If we make the Born Oppenheimer approximation and write the wavefunction for electrons and nuclei as a product, ${}_t(r,R)={}_k(q,Q){}_k^t(Q),$ then the Schroedinger equation separates into two equations,

$H_{\text{e}\text{l}\text{e}\text{c}}{}_k(q$;$Q)=E_k(Q){}_k(q$;$Q)$

and

$[$:$[T_n+E_k(Q)]\}$

For a diatomic molecule $Q$ is just $r$ which is the bond length.

The Born Oppenheimer approximation is accurate if integrals

${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}{}_k(q$;$r)\frac{d}{dr}{}_j(q$;$r)dq$

are small. Explain why these integrals are small when $E_j(r)$ and $E_k(r)$ are far apart. To do this calculate

${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}{}_k(q$;$r)[H_{\text{e}\text{l}\text{e}\text{c}},p_r]{}_j(q$;$r)dq$

where $p_r$ is the momentum operator.