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I'm reading up on molecular orbital theory and LCAO in Ian Fleming's "Molecular Orbitals and Organic Chemical Reactions" and I don't understand the logic here:

"When there are electrons in the orbital, the squares of the c-values are a measure of the electron population in the neighbourhood of the atom in question. Thus in each orbital the sum of the squares of all the c-values must equal one, since only one electron in each spin state can be in the orbital."

I get the first part but i don't see how c values are related to spin and wouldn't total spin in an orbital equal 0? Any help would be appreciated but, since I haven't studied quantum mechanics yet, a simple as possible explanation would be best.

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The short answer is that $\sum |c|^2$ is basically equal to "the number of electrons in the orbital". You might think that this should be 2, not 1. But note that each molecular orbital actually comprises two different "spin orbitals". Each "spin orbital" contains one electron of a particular spin - so there is a spin-up orbital which can hold one spin-up electron, and a spin-down orbital which can hold a spin-down electron.

The value of $\sum |c|^2$ is calibrated with respect to these "spin orbitals", so the value has to be 1.

Obviously, this raises far more questions: What is a spin orbital exactly? What is a molecular orbital exactly? Why does one MO have two spin orbitals? (Well, you might have some intuition as to that.) Why does $\sum |c|^2$ represent electron density? Why is the number taken with respect to the spin orbitals and not the molecular orbitals? I could provide answers for these, but unfortunately, no further detail is possible without going into quantum mechanics. If you want to truly understand this, there is no way around it. Of course, it doesn't mean you have to learn it right now. The only point is that it will only make more sense if and when you do it.

When you do, you will find that these topics are covered in standard QM texts.

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    $\begingroup$ I was always confused when the definition of an MO was a “one-electron wave function” given that an MO can hold 2 electrons. This makes much more sense now. $\endgroup$
    – Josh Mitchell
    Commented Aug 30, 2021 at 22:51

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