The use of the second order perturbation correction, Eq. (4), for the calculation of the  value in the case of PJTE instability is incorrect because in this case , meaning the first perturbation correction is larger than the main term, and hence the criterion of applicability of the perturbation theory in its simplest form does not hold. In this case, we should consider the contribution of the lowest excited states (that make the total curvature negative) in a pseudo degenerate problem of perturbation theory. For the simplest case when only one excited state creates the main instability of the ground state, we can treat the problem via a pseudo degenerate two-level problem, including the contribution of the higher, weaker-influencing states as a second order correction.
In the PJTE two-level problem we have two electronic states of the high-symmetry configuration, ground  and excited , separated by an energy interval of , that become mixed under nuclear displacements of certain symmetry ; the denotations , , and  indicate, respectively, the irreducible representations to which the symmetry coordinate and the two states belong. In essence, this is the original formulation of the PJTE. Assuming that the excited state is sufficiently close to the ground one, the vibronic coupling between them should be treated as a perturbation problem for two near-degenerate states. With both interacting states non-degenerate the vibronic coupling constant  in Eq. (5) (omitting indices) is non-zero for only one coordinate  with . This gives us directly the symmetry of the direction of softening and possible distortion of the ground state. Assuming that the primary force constants  in the two states are the same (for different  see [1]), we get a 2×2 secular equation with the following solution for the energies  of the two states interacting under the linear vibronic coupling (energy is referred to the middle of the  gap between the levels at the undistorted geometry):
It is seen from these expressions that, on taking into account the vibronic coupling, , the two APES curves change in different ways: in the upper sheet the curvature (the coefficient at  in the expansion on ) increases, whereas in the lower one it decreases.  But until  the minima of both states correspond to the point , as in the absence of vibronic mixing. However, if
the curvature of the lower curve of the APES becomes negative, and the system is unstable with respect to the  displacements (Fig. 1). Under condition (8), the minima points on the APES are given by
From these expressions and Fig. 1 it is seen that while the ground state is softened (destabilized) by the PJTE, the excited state is hardened (stabilized), and this effect is the larger, the smaller  and the larger F. It takes place in any polyatomic system and influences many molecular properties, including the existence of stable excited states of molecular systems that are unstable in the ground state (e.g., excited states of intermediates of chemical reactions); in general, even in the absence of instability the PJTE softens the ground state and increases the vibrational frequencies in the excited state.