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     

Electronic properties of oxides: Chemical and theoretical approaches

An original analysis of the electronic and chemical properties of oxides is proposed based on the electronegativity χ and the chemical hardness η. This model which has been applied to various oxide based metals, degenerate semiconductors and optical properties of transition metal oxides allows explaining their electronic behaviors: Strong electronegativity and weak chemical hardness characterize oxides of transition elements with high oxidation state. Strong electronegativity and strong chemical hardness feature insulators with a large optical gap. Weak electronegativity and moderate chemical hardness describe alkali and alkaline earth oxides and weak electronegativity and strong chemical hardness are for ionic oxides with a relatively large optical gap. For a few illustrative case studies, ab intio electronic band structure calculations within the density functional theory framework are used.     

HSAB principle applied to the time evolution of chemical reactions

Time evolution of various reactivity parameters such as electronegativity, hardness, and polarizability associated with a collision process between a proton and an X- atom/ion (X = He, Li(+), Be(2+), B(3+), C(4+)) in its ground ((1)S) and excited((1)P,(1)D,(1)F) electronic states as well as various complexions of a two-state ensemble is studied using time-dependent and excited-state density functional theory. This collision process may be considered to be a model mimicking the actual chemical reaction between an X-atom/ion and a proton to give rise to an XH(+) molecule. A favorable dynamical process is characterized by maximum hardness and minimum polarizability values according to the dynamical variants of the principles of maximum hardness and minimum polarizability. An electronic excitation or an increase in the excited-state contribution in a two-state ensemble makes the system softer and more polarizable, and the proton, being a hard acid, gradually prefers less to interact with X as has been discerned through the drop in maximum hardness value and the increase in the minimum polarizability value when the actual chemical process occurs. Among the noble gas elements, Xe is the most reactive. During the reaction: H(2) + H(+) --> H(3)(+) hardness maximizes and polarizability minimizes and H(2) is more reactive in its excited state. Regioselectivity of proton attack in the O-site of CO is clearly delineated wherein HOC(+) may eventually rearrange itself to go to the thermodynamically more stable HCO(+).     

A tug-of-war between electronic excitation and confinement in a dynamical context

Quantum fluid density functional theory has been used to study the time evolution of various reactivity parameters such as hardness, electrophilicity, entropy, chemical potential, polarizability, electronegativity etc. in a confined environment during time dependent processes like atom-ion collision and atom-field interaction. Responses in the reactivity parameters of the helium atom, in the dynamical context, for ground state as well as in excited state, have been reported. The confinement is incorporated through a Dirichlet type boundary condition. With a decrease in the size of the cylindrical box, the system gets harder and less polarizable. Simultaneous excitation and confinement may bring back the ground state behavior.     

Conceptual DFT based electronic structure principles in a dynamical context

Within the context of quantum fluid density functional theory diverse processes simulating a chemical reaction like interaction of an atom or a molecule with an externally applied electric or magnetic field or its collision with a proton have been analyzed in a dynamical situation. The changes produced in the chemical reactivity parameters namely chemical potential, hardness, polarizability during such processes have been identified and discussed. In addition, confinement is also introduced to observe the necessary variation in the different reactivity parameters.     

DFT studies and AIM analysis of AlN-polycycles

The structure and properties of AlN-polycycles were studied by DFT (density functional theory) method. The results of calculations were obtained at B3LYP/6-311G(d, p) level on model species. Topological parameters such as electron density, its Laplacian, kinetic electron energy density, potential electron energy density, and total electron energy density at the ring critical points (RCP) from Bader’s ‘Atoms in molecules’ (AIM) theory were analyzed in detail. These results indicate a good correlation between ρ(3, +1), G(r), H(r), and V(r) averaged values and hardness of AlN-polycycles. The aromaticity of all molecules has been studied by nucleus-independent chemical shift. There is a linear correlation between ΣNICS(0.0)_molecule values and polarizability.     

Hardness Equation for Ormosils

Hardness of ormosils coating on various kinds of substrates is important, and by considering recent progresses in understanding the hardness of ionic crystals or covalent crystals, new hardness equations for calculating the hardness of glasses or ormosils from chemical compositions were derived. When we applied an indenter to the surface of glass or sol-gel coatings, the surface of indenter is a declined one to the flat surface of glass or coating, thus the applied force should be analyzed by using the shear modulus, S, and Young's modulus, E. This is now well accepted for the analysis of hardness of ionic or covalent bonding inorganic materials. For example, by considering the binding energy and plastic deformation, Gilman showed that Hv of NaCl crystal can be calculated by an equation including elastic stiffness which indicated a good agreement between calculated and observed value. For covalent crystals he reported that the strength of chemical bonds could be correlated with the glide (shear) activation energy for covalent crystals quantitatively. These explanations are basically applied to the hardness of silicate glasses and ormosils. By considering both shear modulus and Young's modulus we have derived equations for calculating the hardness of glasses or ormosils from chemical composition, which includes packing density of atoms and bond energy per unit volume has been taken account. The agreements between calculated and observed hardness values for ormosils were comparatively good.     

