Chemical hardness definition

Note that the factor 2 appearing in above equations is due to the chemical hardness definition bearing 1/2 in the custom definition (3.3) and can be skipped (as will be done below) when the chemical hardness will be considered as not-normalized to its acid-base behavior, see Eq. (4.158) of Section 4.2.3.3 as well as the discussion of Eq. (4.252) in Section 4.5. 

Then, while considering the finite-difference approximations to the electronegativity and chemical hardness definitions, in a Koopmans theorem environment (Koopmans, 1934) they take the woiking forms (Parr Yang, 1989 Chermette, 1999 Geerlings et al., 2003 Putz, 2012b) ... 

The last remark is nothing else than the confirmation of the fact that a more complex way of atomic radii involvement in chemical hardness definition, in the sense of atomic potential and of chemical action influences, may lead with better results. Following this line we may conclude the analysis of Lewis acids of Table 3.1 with the recommendation of the grouping DFE " as the best soft-to-hard ordering this is also the most complex computational approach with the most higher frequency of ordering appearance among the compared models. Moreover, the hard-hard... 

The formal definition of the electronic chemical hardness is that it is the derivative of the electronic chemical potential (i.e., the internal energy) with respect to the number of valence electrons (Atkins, 1991). The electronic chemical potential itself is the change in total energy of a molecule with a change of the number of valence electrons. Since the elastic moduli depend on valence electron densities, it might be expected that they would also depend on chemical hardness densities (energy/volume). This is indeed the case. 

One more step provides an operational definition. The HOMO level lies, I = ionization energy, below the vacuum level, while the LUMO level lies, A = electron affinity, below it. Thus, the chemical hardness lies midway in the gap and usually is given in units of eV. 

Two quantities derived from DFT are the electronic chemical potential /r and the chemical hardness 17 [2]. The definitions of these quantities are... 

Confusion can arise because, in this chapter, both the chemical potential and the hardness will be treated as variables dependent on position. The original articles introducing the chemical hardness gave a useful operational definition [7]. 

Although we will not discuss the finite difference approximations E1 (r) further in this section, it is useful to note that these functions can be defined also as potential derivatives of a gap and a chemical potential. The gap is the usual definition of the chemical hardness [8,9],... 

In the next section we shall recall the definitions of the chemical concepts relevant to this paper in the framework of DFT. In Section 3 we briefly review Strutinsky s averaging procedure and its formulation in the extended Kohn-Sham (EKS) scheme. The following section is devoted to the presentation and discussion of our results for the residual, shell-structure part of the ionization potential, electron affinity, electronegativity, and chemical hardness for the series of atoms from B to Ca. The last section will present some conclusions. 

The HSAB principle can be considered as a condensed statement of a very large amount of experimental information, but cannot be labelled a law, since a quantitative definition of the intuitive concepts of chemical hardness (T ) and softness (S) was lacking. This problem was solved when the hardness found an exact, and also an operational, definition in the framework of the Density Functional Theory (DFT) by Parr and co-workers [2], In this context, the hardness is defined as the second order derivative of energy with respect to the number of electrons and has the meaning of resistance to change in the number of electrons. The softness is the inverse of the hardness [3]. Moreover, these quantities are defined in their local version [4, 5] as response functions [6] and have found a wide application in the chemical reactivity theory [7],... 

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]... 

Nevertheless, it is a great advantage in science to have quantitative definitions so that one can measure what one is speaking about, and express it in numbers. Fortunately this is what has happened to chemical hardness. The means by which this has come about lies in density functional theory. This will be the topic of the next chapter. 

The quantity 77 is called the absolute, or chemical, hardness. From the method of finite differences we obtain the operational definition of 77 ... 

Although the meaning of the term hardness or softness does not necessarily identify the physical properties, using a thermodynamic approach this chemical hardness can be shown to be related to the compressibility factor, which in turn predicts mechanical hardness of minerals [153]. The definition of softness suggests that there would be correlation between softness and polarizability (a). Some workers [154]... 

According to this expression one can see that t, which has been identified as the chemical hardness (the factor of 1/2 in the original definition [7] of the global hardness has been omitted here for convenience), is equal to the derivative of the chemical potential with respect to the total number of electrons, when the external potential is held fixed. 

The necessity for a quantitative definition for chemical hardness was addressed by Parr and Pearson [6], who defined the hardness as... 

The purpose of this work is to start from the basic equations of density functional theory to describe the changes in the energy associated with the transition from one ground-state to another, in terms of different sets of variables. In this process one will find the natural definitions of the hardness and softness kernels, the local hardness, the local softness, the global hardness and the global softness [23]. Then, we will proceed to establish their relation with ionization potentials and electron affinities, in order to confirm their behavior as a measure of chemical hardness or softness [14, 24]. Finally, this theoretical framework will be used to analyze the maximum hardness and the HSAB principles. 

Table 8 The values of chemical power for the CIPAHs of Fig. 6 computed using definition (3) applied to the electronegativity and chemical hardness values from Tables 6 and 7...

However, this is not surprising because Ihe actual definition of electronegativity and chemical hardness reflects the holding power wifii which the whole atom attracts valence electrons to its center. There is therefore natural that as the atom is richer in core electrons down groups lesser is the attractive force on the outer electrons from the center of the atom. In this respect, the actual scales mirror at the best the atomic stability at the valence shell. 

The energetic discourse may be complete with the electronegativity and chemical hardness evaluations by applying the DFT definitions (3.1) and (3.3) to physical and chemical energies, respectively. In the first case, expression (4.147) is applied to provide the following so-called universal forms of (Parr Bartolotti, 1982) ... 

In the second term from the right of Eq. (4.176), it can be recognize the identity (4.175), and the prime term from the right of Eq. (4.176) is recognized as the chemical hardness associated to the electronic system, aecord-ing to the definitions from the Sections 3.2 and 4.2 ... 

Still, if the Parr-Pearson or chemical hardness is abstracted from Eq. (4.250) as the derivative of the Parr s electronegativity (3.1) (Parr Pearson, 1983), as in Eq. (3.3) presented, an operational definition of it from the method of finite differences ean be achieved ... 

We present the main results in Table 4.14 where the basic electronegativity and chemical hardness calculated by finite difference approximations in terms of IP and EA definitions are considered (Lackner Zweig, 1983). This is based on two positive arguments they are based on the Parr s DFT groimd state parabola method that is consistent with definition (4.234) of global electrophilicity electronegativity imder Mulliken... 

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