Numerical evaluation of the internal orbitally resolved chemical hardness tensor in density functional theory

Abstract

The purpose of the present work was to develop a method allowing one to extract the information needed for the construction of the internal chemical hardness tensor at the molecular orbital level from standard density functional calculations. This method is based on the Janak theorem and on the extension of the Slater transition-state concept. A detailed discussion of the current ideas about the validity of the Janak theorem is presented as well as of the established relations of this subject with the ensemble V-representability problem. The internal chemical hardness tensor has been obtained for water molecule as an example system. Its structure is consistent with the criteria for the internal molecular stability.

Local atomic and orbital reactivity indices from density functional calculations for hydrogen-bonded 1,2-dihydroxybenzene

“….w x DFT 20 affords additional opportunities in theoretical studies of such systems. The concept of orbital hardness indices has proven to be useful in describing local reactivity trends of chemical sysw x tems 21,22 . In the present work, we determined and used the orbital hardness matrix to elucidate in more detail the interrelationship between internal hydrogen bonding and the stabilization of neutral and ionic phenolic radicals.…”

Orbital hardness tensors from hydrogen through xenon from Kohn–Sham perturbed orbitals

“…In particular, if virtual orbitals are considered in the hardness tensor the deviation from symmetry can become even more significant as the tensor elements have to be computed using a different definition of the difference quotient: The occupation number must be in the interval of [0,2] (or [0,1] for open‐shell systems), hence a no charge can be added to a fully occupied orbital, and no charge can be removed from an empty orbital. To access a symmetrical hardness tensor that includes virtual states, a scheme employing thermal expansion of DF theory was proposed in which a nonzero temperature reference system is considered, in order to occupy with a small fraction of electron the virtual states; this scheme has been applied to the hardness tensor for water and ferrocene 15.…”

“…A hardness tensor computation within the atoms in molecule (AIM) method was proposed by Komorowski 10. Later several works 11–16 appeared that promoted the idea of defining η ij in its “natural” framework, namely within DFT. Initially, Neshev and Proynov and colleagues 11, 12 adopted the X α method to derive η ij as first derivative of the orbital energy with respect to the orbital occupation.…”

On the applicability of the HSAB principle through the use of improved computational schemes for chemical hardness evaluation

“…From the time hardness was first defined7, 8, 10, 11 within the density function theory (DFT), a huge amount12–22 of work has been devoted to the derivation of operational definitions and scales for different atomic systems, useful for the practical attachment of numbers to the rather intuitive concept. At the same time the two major principles, the HSAB and the maximum hardness (MH) principles,6, 23 associated with hardness have been widely tested using several working formula and computational approaches at different levels of approximation 24–53.…”

“…(3) indicates that hardness is related to the energy gap between occupied and unoccupied orbitals, and, as correctly pointed out by Chermette,57 possible discontinuities in the derivative of the energy with respect to the number of electrons for systems possessing a HOMO‐LUMO energy gap, like molecules (with discrete energy levels), may be encountered. The situation outlined above seems to discourage the use of the last working formulas and other computational definitions12–22 have been proposed to improve it. The articles dealing with this subject published in the last two decades, however, clearly demonstrate that eq.…”