Absolute hardness, table

FIGURE 4.13 The comparative trend of tire atomic finite difference chemical electronegativity Xj, and chemical hardness from Table 4.3 (upper) with respect to the atomic absolute electronegativity zT and and hardness from Table 4.8 (middle),... 

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... 

To an ice-cooled solution of the nitrile, in absolute alcohol (see Table II above) is added dry hydrogen chloride until 1.1 moles has been taken up. The resulting solution is allowed to stand at 0°C for the times shown in Table II, column 2. After this time, ether is added in the amounts shown in column 3 for the purpose of preventing the formation of a hard cake of the salt. In the case of very reactive nitriles such as acetonitrile and chloroacetonitrile it is advisable to have the ether present before the hydrogen chloride is added in order to prevent solidification of the reaction mixture. After allowing the reaction mixture to stand for 15-20 hr in a refrigerator, it is cooled to —30°C to hasten crystallization. The product salt is filtered, washed with cold (—40°C) ether, and dried... 

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

There is no hard boundary between the overlap dependent and independent regions. One can only ask at which point the overlap dependent energies become comparable to the overlap independent. If we take Hj as an example, the separation of the lowest singlet and triplet state depends primarily on overlap (in any model based upon atomic wavefunctions) whereas the absolute energies contain overlap independent terms. Table 2 shows these as a function of interatomic distance 104, IOS with the Coulomb (overlap independent) and exchange (overlap dependent) terms being given by... 

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. 

Table 1 shows a gradual improvement in the calculated a along the series vx A < vxc < vxc°P < vxc- The LDA leads, for these relatively hard systems, to a polarizability that is systematically too high, which is related to the excess energies of the occupied orbitals, see below. Note, that the GGA-BP gradient correction of the xc potential produces only a relatively small reduction in the LDA/ALDA average absolute error from 8.8% to 5.6%. The improved SAOP potential reduces the error substantially to 2.9%. Still, further significant improvement is achieved with the accurate xc potential the error from the combination (accurate vxc)/ALDA is only 1.0% (See Table 1). Therefore, the crucial improvement of the TDDFRT results for the molecules considered is achieved with just an alteration of the xc potential, while keeping ALDA for the xc kernel.

Tables 1 and 2 list the absolute hardness values for a number of neutral atoms and molecules.

Table 1. Theoretical and experimental absolute hardness values (in eV).

The values of absolute hardness, shown in the Tables, are highly satisfactory when compared with the known chemical behaviour of the selected systems. The expected chemical behaviour is well reproduced when the valence shell electrons in the absolute hardness calculation are taken into account. As previously mentioned [3], single atom chemistry is not so significant in terms of Lewis acid-base... 

Table 2 Mean absolute deviations (eV) from "experiment" for the electronegativities and hardnesses calculated at the B3LYP, B3PW91 and CCSD(T) level with the 6-311++G(3df,2p). 15 atoms and 7 cations were considered.

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. 

Confirm the absolute electronegativity and absolute hardness values for the following species, using data from Table 6-6 and Appendix B-5 ... 

TABLE 8.4. Some Absolute Hardness Values in Electron Volts ... 

For electron donors, a simple attempt at classifying hard and soft bases is illustrated Table 6.3. Such classification schemes reveal, not suipris-ingly, that hard and soft are not absolute, but gradually varying qualities. Pearson (1963) has proposed, in generalizing hardness and softness properties for Lewis acids and bases for many kinds of systems, the so-called HSAB (hard and soft acid-base) rules. 

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