Hardness equalization principle

The subsystem resolved analog of the local hardness equalization principle now reads ... 

We, therefore, conclude that the present effort of deriving an algorithm for the evaluation of the hardness of the carbon containing poly atomic molecules is a successful venture. The detailed study would suggest that the paradigm of the hardness equalization principle may be another law of nature like the established electronegativity equalization principle. 

Hardness Equalization Principle Analogous to the Electronegativity Equalization Principle... 

To justify our hypothesis that- The Electronegativity and the Absolute Hardness are two different Appearances of the One and the Same Fundamental Property of Atoms and the Hardness Equalization Principle can be Equally Conceived like the Electronegativity Equalization Principle we have basically launched a search whether the molecular hardness, an important conceptual descriptor of chemistry and physics, can be evaluated in terms of the atomic hardness values. 

To justify the hypothesis and our perception on the hardness equalization principle, we, starting from the semi-empirical radial dependent formula of computing global hardness of atoms suggested by us (Ghosh and Islam 2009), have derived an algorithm of the molecular hardness assuming that the hardness equalization principle is operative and justifiably valid. 

Now invoking hardness equalization principle after the formation of the molecule, the hardness of the individual constituents must be equalized i.e. 

In this essay we are applying our models of new definition of hardness and hardness equalization principle in computing the molecular hardness of some poly atomic carbon compounds. 

In this chapter, we present and analyze three important equalization principles in chemistry viz., electronegativity equalization principle (EEP), hardness equalization principle (HEP), and principle of electrophilicity equalization (PEE). 

The proposal of a hardness equalization principle [33] possibly arises from a dual consequence of the electronegativity equalization principle and the fact that global molecular softness (S) can be expressed as a mean of the corresponding local softness (Sj) of the constituent atoms [34], It is already known that the hardness (t]) varies directly with the electronegativity (%). i.e., T a X, for both atoms [30, 34] and moleeules [34], Therefore, Eq. (1) ean be similarly expressed in terms of hardness as [33,35, 36] ... 

From the early days of theoretical chemistry, the equalization principle and its application in real field are very popular among researchers. In the Chapter 8, Chattaraj and his colleagues have analyzed the three important equalization principles in chemistry viz., electronegativity equalization principle (EEP), hardness equalization principle (HEP), and principle of electrophilicity equalization (PEE). The qualitative nature and validation of three principles have been concluded nicely at the end of this chapter. 

As already mentioned, through DFT, it has been possible to explain the electronegativity equalization principle [1,7,10-13] and the hard and soft acids and bases principle [12,15-22] and, additionally, it has also been possible to introduce new ones like the maximum hardness principle [52,53] and the local hard and soft acids and bases principle [20,54—56]. 

These descriptors have been widely used for the past 25 years to study chemical reactivity, i.e., the propensity of atoms, molecules, surfaces to interact with one or more reaction partners with formation or rupture of one or more covalent bonds. Kinetic and/or thermodynamic aspects, depending on the (not always obvious and even not univoque) choice of the descriptors were hereby considered. In these studies, the reactivity descriptors were used as such or within the context of some principles of which Sanderson s electronegativity equalization principle [16], Pearson s hard and soft acids and bases (HSAB) principle [17], and the maximum hardness principle [17,18] are the three best known and popular examples. 

Just like Sanderson s electronegativity equalization principle, the Hard and Soft Acids and Bases principle was originally introduced without strong theoretical basis. Nevertheless, it was used widely from its formulation on. The principle states that hard acids prefer to coordinate with hard bases and soft acids with soft bases [82], In 1983, Parr and Pearson provided a definition for the chemical hardness [25]... 

Associated with these properties, important chemical reactivity principles have been rationalized within the framework of conceptual DFT the hard and soft acids and bases principle (F1SAB) [9], the Sanderson electronegativity equalization principle (EEP) [11], the maximum hardness principle (MF1P) [9,12,13], and the minimum polarizability principle (MPP) [14], The aim of this chapter is to revise the validity of the last two principles in nontotally symmetric vibrations. We start with a short section on the fundamental aspects of the MF1P and MPP (section 2). Section 3 focuses on the breakdown of these principles for nontotally symmetric vibrations, while section 4 analyses the relationship between the failure of the MF1P and the pseudo-Jahn-Teller (PJT) effect. A mathematical procedure that helps to determine the nontotally symmetric distortions of a given molecule that produce the maximum failures of the MPP or the... 

Popular qualitative chemical concepts such as electronegativity [1] and hardness [2] have been widely used in understanding various aspects of chemical reactivity. A rigorous theoretical basis for these concepts has been provided by density functional theory (DFT). These reactivity indices are better appreciated in terms of the associated electronic structure principles such as electronegativity equalization principle (EEP), hard-soft acid-base principle, maximum hardness principle, minimum polarizability principle (MPP), etc. Local reactivity descriptors such as density, Fukui function, local softness, etc., have been used successfully in the studies of site selectivity in a molecule. Local variants of the structure principles have also been proposed. The importance of these structure principles in the study of different facets of medicinal chemistry has been highlighted. Because chemical reactions are actually dynamic processes, time-dependent profiles of these reactivity descriptors and the dynamic counterparts of the structure principles have been made use of in order to follow a chemical reaction from start to finish. 

The importance of the equalization [48-49] of chemical potential and the maximum hardness [50] principles in chemistry, necessitates the study of isoelectronic changes in p and q, respectively. March [51] showed that in principle Eq. (8) can be employed for the evaluation of the isoelectronic changes in p. According to the March equation,... 

Atoms-in-molecule the induced ionic character of bonds is partially compensated by chemical forces through hardness equalization between the pair regions in a molecule. HSAB principles are therefore involved as a second-order effect in charge transfer. See expansion (30) for an example driven by ionic interactions through bonds. 

For transition metals, the participation of d orbitals must also be taken into account, and values have been tabulated for the first row series [87] At first sight it seems that equations (1), (2-4) and (6) are all what we need to compute partial charge distributions. This is not the case and the main problem lies in the evaluation of the chemical hardnesses. This can be easily understood by applying (1) and (2-4) to a simple diatomic molecule A-B where B is more electronegative than A. If we apply the electronegativity equalization principle XA = XB — 0 )> it comes with qA = —5jj = q ... 

In 2 computational aspects are discussed, with the assessment of DFT methods ( 2.1) in the evaluation of (a) ionization energies and electron affinities, and via eqn. (11), finite difference estimates of electronegativities and hardnesses ( 2.1.1) Mid (b) of dipole and quadrupole moments ( 2.1.2).In the final paragraph 2.2 a problem at the borderline between computational and conceptual DFT is tackled the evaluation of Fukui fimctions "beyond" the finite difference approximation. In 3 conceptual DFT is discussed, where in 3.1 attention is paid to the evaluation and/or use of DFT based concepts as such the shape factor and the local softness as Molecular Similarity indicators ( 3.1.1), and the nuclear Fukui function ( 3.1.2). In the final part of this Section ( 3.2) the role of DFT based concepts in various principles is discussed. The influence of solvent on the acidity of alkylalcohols is discussed within the framework of Sanderson s Electron ativity Equalization Principle [30] ( 3.2.1). The Hard and Soft Acids and Bases Principle and Pearson s Maximum Hardness Principle [31] are used as the guiding prindples in the study of the cycloaddition reactions of HNC to alkenes and aligmes. 

First, on the associate symbol, one could remark that the electronegativity index was historically assigned by X while the chemical hardness, which in above chemical bonding phenomenology follows the electronegativity equalization principle, should be identified by letter Y . Other literal argument was offered elsewhere (Putz, 2008b). 

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