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I understand that pi bonds involve two regions of electron density in a "side-by-side" arrangement, but why exactly does this occur when a double bond is formed? Furthermore, the book states that pi bonds have a "single nodal plane containing the internuclear axis". I am having trouble conceptualizing this. Could this be explained?
The first bond formed between two atoms is always a sigma bond. This creates an head-on overlapping between bonds. The pi bonds are then formed because the next two regions of electron density are perpendicular to the head-on sigma bonds. Think of two right angles facing each other. Imagine that each line in each right angle represents a region of electron density. When two electron densities touch head to head, there are another two electron densities facing parallel to each other. Since electron density is a cloud and not a line, the densities can overlap side by side and form a pi bond.
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