The Born method  extends the Born-Oppenheimer approximation
by including corrections to the effective potential E0(R) in
Eq. 2.4. The full Hamiltonian, separated in two parts,
includes the term
describing the relative motion of the two
nuclei as well as the coupling between the electronic (muonic) and
the nuclear motions, in addition to the term
for the fixed
The validity of the Born-Oppenheimer approximation is dependent on the smallness of the expansion parameter where is the reduced mass of the two nuclei, and m the mass of electron or muon. The parameter reflects the ratio of the amplitudes of the nuclear motion in comparison to the equilibrium internuclear distance, hence the assumption that allows one to neglect the term in (2.8), which contains derivatives of with respect to R . Dependence of the Born approximation on is less clear and somewhat qualitative, and the accuracy depends on specific characteristics of the problem including the strength of off-diagonal coupling in Eq. 2.8.
In the case of the muonic molecule, is not small. For its loosely bound states, which are our main interest, the vibrational amplitudes are not small compared to the internuclear distance. Hence both the Born-Oppenheimer and the Born Adiabatic approximation fail to describe even qualitative characteristics of the muonic molecule, such as the number of the bound states . As for loosely bound states, the former method gives a state bound much too deeply, while with the latter method the state is not bound at all .