Absolute hardness defined

The resistance to penetration defined in this sense is called by F. Auerbach the absolute hardness of a body, and he has designed an apparatus for measuring this property. 

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. 

Using density functional theory, the electronic chemical potential, fi and absolute hardness, r] have been introduced by Pearson [14]. A chemical system is characterized by its electronic chemical potential, fi, and by its absolute hardness, r. These are exactly defined... 

We define two parameters the absolute electronegativity, which is approximately the same as electronegativity as Mulliken originally defined it for atoms, namely the average of the ionisation potential I and the electron affinity A (Equation 3.1). The other is called the absolute hardness, rj, which is identified with the difference in energy between the ionisation potential I and the electron affinity A (Equation 3.2). 

The second approach to defining the absolute hardness rj has a companion parameter taken from density functional theory, called the electronic chemical potential p. The value of /i is essentially the same as the negative of X, as defined in Equation 3.1, and the value of 77 is essentially the same as in the more approximate definition in Equation 3.2. Tables 3.1-3.5 record some useful values for radicals, molecules and ions based on this definition. 

There are two major approaches to quantitative measures of acid-base reactions. One, developed by Pearson, uses the hard-soft terminology, and defines the absolute hardness, ll, as one-half the difference between the ionization energy and the electron affinity (both in eV) ... 

The second derivative of the energy with respect to the number of electrons is called absolute hardness t] (or chemical hardness) [Parr and Pearson, 1983], which for a molecule with Nd electrons is defined as ... 

From the frontier orbital energies an approximated absolute hardness is also obtained. In fact, under a finite difference approximation and a quadratic dependence of the energy on the number of electrons, absolute hardness is defined as... 

The third chapter, Quantitative Structure-Cytotoxicity Relationship of Bioactive Heterocycles by the Semi-empirical Molecular Orbital Method with the Concept of Absolute Hardness by Mariko Ishihara, Hiroshi Sakagami, Masami Kawase, and Noboru Motohashi, presents the relationship between the cytotoxicity (defined as 50% cytotoxic concentration) of heterocycles such as phenoxazine, 5-trifluoromethyloxazoles, O-heterocycles such as 3-formylchromone and coumarins, and vitamin K2 derivatives against some tumor cell lines and 15 chemical descriptors. The results suggest the importance of selecting the most appropriate descriptors for each cell type and compound. The review is of interest as it represents the relationship of the molecular structures with the cytotoxic activity of these heterocycles. 

R. G. Pearson has reported a scale of absolute hardness for acids and bases, defining hardness to be one-half the difference between the ionization potential and the electron affinity of an atom or species.21 He has obtained good agreement in many cases between the calculated values for hardness and the experimental behavior of the acid or base. 

Parr and Pearson introduced the concept [10-13] of the absolute hardness ( y) of a species defined as half the negative rate at which its electronegativity change with a change in its electron population at constant potential ... 

The absolute electronegativity is now defined by Equation 3.5, which is the negative of the slope of the E vs. N curve. This is a continuous function, which allows for nonintegral electron populations, a familiar concept in organic chemistry. The absolute hardness is then defined as the second integral of the same curve in Equation 3.6, which is therefore the curvature. 

The concept of topological resonance energy was introduced to provide a relative measure of aromaticity <77JA1692>. In a newer approach, the absolute hardness, or HOMO-LUMO gap, as the measure of aromaticity is defined to be in the ionization potential of the species minus its electron affinity. The parameters of the absolute hardness and a related relative hardness for (152) are consistent with the high reactivity of this compound <89JA737l>. 

The first density derivative of E, the chemical potential, p (electronegativity x) of the density functional theory is given by Eq. (7). The absolute hardness, q, is defined [13]... 

Another definition of aromaticity that does not depend explicitly on either experimental results or on comparison with reference compounds is derived from the concept of absolute hardness, 17, which is defined as one-half of the energy difference between the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) of a system (equation 4.70). According to Koopmans theorem, Ehomo is related to the ionization potential of the species, while Elumo is related to its electron affinity. A large gap between these two orbitals implies resistance to both oxidation and reduction, and low chemical reactivity is one of the defining characteristics of aromaticity. 

Zhou, Parr, and Garst showed that absolute hardness correlates well with theoretical measures of aromaticity but that the value of tj by itself does not allow the categorization of aromaticity. Zhou and Parr later defined the relative hardness of a species as the difference between its hardness and the hardness of an acyclic reference compound. Based on these correlations, the authors proposed that a compound is aromatic if its Hiickel absolute hardness (determined from the Hiickel HOMO-LUMO gap) is less than -O.ip, antiaromatic if the value of tj is less than -0.15j8, and nonaromatic if rj is between those two values. The corresponding division based on relative hardness is 0. That is, a cyclic molecule that is harder than an acyclic analog is aromatic, while one that is not as hard as an acyclic analog is antiaromatic. 

The absolute hardness is not to be confused with the Mulliken-Jaffe definition of electronegativity, given by Equation (14.25), where the electronegativity / is defined as the average of the IE and EA (for the valence electrons). The absolute hardness and... 

Mulliken-Jaffe electronegativity x) and absolute hardness ( /) for the halide ions, defined using the frontier molecular orbital model. 

In a first contribution, the relationship was studied between the absolute hardness, obtained from Huckel theory, and the resonance energy per jr-electron (REPE), defined as... 

It was not until 5 years later that Parr and Pearson defined absolute hardness q as the second derivative of energy with respect to the number of electrons... 

Lewis [5] was the first to describe acids and bases in terms of their electron accepting and electron donating properties. Mulliken [6] further refined the understanding of the acid base interactions for which he was awarded the Nobel Prize for Chemistry. His quantum mechanical approach introduced the concept of two contributions, an electrostatic and a covalent, to the total acid-base interaction. Pearson [7] introduced the concept of hard and soft acids and bases, the HSAB principle, based on the relative contributions from the covalent (soft) interaction and the electrostatic (hard) interaction. In his mathematical treatment he defined the absolute hardness of any acid or base in terms of its ionisation potential and electron affinity. Pearson s is probably the most robust approach, but the approaches in most common use are those developed by Gutmann [8] and Drago [9], who separately developed equations and methods to quantify the acid or basic strength of compounds, from which their heats of interaction could be calculated. 

Based on the research of Klopman, and Parr and Pearson (Klopman 1968 Parr and Pearson 1983), Pearson described the absolute hardness (ri) quantitatively as being proportional to the difference between I (ionization potential) and A (electron affinity) of the species (Pearson 1988). Absolute softness is defined as t). The absolute electronegativity (x) and the absolute hardness (Tj) are applied quantitatively to any given acid-base reaction. Table 3.10 presents x and Tj values for some representative metal ions. 

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