Local hardness: a critical account

This paper gives a critical account on the present status of the local hardness. We analyze the behavior of hardness related global, local, and nonlocal properties, paying particular attention to local hardness. Although this reactivity index has become very useful in predicting the regioselectivity of chemical reactions, the lack of a rigorous definition in the literature has shown that different approximations can yield different and sometimes contradictory conclusions. The present paper tries to provide insights into this controversial issue, starting from an approximate model of the hardness kernel and analyzing the features of the different models to evaluate the local and global hardnesses.     

Internal rotation of fluorinated butane compounds: The maximum hardness principle and carbon-carbon rotational barriers

A density functional study of the internal rotation, about the central carbon-carbon bond, of butane, 1,1,1,3,3-pentafluorobutane (PFB) and perfluorobutane (PerFB), has been investigated. The bond length, torsional potential energy and hardness profiles were obtained using the B3LYP density functional method with the basis set 6-311G. The maximum hardness principle (MHP) is only verified for butane. It was also found that for butane and PerFB there is a reciprocal relationship between the central carbon-carbon bond length variations and the hardness profile, being the agreement for butane excellent. This could provide an alternative approach for studying the MHP.     

Analogies and differences between two ways to evaluate the global hardness

The weight of the energetic components (electronic kinetic, electron-nucleus and electron-electron Coulombic, and correlation energies) of the ionization potential, electron affinity, chemical potential, and global hardness is evaluated and contrasted with the energetic components of the hardness kernel and the experimental values of these properties for 40 systems. The contrast of the hardness terms obtained from finite difference and hardness kernel gives some insight on the possible implications to differentiate the electronic energy with respect to the number electrons or the electron density.     

A dynamical study of the principle of maximum hardness

Temporal evolution of local and global hardness during an ion-atom collision process has been studied within a quantum fluid density functional framework. A dynamical variant of the maximum hardness principle has been found to be operative. Entropy maximises in the encounter regime. Time dependence of density and its laplacian provides important insights into the collision processvis-a-vis the hardness maximisation.     

Study of the orbital hardness and the Kohn‐Sham radius on single monoatomic anions

Within the Kohn‐Sham framework and for a series of single charged monatomic anions, the orbital hardness is calculated as a change in the frontier eigenvalue, which is equivalent to integrate the local hardness function obtained through the derivative of the KS effective potential respect to the occupation number. The local hardness function is composed by the sum of two terms with opposite sign that describe the electrostatic and exchange‐correlation interactions. Moreover, it is found that, at the KS radii, the last term vanishes with the result that the orbital hardness of the anion is a measure of the electrostatic potential exerted by the frontier density at the KS radii. A further derivation leads to establish a direct relationship between hardness and the inverse of the KS radii. The polarizability of the anion is also examined by computing it from the volume of a sphere having the KS radii. These results show that anions from the halide group are hard and little polarizable, whereas anions from the alkali group are soft and highly polarizable. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011     

Semiempirical evaluation of the global hardness of the atoms of 103 elements of the periodic table using the most probable radii as their size descriptors

Relying upon the fact that the density functional computation of the global hardness of the atoms of the elements are still at large and there is some mathematical in congruency between the theory and operational formula of finite difference approximation, we have suggested a radial‐dependent ansatz for evaluating global hardness of atoms as: η=a(7.2/r)+b (in eV), where, “a” and “b” are the constants and r is the absolute radius of atoms in angstrom unit. The ansatz is invoked to evaluate the global hardness of atoms of 103 element of the periodic table. The evaluated new set of global hardness is found to satisfy the sine qua non of a reasonable scale of hardness by exhibiting perfect periodicity of periods and groups and correlating the gross physicochemical properties of elements. The inertness of Hg and extreme reactivity Cs atoms are nicely correlated. The chemical reactivity and its variation in small steps in the series of lanthanide elements are also nicely reproduced. The results of the present semiempirical calculation also have strong correlation with the result of some sophisticated DFT calculation for a set of atoms. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2010     

