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On the Method of the Determination of the Global Hardness of Atoms and Molecules INTRODUCTION The hardness is an important conceptual constructs of chemistry and physics. It has equal rank and status with other two very important conceptual constructs viz. atomic radius and the electro- negativity. The importance of the hypothetical constructs is self evident from the statement that, without the concept and operational significance of radius, hardness and electronegativity, chemistry and many aspects of condensed matter physics, become chaotic and the long established unique order in chemico-physical world would be dis- turbed. The notion of hardness was first introduced by Mulliken (1952) when he pointed out that the `Hard' and `Soft' behavior of various atoms, molecules and ions can be conceived during acid- base chemical interaction. Soon after Mulliken's classification, the terms "hardness" and "softness" were in the glossary of conceptual chemistry and implicitly signified the resistance towards the deformability of atoms, molecules and ions under small perturbation usually developed during the event of chemical reaction. Thereafter, Pearson (1963) and Klopman (1964) tried to systematize and rationalize this intrinsic property of atoms and molecules. Pearson (1963) qualitatively clas- sified molecules, atoms and ions in three classes, hard, soft and borderline- known as the HSAB principle and Klopman (1964) had drawn a link to Hard ­Soft behavior with the HOMO-LUMO gap of the frontier orbital theory. It is unequivocal that the hardness as conceived in chemistry fundamentally signifies the resistance towards the deformation or polarization of the electron cloud of the atoms, ions or molecules under small perturbation generated during the process of the chemical reaction. Thus, the general operational significance of the hard-soft behav- ior of a chemical species may be understood in the following statement. If the electron cloud is strongly held by the nucleus, the chemical species is `hard' but if the electron cloud is loosely held by the nucleus, the system is `soft'(Klopman, 1964; Pearson, 1963). The quest for the theoretical basis of the hard- ness and softness of atoms and molecules has created such a surge of fundamental research in chemistry that it gave birth of a new branch of density functional based theoretical science known as `Conceptual Density Functional Theory, CDFT' (Geerlings, Proft, & Langenaeker 2003). The conceptual density functional theory has added Maximum Hardness Principle, (MHP) (Pearson 1987, 1993; Parr et al., 1991) and Mini- mum Polarizability Principle, (MPP) (Chattaraj et al 1996) to the list of the fundamental laws of nature. The CDFT has been successfully exploited in elucidating and correlating mechanistic aspects viz. regio-selectivity, catalysis, aromaticity, intra- molecualr rotation, inversion and isomerization reactions (Zhou et al., 1989; Parr et al., 1991; Chattaraj et al. 1994; Pearson et al., 1992; Pal et al 1993, Chattaraj et al 1995, Ayers et al 2000, Ghosh et al 2000, 2002). It is important to mention here some outstand- ing fundamental works of Putz and his coworkers (Putz et al 2003,2005,2006,2007,2008,2009,20 10) on electronegativity and hardness and their usefulness for the theoretical prediction of several physicochemical properties-like the fundamentals of chemical bonding and aromaticity. It is shown that the aromaticity of peripheral topological path may be well described by superior finite differ- ence schemes of electronegativity and chemical hardness indices in certain calibrating conditions. THE PHYSICAL HARDNESS The materials engineers have used for centuries the legend "hardness" to describe mechanical stability (Gilman 1997). In solid mechanics, hard- ness means the resistance to deformation, both elastic and plastic. The particular properties are the bulk modulus which measures the resistance to elastic volume changes, the shear modulus which 251