Low-energy unphysical saddle in polynomial molecular potentials

Vibrational spectra of polyatomic molecules are often obtained from a polynomial expansion of the adiabatic potential around a minimum. For several molecules, we show that such an approximation displays an unphysical saddle point of comparatively small energy, leading to a region where the potential is negative and unbounded. This poses an upper limit for a reliable evaluation of vibrational levels. We argue that the presence of such saddle points is general.