BornOppenheimerApproximation

The Born–Oppenheimer approximation is a fundamental method in quantum chemistry and molecular physics for simplifying the molecular Schrödinger equation by exploiting the large mass difference between electrons and nuclei. Proposed by Max Born and R. Oppenheimer in 1927, it treats electronic and nuclear motions separately: electrons move rapidly in the field of comparatively slow-moving nuclei.

In mathematical terms, the molecular Hamiltonian is H = Tn + Te + V(r,R). The molecular wavefunction Ψ(r,R) is approximated as a product Ψ(r,R) ≈ ψe(r; R) χ(R), where r denotes electronic coordinates and R nuclear coordinates. The electronic factor ψe solves the electronic Schrödinger equation He ψe = Ee(R) ψe with nuclei fixed at R, where He = Te + V(r,R). The resulting potential energy surface Ee(R) acts as the potential for the nuclear motion, which is governed by [Tn + Ee(R)] χ(R) = E χ(R). Additional corrections, such as the diagonal Born–Oppenheimer correction, refine the surface.

The approximation is widely used to compute potential energy surfaces, facilitate vibrational and rotational spectra, and enable Born–Oppenheimer molecular dynamics. However, it has limits: nonadiabatic couplings between electronic states can be important, especially near conical intersections or during fast photochemical processes, where the product Ψ(r,R) cannot be cleanly separated. In such cases, beyond-Born–Oppenheimer methods, including surface hopping and multiconfigurational approaches, are employed.