Maximum hardness principle

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. 

Chattaraj, P. K. 1996. The maximum hardness principle An overview. Proc. Indian Natn. Sci. Acad., Part A 62 513-519. 

On the conceptual side, the powers of DFT have been shown to be considerable. Without going into detail, I mention only that the Coulson work referred to above anticipated in large part the formal manner in which DFT describes molecular changes, and that the ideas of electronegativity and hardness fell into place, as do Ralph Pearson s HSAB and Maximum Hardness Principles. 

The Maximum Hardness principle [41] further extends HSAB principle by stating that molecules try to arrange themselves to be as hard as possible . 

The performance of the method proposed above in the calculation of absolute hardness values of a set of neutral atoms and molecules is investigated. The Fukui indices and the polarization functions for the a-bonds of test molecules are also reported. Finally, the maximum hardness principle was checked by studying the "hardness profile" along the reaction path for the isomerization of HCN and 03H+ systems. 

The reported results show that the inclusion of the gradient corrected nonlocal effects is recommended to obtain data consistent with the maximum hardness principle. In fact, in the case of isomerization of HSiN the calculated hardness value for TS is higher than that of the minimum when local VWN potential is used. The introduction of the nonlocal corrections removes this error. The results for all... 

In this chapter, we review our latest results on the validity of the maximum hardness and minimum polarizability principles in nontotally symmetric vibrations. These nuclear displacements are particularly interesting because they keep the chemical and external potentials approximately constant, thus closely following the two conditions of Parr and Chattaraj required for the strict compliance with the maximum hardness principle. We show that, although these principles are obeyed by most nontotally symmetric vibrations, there are some nontotally symmetric displacements that refuse to comply with them. The underlying physical reasons for the failure of these two principles in these particular nuclear motions are analyzed. Finally, the application of this breakdown to detect the most aromatic center in polycyclic aromatic hydrocarbons is discussed. 

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 maximum hardness principle also demands that hardness will be minimum at the transition state. This has been found to be true for different processes including inversion of NH3 [147] and PH3 [148], intramolecular proton transfer [147], internal rotations [149], dissociation reactions for diatomics [150,151], and hydrogen-bonded complexes [152]. In all these processes, chemical potential remains either constant or passes through an extremum at the transition state. The maximum hardness principle has also been found to be true (a local maximum in hardness profile) for stable intermediate, which shows a local minimum on the potential energy surface [150]. The energy change in the dissociation reaction of diatomic molecules does not pass through a... 

The study of reaction paths, in DFT, is not a new 1114,115. Thus we have chosen to explore the potential energy surfaces (PES) introducing the possibility to rationalize the results through the computations of the global hardnesses along the whole reaction path, with the aim to verify if, for the studied processes, the maximum hardness principle (MHP) 53 is satisfied. 

In the present article we review the work on the hardness and related concepts done at the University of North Carolina at Chapel Hill. Section 2 introduces the global chemical hardness and softness. Several related local quantities arc described in Sect. 3. The Maximum Hardness Principle and the HSAB Principle are stated and proved in Sect. 4. Various applications of the hardness and related concepts in understanding chemical problems are described in Sect. 5. Finally, Sect. 6 contains a summary and some comments on the future. 

Before turning to specific uses of the various hardness and softness quantities, we state and outline the proofs of two important rules of nature, the Maximum Hardness Principle and the HSAB Principle. 

The maximum hardness principle requires the hardness of the equilibrium state to be maximum. In other words, it amounts to the following inequality ... 

Two formal proofs of the HSAB principle with a restriction of common chemical potential for the partners have been provided very recently [15]. The first proof makes use of the maximum hardness principle. The energy change (to first order) associated with the charge transfer process described above is given by... 

Perhaps little did Ralph Pearson realize when he proposed the hardness concept that it would encompass such a multitude of physico-chemical problems and that it would spawn so many new concepts. The Maximum Hardness Principle and the HSAB Principle, if they prove out, should be cornerstones for understanding molecular structure and molecular reactivity. The complex of ideas related to hardness and softness deserve extensive further application and careful further theoretical study. 

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