I'd say it's more of a physical approximation than a mathematical one. For low vibrational states (shorter internuclear distances), the region of the dipole moment function is relatively flat. So just picking the equilibrium distance actually approximates it pretty well. At high vibrational states, where you'd sample large internuclear distances, the curve starts to get wobbly on the outskirts and the approximation breaks down. This makes sense because you'd expect BO breakdown at higher vibrational energies.I guess I don't understand what is the mathematical approximation used that allows you to assume that the electronic integral, which is a function of ##R## can be approximated to be constant. In the BO approximation you would use the adiabatic approximation, but I am not sure here, formally, what allows you to do that. Intuitively, if you have, say, the function ##cos^2##, but your response time to this oscillation is too slow, what you see is the average over many periods which is ##1/2##. Given that that electronic integral sees just the average internuclear distance, I assumed it is something similar to this i.e. the electrons see just an average of the internuclear distance.