Polaron activation energy

In the simplest scenario, we can assume two parabolic functions as the potential wells for the two lattice cites.

V₁=(x-x₀)²
V₂=(x+x₀)²

then the activation energy is where the two potential cross: V1=V2.

Here we get E=0 and E_a=V₁=V₂=x₀².

In the case of optical process, we are talking about vertical excitation:
then the excitation is from the bottom of a well to the edge of the other well.

Let x=x₀, V₂=4*x₀²=4E_a, this is where the E_opt=4*E_a comes from.