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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

\begin{displaymath}\Psi (r,R)= \chi (R) \psi (r,R),
\end{displaymath} (16)

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

 \begin{displaymath}{\cal H} ^{0} \psi (r,R) = E^{0} (R)\psi (r,R),
\end{displaymath} (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;

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