"Hardness profile" of a reaction path

Quantum chemical study of the umbrella inversion of the ammonia molecule

“…The dynamics of the process of evolution of the transition state from the ground state prior to the inversion of ammonia can be quantitatively followed by computation of µ and η and other related parameters as a continuous function of reaction coordinates for the transformation. Datta [37] has reported an INDO calculation of inversion of ammonia within the paradigm of DFT. Now, using Koopmans' theorem, the operational approximate working formula of µ and η are obtained in terms of energy of the HOMO and LUMO, the highest occupied and the lowest unoccupied molecular orbitals or the frontier orbitals of a molecule [32,34].…”

Are the maximum hardness and minimum polarizability principles always obeyed in nontotally symmetric vibrations?

“…Both principles have been applied successfully to the study of molecular vibrations, 9,15-18 internal rotations, 6͑b͒,19 excited states, 20,21 aromaticity, 22 and different types of chemical reactions. 12,[23][24][25][26][27][28][29][30][31][32][33] It has been found in most of these cases that the conditions of maximum hardness and minimum polarizability complement the minimum energy criterion for molecular stability.…”

“…However, relaxation of these constraints seems to be permissible, and in particular, it has been found that in most cases the MHP still holds even though the chemical and external potentials vary during the molecular vibration, internal rotation or along the reaction coordinate. 12,[15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33] Hereafter, we will refer to the generalized MHP ͑GMHP͒ or MPP ͑GMPP͒ as the maximum hardness or minimum polarizability principles that do not require the constancy of chemical and external potentials during molecular change. It is worth emphasizing that the generalized versions of these principles have not been proven.…”

Imaginary Vibrational Modes in Polycyclic Aromatic Hydrocarbons: A Challenging Test for the Hardness Profiles

“…[13,18] Even though these constraints are never fulfilled in any kind of nuclear displacement, the literature has shown that the relaxation of these constraints is permissible and it has been found that the MHP holds for molecular vibrations, [19][20][21] internal rotations, [22] and different types of chemical reactions. [23] However, some failures of the MHP have also been reported, [24] with hardness profiles violating the MHP. Thus, hardness profiles can follow or break the MHP, but in both cases the stationary points of the hardness profiles are usually very close to the stationary points of the energy profiles.…”