Hardness Kernel Research Articles

Global Hardness Evaluation Using Simplified Models for the Hardness Kernel

Two simplified models of the hardness kernel, η(r,r‘) ≅ 1/|r − r‘| and η(r,r‘) ≅ δ(r − r‘), have been tested to calculate the global hardness for a set of 18 molecules using the hybrid B3LYP functional. It is found that the simplest model, η(r,r‘) ≅ δ(r − r‘), yields the best ordering of the systems by hardness when compared to experimentally available hardnesses. However, it is worth noting that this approximation provides correct estimates of global hardnesses only after empirical corrections. Finally, it is also shown that the B3LYP method gives results close to conventional ab initio correlated methods.

Calculation of the nuclear Fukui function and new relations for nuclear softness and hardness kernels

Calculations of Cohen’s nuclear Fukui function and softness are presented for a sample set of diatomic molecules. The obtained results were interpreted using Berlin’s theorem of binding and antibinding molecular regions. Moreover, new relations among the nuclear reactivity descriptors were derived within the four ensembles of density functional theory; a definition was provided for the nuclear hardness kernel, in accordance with Cohen’s nuclear softness kernel. It turned out that local hardness can be connected with this nuclear hardness kernel, strengthening the idea that local hardness should be considered as a nuclear reactivity index, whereas local softness is an electronic reactivity index.

ENERGY REQUIREMENTS FOR SIZE REDUCTION OF WHEAT USING A ROLLER MILL

An experimental two-roll mill was developed and instrumented for computerized data acquisition. Milling testswere performed on three classes of wheat. Included in the study were six independent variables each with three levels,namely, class of wheat, moisture content, feed rate, fast roll speed, roll speed differential, and roll gap. Two covariates,single kernel hardness and single kernel weight, were also included in the statistical analysis. Prediction models wereconstructed for five dependent variables (fast roll power, slow roll power, net power, energy per unit mass and specificenergy). The prediction models fitted the experimental data well (r2 = 0.88 ~ 0.95). The power and energy requirementsfor size reduction of wheat were highly correlated with the single kernel characteristics of wheat. Feed rate affected fastroll power, slow roll power and net power significantly. Roll gap had a significant effect on roller mill grinding.Additional milling tests were conducted by randomly selecting independent variables and covariates to verify therobustness and validity of the prediction models.

Simplified Models for Hardness Kernel and Calculations of Global Hardness

Reported in this paper are simplified models for the hardness kernel η(r,r‘) and theoretical values of the global hardness for the first 54 neutral atoms calculated using these models. It is found that a particularly simple model for the hardness kernel, η(r,r‘) = 1/|r − r‘| + C, where C is a constant, generates good results for the global hardness η. Both main-group elements and transition metal elements are considered, as are various conventional models, in approximating energy components.

Chemical potential, hardness, hardness and softness kernel and local hardness in the isomorphic ensemble of density functional theory

New relations among reactivity descriptors are provided within the recently introduced modified isomorphic ensemble of density functional theory. In addition, expressions for the softness and hardness kernel are derived in the canonical, grand canonical, isomorphic, and grand isomorphic ensemble. There results a new definition for the local hardness, η(r)=[∂u(r)/∂N]σ=ησ−g(r), where g(r)=[∂υ(r)/∂N]σ and σ=ρ/N is the shape factor. This identifies the local hardness as a function measuring the response of the system’s external potential to a perturbation in electron number at a constant shape factor. Furthermore, it is shown that one cannot represent both local softness and local hardness unambiguously with one representation.

Orbital hardness matrix and Fukui indices, their direct self-consistent-field calculations, and a derivation of localized Kohn–Sham orbitals

Formulas governing fixed orbital hardnesses and their relation to the hardness kernel are derived. It is shown how the orbital hardness matrix and its inverse matrix, the orbital softness matrix, may thus be directly calculated, and then the total chemical hardness, softness, and electronegativity of a molecular species. These quantities are calculated for the molecule HCN, using Dirac exchange and von Barth–Hedin correlation in the local density form of Kohn–Sham theory. The result complies with the frontier orbital theory. As quantitative indicators of orbital reactivity, the frontier orbital softness and Fukui indices generally have larger values than inner electron orbitals. The relation of orbital hardness matrix elements to the two-electron orbital integrals in a typical molecular orbital calculation is discussed, and it is demonstrated that diagonalization of the orbital hardness matrix leads to orbitals more localized than conventional Kohn–Sham orbitals.

Hückel-type semiempirical implementation of a variational method for determining electronic band gaps

A previously derived [J. Chem. Phys. 103, 7645 (1995)] general variational principle, for determining the hardness (or band gap) of electronic systems, is applied to π-electron systems by a straightforward simple parameterization of the hardness kernel. For conjugated polyenes and annulenes, the hardness (band gaps) are essentially the same as (although no identical with) those predicted by the Hückel method. In contrast with the Hückel method, one need not assume that the total electronic energy is a sum of one-electron energies. Comparisons are made with Nalewajski’s Charge Sensitivity Analysis.

A local model for the hardness kernel and related reactivity parameters in density functional theory

The local density approximation is used to construct a model for the reactivity parameters defined in density functional theory. Starting from a model for the hardness kernel, equations are derived to compute the softness kernel, local softness and local hardness, global softness and global hardness, the Fukui function and also the electric dipole polarizability. A very simple functional is used to test the equations in some atomic systems. The results are promising. Extensions and applications of the model are discussed.

Aspects of the Softness and Hardness Concepts of Density‐Functional Theory

AbstractIn the density‐functional theory of the ground state of an electronic system there arise the concepts of softness, hardness, local softness, and local hardness. Definitions of these quantities are reviewed, and then local softness and local hardness are discussed in some detail. The local softness of a species, the derivative magnified image, is a measure of the chemical reactivity of a site in the molecule. From it can be obtained the total global absolute softness in the sense of Pearson and a normalized chemical reactivity index of frontier type. Several formulas for s(r) are obtained, including new fluctuation formulas, and its determinative role in chemisorption, catalysis, and frontier‐controlled charge‐transfer processes is briefly discussed. Local hardness is a corresponding appropriately defined functional derivative η(r) = [δμ/δp(r)]v(r). Difficulties associated with ambiguities in this definition are discussed and resolved. It is concluded that for most purposes the best working formula for local hardness is magnified image, where η(r, r′) is the hardness kernel; magnified image, where F[p] is the usual Hohenberg‐Kohn functional and f(r) is the Fukui function. With this definition, η(r) = η, a constant which is the global hardness. Just as the chemical potential equalizes in the ground state, so does the hardness. It is demonstrated that hardness can be taken to be an average of orbital contributions.