Electrophilicity kernel and its hierarchy through softness in conceptual density functional theory

In the context of long‐range density functional theory of softness kernel, the concepts of local and kernel electrophilicity are developed to obtain the local to global hierarchical criteria such as bilocal symmetry, asymptotic behavior, and integral local to global relationships. Further development into potential‐density dependence is also provided while considering the local plus nonlocal specification of softness kernel and the associated atomic scales for the derived kernel; local and global electrophilicity are provided whose reliability is judged through the periodical features along the periodic table especially for the alkali metal atoms and the halogen atoms. © 2013 Wiley Periodicals, Inc.     

Calculation of dissociation constants and chemical hardness of some biologically important molecules: A theoretical study

The gas‐phase geometries of neutral, protonated, and deprotonated forms of some biologically important molecules, alanine (Ala), glycine (Gly), phenylalanine (Phe), and tyrosine (Tyr), were optimized using density functional theory at B3LYP/6‐311++G(d) and the ab initio HF/6‐311++G(d) level of theories. The neutral and different stable ionic states of Ala, Gly, Phe, and Tyr have also been solvated in aqueous medium using polarizable continuum model for the determination of solvation free energies in the aqueous solution. The gas‐phase acidity constants of above four molecules have been also calculated at both levels of theories and found that the values calculated at HF/6‐311++G(d) method are in good agreement with experimental results. A thermodynamic cycle was used to determine the solvation free energies for the proton dissociation process in aqueous solution and the corresponding pK_a values of these molecules. The pK_a values calculated at B3LYP/6‐311++G(d) method are well supported by the experimental data with a mean absolute deviation 0.12 pK_a units. Additionally, the chemical hardness and the ionization potential (IP) for these molecules have been also explored at both the level of theories. The Tyr has less value of chemical hardness and IP at both levels of theories compared with other three molecules, Ala, Gly, and Phe. The calculated values of chemical hardness and IP are decreasing gradually with the substitution of the various functional groups in the side chain of the amino acids. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011     

Studies of chemical hardness and Fukui function using the exact solution of the density functional theory

Neal's procedure has been applied to determine the electron density ρ(x) for the H₂ molecule. The chemical hardness has been calculated employing the ab initio and density functional theory methods and the values are found to be reasonably good. The principle of maximum hardness (PMH) was tested. Fukui functions and the distribution of electron density around the internuclear distance were studied employing the electron density of the H₂ molecule. © 2001 John Wiley & Sons, Inc. Int J Quant Chem 81: 4–10, 2001     

A new approach to local hardness

The applicability of the local hardness as defined by the derivative of the chemical potential with respect to the electron density is undermined by an essential ambiguity arising from this definition. Further, the local quantity defined in this way does not integrate to the (global) hardness-in contrast with the local softness, which integrates to the softness. It has also been shown recently that with the conventional formulae, the largest values of local hardness do not necessarily correspond to the hardest regions of a molecule. Here, in an attempt to fix these drawbacks, we propose a new approach to define and evaluate the local hardness. We define a local chemical potential, utilizing the fact that the chemical potential emerges as the additive constant term in the number-conserving functional derivative of the energy density functional. Then, differentiation of this local chemical potential with respect to the number of electrons leads to a local hardness that integrates to the hardness, and possesses a favourable property; namely, within any given electron system, it is in a local inverse relation with the Fukui function, which is known to be a proper indicator of local softness in the case of soft systems. Numerical tests for a few selected molecules and a detailed analysis, comparing the new definition of local hardness with the previous ones, show promising results.     

