The Born-Oppenheimer Approximation

where R is the set of nuclear coordinates and r is the set of electronic coordinates. If spin-orbit effects are important, they can be added through a spin-orbit operator Ĥso. Unfortunately, the V̂eN(r,R) term prevents us from separating Ĥ into nuclear and electronic parts, which would allow us to write the molecular wavefunction as a product of nuclear and electronic terms, Ψ(r,R) = Ψ(r)χ(R). We thus introduce the Born-Oppenheimer approximation, by which we conclude that this nuclear and electronic separation is approximately correct. The term V̂eN(r,R) is large and cannot be neglected; however, we can make the R dependence parametric, so that the total wavefunction is given as Ψ(r;R)χ(R). The Born-Oppenheimer approximation rests on the fact that the nuclei are much more massive than the electrons, which allows us to say that the nuclei are nearly fixed with respect to electron motion. We can fix R, the nuclear configuration, at some value Ra, and solve for the electronic wavefunction Ψ(r;Ra), which depends only parametrically on R. If we do this for a range of R, we obtain the potential energy curve along which the nuclei move.