Absolute Hardness rj

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 ... 

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. 

The relationship between the electron structure and the cytotoxicity of imidazole derivatives was next investigated using the absolute hardness (rj) vs. absolute electron negativity (x) (Fig. 4) [16-18]. By applying this r/-/ activity... 

In conclusion, the present study demonstrates that the cytotoxic activity of coumarin derivatives show a strong linear relationship with the absolute hardness, rj. Hardness and softness, apart from the electron accepting and donating properties of the molecule, are important factors for estimation of their cytotoxic activity. From the rj value, the CC50 value of the novel coumarin compounds can be estimated. CONFLEX is useful for calculating the hardness and softness of the molecule using the PM3 method. 

FIGURE 6.16 Energy Levels for Halogens. Relationships between absolute electronegativity (x), absolute hardness (rj), and HOMO and LUMO energies for the halogens. 

Pearson appbed the concept of hard and soft acids and bases to nucleo-philicity in the 2 reaction by using the concept of absolute hardness, rj ... 

Berkowitz and Parr (1988) give a theoretical support to the absolute hardness, rj. The absolute hardness determines the resistance of species to lose electrons. According to frontier orbital method, the relationship between r] and the HOMO and LUMO energies is reduced to ... 

The relationship 5.20 is shown in Figure 5.13. It is difficult to assess the relative importance of model and experimental errors in these correlations. However, it is clear that the Ai7(Bl2)-Av(I—CN) correlation is less family dependent than the pAlBi2- i (I I) and p7fBicN-Av(I—CN) relationships shown in Figures 5.12 and 5.10. Hence the relationships 5.19 and 5.20 may support the use of Av(I—CN) as a spectroscopic scale of soft affinity. Indeed, diiodine is the archetype of soft Lewis acids in the Pearson classification since it has a very low absolute hardness (rj = 3.4 eV). Moreover, Av(I—CN) values obey the HSAB principle (soft acids prefer soft bases) since they decrease with the absolute hardness of the donor atom (in a given column of the periodic table), as shown in Table 5.28. 

Later on, Pearson tried to establish a quantitative scale of absolute hardness and softness.Absolute hardness is characterized by the value 17 and absolute electronegativity by the value %. The absolute hardness (rj) (is given by the expression r/ = (/ - A)/2, and the absolute softness (cr) by the expression (7 = 1/r/. Table 2.7.5. contains a selection of Pearson s absolute hardness parameters. 

In this equation tj, the absolute hardness, is half the difference between I, the ionization potential, and A, the electron affinity. The softness, cr, is the reciprocal of r. Values of ij for some molecules and ions are given in Table 8.3.1,1 Note that the proton, which is involved in all BrOnsted acid-base reactions, is the hardest acid listed, with t] = 00 (it has no ionization potential). The above equation cannot be applied to anions, because electron affinities cannot be measured for them. Instead, the assumption is made that rj for an anion X" is the same as that for the radical X. 112 Other methods are also needed to apply the treatment to polyatomic cations.112... 

A Antisymmetrizing operator A Vector potential P First hyperpolarizability P Resonance parameter in semi-empirical theory B Magnetic field (magnetic induction) X, /r, A, cr Basis functions (atomic orbitals), ab initio or semi-empirical methods rraiipp inrliiflinp basis fiinrHon 7] An infinitesimal scalar rj Absolute hardness h Planck s constant H hjl K h Core or other effective one-electron operator hap Matrix element of a one-electron operator in AO basis Matrix element of a one-electron operator in semi-empirical theory... 

We performed the QSAR analysis of the CC50 values of 5-trifluoromethyloxazole derivatives. Higher relationship coefficients were found in COSMO than in non-COSMO. This suggests that calculation should be performed under biomimetic conditions, although longer calculation time is required. The concept of absolute hardness is applicable to estimate the cytotoxicity of 5-trifluoromethyloxazoles using chemical descriptors 77,00, or the rj x activity diagram. 

Parr and I had originally called t] the absolute hardness. The reason was that it was a companion parameter to Xu (or —p), called the absolute electronegativity, because it had a sound basis in fundamental theory. While this made sense for ju, it seemed unnecessary for rj. The other scientific use of the term hardness would be for physical or mechanical hardness. Thus the name chemical hardness seems more appropriate for rj. 

Other theoretical tests of aromaticity have been studied. The graph theory has predicted thiepine to be antiaromatic <76JA2750>. A new measure of aromaticity has been studied on the basis of absolute hardness >]) known to be an index of stability and reactivity of a molecule. The operation formula is the equation, tj = (/-/l)/2 where / is the ionization potential and A the electron affinity. Alternatively, the equation is rj = (Fhomo- lumo)/2. Absolute hardness has also predicted thiepine (1) and 2-benzothiepine (5) to be antiaromatic and 3-benzothiepine (6) to be nonaromatic. However, the prediction by this approach fails for 1-benzothiepine (2) <89JA737l>. 

Table 3. Acidic hardness (r),) and basic hardness (rj ) indices as compared to absolute hardness, r. After Ref. L14 and 4]...

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. 

Since rj is defined in terms of frontier orbitals, the absolute hardness is considered to be a good reactivity criterion. The harder a molecule (large tj value). 

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