A pictorial (and hopefully pedagogical) discussion on the Born–Oppenheimer approximation

The Born–Oppenheimer approximation is one of the central tenets of chemical dynamics and reactivity. Despite its central importance in chemistry, the Born–Oppenheimer approximation is often severely misinterpreted or misrepresented. More specifically, widespread claims about the Born–Oppenheimer approximation within the chemistry community imply that this approximation enforces the nuclei of a molecule to be (1) frozen and (2) treated as classical particles. Both claims are wrong. This article aims to discuss the derivation of the Born–Oppenheimer approximation in a pedagogical way, representing the main steps of this derivation pictorially via a model molecular system. The pictorial derivation is then connected to the formal mathematical derivation for the interested reader. We hope that this pictorial derivation can help chemists to better understand the main steps leading to this key approximation and why the two claims mentioned above are incorrect. We also believe that a detailed understanding of the Born–Oppenheimer approximation is necessary to tackle problem in chemistry where this approximation breaks down, like photochemistry for example.