"First, once there is more than a single electron, the combined effect of the electrical shielding of the nuclear charge and interelectron electrical repulsion jumbles things up. It turns out that these effects of nuclear shielding and electron-electron repulsion can be taken into account to a good approximation as relatively simple corrections to what we would expect if each electron were in the atom all by itself. This is known as the orbital approximation.
Second, and far more important, is that an amazing new property of the physical world comes into play, tracing to the fact that all electrons are, as far as we know, indistinguishable from one another. Once this indistinguishability is taken into account, through what is known as the Pauli exclusion principle, the result is a profound restriction on the way electrons can arrange themselves in an atom. It is this restriction that accounts for the periodic properties of the elements and so, ultimately, the structure of the physical world as we know it.
You may have learned the "rule" that no more than two electrons can be in the same orbital. If you have, you may also have puzzled about why such a rule is so. If you have decided, like many people who have been presented with just the rule without any explanation, that it has to do with electrical repulsion—that it reflects the electrons repelling one another due to their electrical charge—then you are in for a neat surprise. The "rule" instead traces to a deep algebraic property of nature that has nothing whatsoever to do with the charge on electrons!........................"
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