The unconstrained local hardness: an intriguing quantity, beset by problems

Abstract

Developing a mathematical approach to the local hard/soft acid/base principle requires an unambiguous definition for the local hardness. One such quantity, which has aroused significant interest in recent years, is the unconstrained local hardness. Key identities are derived for the unconstrained local hardness, δμ/δρ(r). Several identities are presented which allow one to determine the unconstrained local hardness either explicitly using the hardness kernel and the inverse-linear response function, or implicitly by solving a system of linear equations. One result of this analysis is that the problem of determining the unconstrained local hardness is infinitely ill-conditioned because arbitrarily small changes in electron density can cause enormous changes in the chemical potential. This is manifest in the exponential divergence of the unconstrained local hardness as one moves away from the system. This suggests that one should be very careful when using the unconstrained local hardness for chemical interpretation.

An Estimation of Hybrid Quantum Mechanical Molecular Mechanical Polarization Energies for Small Molecules Using Polarizable Force-Field Approaches

“…10, the PAC coupling matrix can be written as $$S_{AB}=\frac{{\partial }^{2}E}{\partial {\varphi }_A\partial {\varphi }_B}=\frac{\partial q_A}{\partial {\varphi }_B},$$ which is closely related to the concept of local softness. 77–80 Due to the difference between the linear-response kernel and the local softness (see, for example, Eq. 21 in Reference 80 ), we will call S AB the pseudo-softness supermatrix.…”

“…77–80 Due to the difference between the linear-response kernel and the local softness (see, for example, Eq. 21 in Reference 80 ), we will call S AB the pseudo-softness supermatrix.…”

Conceptual density functional theory based electronic structure principles

“…Attempts have been made to dene a local variant of hardness under certain conditions. [60][61][62] Because of the inter-dependence of v(r) and r(r) within the premise of DFT, ambiguities arise [63][64][65] for the simultaneous denition of local hardness and local soness when local hardness is described as follows: 66,67 hðrÞ ¼ dm drðrÞ vðrÞ (23) h(r) could be evaluated from the hardness kernel h(r,r 0 ) also in the following way:…”

Alternative Ornstein–Zernike models from the homogeneous electron liquid for density functional theory calculations

“…Currently, we evaluate the matrix exponential on the integration grid by diagonalizing a matrix with the dimension of the grid. The computational cost can be reduced by expanding the DCF in a basis set [100] and using a more efficient six-dimensional integration grid, [101,102] but this method will remain more expensive than conventional approximate exchange-correlation functionals. The second difficulty is that it can be very difficult to find a satisfactory parameterization for the DCF.…”