Hardness, absolute

Sum of the squared atomic charge densities Sum of the absolute values of charges Absolute hardness... 

In this equation, r) the absolute hardness, is one-half the difference between /, the ionization potential, and A, the electron affinity. The softness, a, is the reciprocal of T]. Values of t) for some molecules and ions are given in Table 8.4. Note that the proton, which is involved in all Brdnsted acid-base reactions, is the hardest acid listed, with t — c (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 t) for an anion X is the same as that for the radical Other methods are also needed to apply the treatment to polyatomic... 

The value j a is called the absolute hardness and can be obtained as half of the difference between the ionization potential and the electron affinity [129],... 

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

N. Vaidya and B. S. Shah, Absolute Hardness of Phenanthrenedoped Anthracene Single Crystals by Knoop Indentation Technique, Indian J. Phys., 77A, 59 (2003). 

Parr and Pearson have introduced the absolute hardness as the second derivative of the energy EA with respect to the number of electrons NA at constant external potential [18]... 

In the molecular orbital theory, the absolute hardness is given by (88TL4843 89JA7371)... 

Figure 3 displays the results for a wet granulated product run at three speeds and three forces in production. The same speeds and similar forces were used on the Presster. Thirty tablets were produced at each speed. It can be seen that although the absolute hardness values differ by 1-2 kP in some cases (most likely due to differences in the two hardness testers used), the range of the window for the hardness values is comparable between the two machines. 

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 ... 

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. 

TABLE 8.3 Some absolute hardness values in electron volts 10... 

Absolute configuration of complexes. 495-496 Absolute electronegativity, 351 Absolute hardness. 351 Absorption spectra of lanthanide and actinide ions. 604-607 Acceptor number CAN). 370 Achiral molecules, and point groups, 64 Acids. 2... 

Moreover, the accuracy in calculating the hardness of a mineral from crystallochemical data, both in relation to Mohs numbers and Khrushchev hardness classes, follows from Table 4.2.2. The hardness classes proposed by Khrushchev are sometimes described as absolute hardness (Bolewski, 1963 Hurlbut and Klein, 1977), which in our view does not appear justifiable, since with the hardness determination results, the conditions in which the measurement has been made are not specified, and in interpreting the results, corrections consequent upon deformations of the indent caused by hardness anisotropy or by brittleness are not considered. 

Figures 7.6e and 7.6f give variation in the length of diagonals compared with hardness values on (010). Parallel to [001], hardness equals 1371 and parallel to [001], 1308. The absolute hardness minimum lies at 1252 in [001] + 10°. Two relatively high hardness maxima equal 1564 and 1556 in [100]+ 10° or [100]+ 30°.

The Principle of Hard and Soft Acids and Bases states that hard acids form more stable complexes with hard bases and soft bases form more stable complexes with soft acids. In orbital terms hard molecules have a large gap between the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO), and soft molecules have a small HOMO-LUMO gap. In recent years it has been possible to correlate the hardness with the electronic properties of the atoms involved. Thus, if the enthalpy of ionisation (I) and the electron affinity (A) are known the so-called absolute hardness (t ) and absolute electronegativity (%) can be found from r = (I - A) / 2 and % = (I + A) / 2. For example, the first and second ionisation enthalpies of sodium are 5.14 and 47.29 eV. Thus for Na+, I = + 47.29 and A = + 5.14, so r = (47.29 - 5.14) / 2 = 21.1. Similarly for silver the first and second ionisation enthalpies are 7.58 and 21.49eV, so for Ag+ we have, n = (21.49 - 7.58) 12 = 6.9. 

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. 

The next reactivity index in importance is the second energy derivative, the absolute hardness [3,24] ... 

The equations 7, 8 and 9 fail to operate when the HOMO - LUMO energy gap becomes too small and do not consider the influence on chemical properties of other orbitals, besides the HOMO and LUMO s. Moreover, it is not possible to study the site selectivity of a chemical species considering only absolute hardness other than space-dependent (local) versions of hardness/softness concepts [5]. Thus in addition to the global definition of r) and S, the local hardness [4] and local softness [5] have been introduced as follows ... 

Brel 0.9r[unrel for 4s and 5s ones. Here, in the framework of DFT, we are interested in rationalizing the relaxation effects on the absolute hardness values and the other reactivity indices, derived from the hardness tensor. 

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