In all the introductions to the Born-Oppenheimer approximation I've seen, they make the following claim: "If you write out the stationary Schrodinger equation for the simplest molecule -- H[itex]_2^+[/itex] -- even it cannot be solved analytically, so we are forced to make an approximation." But how do we know it can't be solved analytically? Is that something that can be proved, or is it just the case that no one has been clever enough to figure out the analytical solution? My question probably stems from a misunderstanding of what the word "analytical" means. I'd love for someone to clear this up for me!