Evaluation of the analogy between exceptions to the generalized maximum hardness principle for non-totally-symmetric vibrations and the pseudo-Jahn-Teller effect

Exceptions to the generalized maximum hardness principle (MHP) for nitrogen heterocycles, along non-totally-symmetric vibrations (bond-length alternating modes), are caused by pseudo-Jahn-Teller (PJT) coupling between the ground and the excited state along those modes. Thus, the PJT effect causes an increase of the energy gap between the ground and the excited state, and this is reflected in an increase of hardness. This relationship is based on the assumption that the hardness (defined as difference between ionization potential and electron affinity) is approximately equal to the HOMO-LUMO gap (Koopmans' approximation), and this is in turn assumed to be approximately equal to the energy difference between the ground and excited state. The analogy is valid here because the excited state and the relevant vibration have the same symmetry, and the geometry and force constants of the optimized minima for the two coupled states are reasonably similar.