Simple Model for Calculating the Rate Constants of Plasma-Chemical Reactions in Low-Pressure DC Magnetron Discharges

A simple model which allows one to calculate the rate coefficients of plasma-chemical reactions in low-pressure DC magnetron discharges is presented. In this model, the electron cyclotron frequency is assumed to be much greater than any electron collision frequency. We also assume that plasma-chemical reactions take place in the near-cathode bright region where the magnetic field, the electric field, the electron density, and the electron energy are maximum. The collision probabilities have been calculated for an electron moving in crossed E × B fields by averaging the cross-sections of plasma-chemical reactions along its trajectory and over all its possible initial pitch angles. Based on this model we calculated the rate constants of the plasma-chemical reactions taking place in DC magnetron reactive sputtering in argon–oxygen gas mixtures.     

Nonadiabatic collision model calculations of ion-pair formation cross sections of Cs+O2→Cs++O

This paper has improved Hickman's nonadiabatic collision model by substituting Hickman's constant velocity classical straight line trajectory approximation with the solution of motion equation mR=?dV(R)/dR, and has calculated the cross sections of ion-pair formation Cs+O₂→Cs⁺+O^?₂ with the improved nonadiabatic collision model (INCM). A comparison of our results with other theoretical and experimental results has been made.     

On the quality of the hardness kernel and the Fukui function to evaluate the global hardness

An approximated hardness kernel, which includes the second derivative with respect to the density of the kinetic energy, the electron-electron coulomb repulsion, and the exchange density functionals, has been tested for the calculation of the global hardness. The results obtained for a series of 40 cations and neutral systems and 16 anions represent in most cases an improvement of the results obtained using the HOMO-LUMO gap approach and indicate the viability of this approach to evaluate global hardness. In addition, the relevance of the Fukui function approximation and the role of the three components of the hardness kernel in the evaluation of the global hardness have been analyzed.     

Electronegativity and hardness in the chemical approximation

The chemical electronegativity of an atom (Mulliken definition) has been identified with the average value of χ, the electronegativity function given by the rigorous density functional theory. An appropriate definition of hardness is developed, and a scale of hardness for bonded atoms is proposed. The electrodynamical atom model is demonstrated to produce a simple relation between atomic hardness and size. Electronegativity has been calculated for bonded atoms in a variety of molecules and crystals, covalent and ionic, without any specific approximation for the energy function E(q). Expressions for the electronegativity of a molecule have been derived and critically discussed.     

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     

Determination of the energy distribution in unimolecular reactions under soft collision conditions

The effective energy distribution of activated molecules at the time of reaction under soft collision conditions ?′(E, ω) is pressure dependent and therefore difficult to recover from unimolecular decomposition data obtained at different pressures. We show in this work that the part of this function restricted to the condition that the collision frequency ω has to be equal to the microscopic rate constant k(E) at the energy given ?″[E,ω = k(E)] is a reasonable approximation to the input energy distribution ?(E) for quite soft collisions. This function is not pressure dependent and then recoverable at least in principle and as a matter of fact is not conceptually far from the function that the already reported deconvolution methods based in physical approximations attempt to recover. The deconvolution methods have been checked under soft collision conditions. We have found that the input energy distribution is recovered with reasonable accuracy for energies transferred by collision 〈ΔE〉 above 5 kcal mol^?1, conditions common in polyatomic systems.     

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     

Kekulé’s Oscillating D3h Cyclohexatriene Structure of Benzene

Kekulé postulated that neighbouring carbon atoms in benzene undergo incessant collision with each other, thereby leading to the interchange of double and single bonds, which amounts to an oscillation between two cyclohexatriene structures in dynamic equilibrium. It has been claimed that Kekulé arrived at a fully symmetric D_6h structure of benzene and that the oscillation hypothesis should not be attributed to him. However, Clausius’ collision theory, which was known at the time, implies that, when the absolute temperature approaches zero, the collision frequency will tend toward zero too, i.e. collisions will stop, and a static, D_3h cyclohexatriene obtains. The classical collision theory did not allow Kekulé to construct the desired D_6h structure as the energy minimum. The theory of harmonic oscillators would have allowed it, but that was not attempted at Kekulé’s time.     

A time-dependent molecular orbital approach to electron transfer in ion–metal surface collisions

A time-dependent molecular orbital method has been developed to study charge transfer in collisions of ions with metal surfaces at energies between 1 and 100 au. A set of localized basis functions consisting of generalized Wannier functions for the surface and s- and p-atomic functions for the ion, is used to separate the system into primary and secondary regions. An effective Hamiltonian and time-dependent equations for the electron density matrix are obtained in the primary region, where most charge transfer occurs. The equations for the electron density matrix are solved with a linearization scheme. The method is suitable to study atomic orbital orientation for collisions of ions and surfaces. A model calculation for Na⁺ + W(110) collisions with a prescribed trajectory is presented. The interaction potentials between the W(110) surface and Na⁺ 3s and 3p orbitals are calculated from Na⁺ pseudopotentials. Results show that the yield of neutralized atoms in 3p states changes as the collision energy is lowered.     

An interpretation of amphoteric gel hardness variation through potential and hardness measurement

The hardness variation of amphoteric gel according to the surrounding solution conditions is quite unique. It hardens and softens reversibly regardless of its molecular network density. But this has been understood merely qualitatively. For the purpose of elucidation of the details of its behavior, we performed quantitative potential and hardness measurements on it. We observed the constant potential of amphoteric gels, approximately -60 mV, regardless of their swelling ratio and hardness. Such observations can be interpreted as the maintenance of the constant charge density of *COO- for any amphoteric gel, and they are further interpreted as intermolecular salt-linkage formation/disruption dominating the hardness of amphoteric gels.