Parr-Pearson hardness

Having a viable EN quantum formulation, its natural companion named chemical hardness, /, can be immediately introduced (Parr Pearson, 1983)... 

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 ... 

Usually, the hardness is seen only as a companion to electronegativity, i.e., associated with the second order effects, as its Parr-Pearson basic (3.3) and (4.251) definitions reflect. This is also the case when the absolute actual picture is employed, a statement supported also by the close quantitative atomic rff and r p scales as the representations from the Figure 4.13 indicate. Instead, when the absolute approach is performed the situation regarding both qualitative and quantitative absolute hardness scale completely changes. Actually, the //J/ values predict a smooth increase of the hardness effects paralleling those of compare the Figures 4.13 and 4.12, respectively. Remarkably, the linked behavior of with is achieved even their basic definitions, Eqs. (4.254) and (4.256), respectively, are not directly related, as there are, for instance, the definitions of... 

Here R is the reactant P is the product Present work ab initio or Parr and Pearson Hardness data taken from (Pearson 1988)... 

However, the present (often called Parr-Pearson) approach does not provide a direct evaluation of molecular hardness from the atomic ones, even a formula can be laid down if assumed the same proportionality between electronegativity and hardness at both atomic and molecular levels ... 

However, one just notes that the assumptions made claims that all pairs electronegativity-hardness, either for atoms or molecules, are correlated by the same factor 9. Therefore, this treatment seems that doesn t properly correlate the average electronegativity X with the average of the chemical hardness t] without such universal proportionality. Further expansions of this Parr-Pearson model were made (Nalewajski, 1998), but the difficulties concerning the inherent correlation between X and T] still remain. 

However, EE-eneigy based relation pemiits the immediate recovery of the Hard and Soft Acids and Bases (HSAB) principle (Pearson, 1985 Parr Pearson, 1983 Klopman, 1968 Chandrakumar Pal, 2002a Chattaraj Maiti, 2003), an important conceptual principle to treat the molecular binding and reactive processes. Writing the energy variation at transfer under the following forms (Chattaraj Parr, 1993) ... 

The low stability of the complex 3.6 is consistent with the hard and soft acids and bases principle of Pearson (1963, 1968 Parr and Pearson, 1983 theoretical aspects Pearson, 1989 Chatteraj et al., 1991 monograph Ho, 1977). According to that principle hard acids will tend to complex with hard bases and soft acids with soft bases. Water is a hard base, whereas the nitrosyl ion is classified by Pearson as a borderline acid with a tendency to be soft. 

Parr RG, Pearson RG (1983) Absolute hardness companion parameter to absolute electronegativity. J Am Chem Soc 105(26) 7512-7516... 

An antecedent of the bond modulus is the chemical hardness of Pearson (1997) which measures the stabilities of molecules. Also, bond moduli are proportional to the physical hardnesses of Yang, Parr, and Uytterhoeven (1987) which they proposed for minerals. 

Equation 16.12 expresses a relation between q and B.This is not a universal relation, but it does apply to the sp-bonded elements of the first four columns of the Periodic Table. Using chemical hardness values given by Parr and Yang (1989), and atomic volumes from Kittel (1996), it has been shown that the bulk moduli of the Group I, II, III, and IV elements are proportional to the chemical hardness density (CH/atomic volume) (Gilman, 1997). The correlation lines pass nearly through the coordinate origins with correlation coefficients, r = 0.999. Thus physical hardness is proportional to chemical hardness (Pearson, 2004). 

It is shown that the stabilities of solids can be related to Parr s physical hardness parameter for solids, and that this is proportional to Pearson s chemical hardness parameter for molecules. For sp-bonded metals, the bulk moduli correlate with the chemical hardness density (CffD), and for covalently bonded crystals, the octahedral shear moduli correlate with CHD. By analogy with molecules, the chemical hardness is related to the gap in the spectrum of bonding energies. This is verified for the Group IV elements and the isoelec-tronic III-V compounds. Since polarization requires excitation of the valence electrons, polarizability is related to band-gaps, and thence to chemical hardness and elastic moduli. Another measure of stability is indentation hardness, and it is shown that this correlates linearly with reciprocal polarizability. Finally, it is shown that theoretical values of critical transformation pressures correlate linearly with indentation hardness numbers, so the latter are a good measure of phase stability. 

The next step is the identification of the concept of chemical hardness, 17, with the second derivative of the energy with respect to the number of electrons, formulated by Parr and Pearson [14]... 

Ayers, P. W., Parr, R. G., and Pearson, R. G. 2006. Elucidating the hard/soft acid/base principle A perspective based on half-reactions. J. Chem. Phys. 124 194107. 

This concept was introduced qualitatively in the late 1950s and early 1960s by Pearson, in the framework of his classification of Lewis acids and bases, leading to the introduction of the hard and soft acids and bases (HSAB) principle [19-21]. This principle states that hard acids prefer to bond to hard bases and soft acids to soft bases. In many contributions, the factor of 1/2 is omitted. The inverse of the hardness was introduced as the softness S=l/rj [22]. A third quantity, which can be expressed as a derivative with respect to the number of electrons is the Fukui function, was introduced by Parr and Yang [23,24] ... 

Pearson35,36 and Parr and co-workers366 c developed the principle of maximum hardness, which states that reacting molecules will arrange their electrons so as to be as hard as possible. Chemical equilibrium, then, is the state of maximum hardness. Soft donors prefer soft acceptors because both partners can increase their hardness by reacting with one another—the shared electrons flow to become less polarizable. To implement this theory quantitatively, Pearson et al. introduced scales of absolute hardness rj and its reciprocal, softness a ... 

Chemical hardness and softness are much newer ideas than electronegativity, and they were quantified only fairly recently. Parr and Pearson (1983) proposed to identify the curvature (i.e. the second derivative) of the E versus N graph (e.g. Fig. 7.10) with hardness, rj [151]. This accords with the qualitative idea of hardness as resistance to deformation, which itself accommodates the concept of a hard molecule as resisting polarization - not being readily deformed in an electric field if we choose to define hardness as the curvature of the E versus N graph, then... 

Parr and Pearson 1301 defined a parameter 17, which they called "absolute hardness" (17 V4[IP-EA]), and calculated 17 for a variety of neutral and ionic Lewis acids and bases possessing from one to four atoms. These authors showed that the qualitative predictions of the HSAB model regarding the relative reactivities of these species toward one another may be obtained using the results from simple calculations of stabilization energies using 17 and electronegativity values. 

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]... 

Another way of characterizing the readiness of molecules to gain or lose electrons upon interaction is based on the concepts of molecular electronegativity and hardness (Berkowitz and Parr, 1988 Parr et al., 1978 Pearson, 1986 1991). The starting point is the consideration that both the extent and ease of electronic deformation will affect the reactivity of a chemical compound (cf. Schiiiirmann, 1998a). The electronegativity (EN) characterizes the tendency of atoms and molecules... 

A parameter akin to electronegativity is hardness, introduced by Pearson and Parr in 1983 ... 

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