The minimal necessary and sufficient condition for Hamiltonians to yie
ld so-called adiabatic energies of molecules is determined and used to
formulate the non-perturbative non-variational Minimal Adiabatic Appr
oximation for obtaining them. No finite electronic energy eigenvalues
which result from this theory can become accidentally degenerate at an
y localized configuration of their nuclei, thus establishing a general
'non-crossing' rule of overall molecular potentials, which obviates t
he Jahn-Teller effect entirely on an adiabatic basis. The resulting el
ectronic energy eigenfunctions are single-valued continuous bounded ap
propriately differentiable functions of their parametric nuclear coord
inates and cannot have any geometric phase factor ascribed to them. Th
e ground-state electronic energy eigenfunctions cannot have any phase
factors which depend on nuclear configuration affixed to them and stil
l retain their ground-state status. The ground-state electronic energy
eigenvalues are bounded from below by those of the Born-Oppenheimer a
pproximation and from above by those of the Born-Huang approximation,
as are their corresponding ground-state total energies.