Born-Oppenheimer approximation

In the conventional method for a molecular system, known as the Born-Oppenheimer approximation, nuclear motions are decoupled from electronic (or muonic) motions, i.e., the total wave functions are assumed to have the form

$$\Psi (r,R)= \chi (R) \psi (r,R),$$ (16)

where $\psi (r,R)$, which is obtained from

$${\cal H} ^{0} \psi (r,R) = E^{0} (R)\psi (r,R),$$ (17)

are the eigenfunctions for the Hamiltonian for fixed nuclei, ${\cal H}^{0}$, with E⁰ its eigenvalues. Here, the internuclear distance R is just a parameter. The nuclear wave function, $\chi (R)$, is solved using E⁰(R) as the effective potential for the nuclear motion;

$$\left[ -\frac{\hbar}{2\mu}\nabla _{R}^{2} + E^{0} (R) - E \right] \chi (R) = 0.$$ (18)