Third-order energy derivative corrections to the Kohn-Sham orbital hardness tensor

The third term in the Taylor expansion of the total energy functional around the number of electronsN is evaluated as the second-order derivative of orbital Kohn-Sham energies with respect to orbital occupancy. Present approach is an extension of an efficient algorithm to compute densityfunctional based orbital reactivity indices. Various energy derivatives used to approximate orbital reactivity indices are defined within the space spanned by the orbital occupation numbers and the Kohn-Sham one-electron energies. The third-order energy functional derivative has to be considered for singular hardness tensor ([η]). On the contrary, this term has negligible influence on the reactivity index values for atomic or molecular systems with positively defined hardness tensors. In this context, stability of a system in equilibrium state estimated through the eigenvalues of [η] is discussed. Numerical illustration of the Kohn-Sham energy functional derivatives in orbital resolution up to the third order is shown for benchmark molecules such as H₂O, H₂S, and OH⁻.     

Charge transfer associated with the physical process of hardness equalization and the chemical event of the molecule formation and the dipole moments

In this article, we have basically launched a search whether the dipole charge and dipole moment of heteronuclear diatomics can be justifiably evaluated in terms of charge transfer kernel using the hardness equalization principle as basis. We have derived a formula for computing dipole charge (q) on the basis of hardness equalization principle as q = aδ + b, where “a” and “b” are the constants and “δ” is the kernel of charge transfer from less hard atom to more hard atom during the rearrangement of charge on molecule formation. We have computed the dipole charges and dipole moments of as many as six different sets of compounds of widely diverse physicochemical behavior in terms of the algorithm derived in the present work. The computed dipole charge nicely reveals the known chemicophysical behavior of such compounds as are brought under the study. A comparative study of the nature of variation of theoretically evaluated and experimentally determined dipole moments reveals that there is an excellent agreement between the two sets of dipole data. Thus, the new algorithm derived for the calculation of the dipole charge using the hardness equalization principle as a basis is efficacious in computing the distribution and rearrangement of charge associated with the chemical event of molecule formation. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011     

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

Finite difference schemes, named Compact Finite Difference Schemes with Spectral-like Resolution, have been used for a less crude approximation of the analytical hardness definition as the second-order derivative of the energy with respect to the electron number. The improved computational schemes, at different levels of theory, have been used to calculate global hardness values of some probe bases, traditionally classified as hard and soft on the basis of their chemical behavior, and to investigate the quantitative applicability of the HSAB principle. Exchange acid-base reactions have been used to test the HSAB principle assuming the reaction energies as a measure of the stabilization of product adducts.     

Negative and Infinite Fukui Functions: The Role of Diagonal Dominance in the Hardness Matrix

The possible genesis of negative atom condensed Fukui functions is discussed based on hardness kernel matrix relationships. The recent hypothesis that diagonal dominance of the hardness matrix is a requirement for positive Fukui functions is proven, and general considerations also predict the possibility of regions with numerically unstable Fukui functions, including discontinuities.     

Modeling the electron density kernels

Existing approximation to the softness kernel, successfully explored in earlier work, has been extended; the normal Gauss distribution function has been used instead of the Dirac delta. The softness kernel becomes continuous functions in space and may be used to calculate the linear response function of the electron density. Three-dimensional visualization of the softness kernel and the linear response function are presented for a nitrogen atom as a working example. By using a single parameter of the spatial Gauss distribution, the novel softness kernel has been adjusted to be consistent with the standard form of the hardness kernel, representing the leading fraction of the electronic interactions in the system.     

Quantum similarity study of atoms: a bridge between hardness and similarity indices

A hardness based similarity index for studying the quantum similarity for atoms is analyzed. The investigation of hardness and Fukui functions of atoms leads to the construction of a quantum similarity measure, which can be interpreted as a quantified comparison of chemical reactivity of atoms. Evaluation of the new measure reveals periodic tendencies throughout Mendeleev's table. Moreover on the diagonal the global hardness was recovered. Considering a corresponding quantum similarity index reveals that renormalization of the measure can mask periodic patterns. The hardness was calculated for atoms with nuclear charge 3相似文献   

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

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.