MO diagram for allyl cation
(1). No, the two orbitals have to be of opposite phase relative to each other. Recall that the MO wavefunctions for $\pi$-delocalization are very much like the wavefunctions for the particle-in-a-box, which requires that the phase alternate. Furthermore, a general rule is that molecular orbitals higher up in energy have more nodes, and the phases of the orbitals in $\pi_2$ must alternate for the middle lobe to be a node.
(2). We can speak of double bonds only for the molecule, not for molecular orbitals. Molecular orbitals do, however, have bonding character, and we characterize $\pi_2$ as non-bonding (little interaction between orbitals) and $\pi_3$ as anti-bonding (unfavorable interaction between orbitals).
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Aditya Dev about 19 hours
Given below is an image from google
In $\pi_2$, isn't there two possibilities? On the third carbon, The green coloured lobe can be either on top or on bottom relative to the lobes of the first carbon.
Does $\pi_3$ mean that there are no double bonds? Because it would result in an out of phase combination. What type of bonding will be present for $\pi_2$? Will it be same as $\pi_3$?
orthocresol over 6 yearsAre you missing the fact that $\pi_3$ is not filled? The $\pi$ bond arises from the filled $\pi_1$ orbital
Aditya Dev over 6 yearswhere can I read more about (1)? I have never heard something like that
a-cyclohexane-molecule over 6 yearsI like Clayden's Organic Chemistry. In the first edition, I think this is chapter 7.