a second electron, we can place it in the n=1, l=0, m_l orbital only if it has m_s = -1/2. Both electrons have a unique set of quantum numbers: n,l, and m_l are the same, but m_s is different.

If we try to add a third electron, however, there is no way to place it in the n=1, l=0, m_l orbital without duplicating either electron 1 or electron 2's set of quantum numbers. It must therefore go into another orbital. The next lowest energy orbital has n=2, l=0, m_l = 0. (See here for a diagram of the energy of various orbitals) We can place an electron in this orbital with spin up. The fourth electron can also go in this orbital with spin down. That's the last electron that can fit into this orbital, and we have to put the fifth and beyond into a different orbital.

It turns out each type of sublevel (s,p,d,f) has a unique capacity of electrons.

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