I've recently noticed that different sources on the internet and in books have different ideas of whcih atomic orbital influences which molecular orbital. In the diagram, where the lines connect the AO to the MO to show their influence, I noticed that in some of the MO diagrams, the $\mathrm{2p}$ orbital does not connect to the $\sigma_1$ orbital (the lowest in energy) and the $\mathrm{2s}$ orbital does not connect to the $\sigma_4$ orbital (the highest in energy). On the other hand, I also see the case where the opposite is true and they DO connect (this is the case for my own chemistry lecturer's notes).
I asked her if she could explain this difference, and she said that the $\sigma_1$ should have some $\mathrm{p}$ character and that the $\sigma_4$ should have some $\mathrm{s}$ character, so they do connect. She didn't however say that the other way was wrong. As a result I'm not sure whether the other way is wrong, or whether it's just like a difference in notation.
I hope someone can help clarify for me. I've attached two photos of the different cases.
Thank you.
$\hspace{15 mm}$
(source)
versus
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