The Born method [94] extends the Born-Oppenheimer approximation
by including corrections to the effective potential ** E^{0}(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
nuclear problem;

Note that includes nuclear repulsion in addition to the electronic interaction with the nuclei. The total wave function is expanded in the complete set of :

where are the eigenstates of the fixed nuclei problem Eq. 2.3. The solution to the problem

is obtained by substituting (2.6) into (2.5), multiplying the result by , and integrating over

where

Note that in (2.8), the diagonal term is moved to the left-hand side, hence the summation is for off-diagonal terms (). In the Born adiabatic approximation, the right hand side of (2.8),

with the effective potential for a given electronic state

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which includes the diagonal correction to the zero order potential, taking partly into account the coupling between the nuclear and electronic (muonic) motion. The Born-Oppenheimer approximation, on the other hand, neglects this coupling, and assumes

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

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 [3]. 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 [4].