Hardness parameter

Waals radius rj for each atom type and a hardness parameter ej that determines the depth of the attractive well and how easy (or difficult) it is to push atoms close together. There are interactions for each nonbonded ij pair, including all pairs. The parameters for a pair are obtained from individual atom parameters as follows ... 

The default exp-6 van der Waals interaction requires, as shown in equation (37) and (38) on page 188, a van der Waals radii, r and a hardness parameter e. The default values for these parameters are based strictly on the atomic number and are given below. 

The hardness parameters for MM+ are the dissociation energies of the nonbonded interactions of two identical atoms divided by 1.125, 8=Dq/1.125. The values of Dq used are ... 

Fig. 8. Correlation between Pearson s hardness parameter (7P) derived from gas-phase enthalpies of formation of halide compounds of Lewis acids (19), and the hardness parameter in aqueous solution (/A), derived from formation constants of fluoride and hydroxide complexes in aqueous solution (17). The Lewis acids are segregated by charge into separate correlations for monopositive ( ), dipositive (O), and tripositive ( ) cations, with a single tetrapositive ion (Zr4+, ). The /P value for Tl3+ was not reported, but the point is included in parentheses to show the relative ionicity of Tl(III) to ligand bonds. 

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. 

It should be noticed that for the same held direction, the hardness as well as the CFF decreases (discussed above). A qualitative explanation for the above results is that the variation of the hardness parameter in the presence of external perturbation is actually dependent on the net effect exhibited by all the atoms present in the molecule [40]. 

Mohs was first to call attention to the significance of the hardness parameter for identification of minerals, and he provided a sufficiently precise and generally accessible tool for this purpose. The 10-degree, empirical scale of hardness he devised, Figs. 1.1 and 4.1.1 (10—diamond, 9—corun-... 

The hardness parameters specified in Table 8.1 apply to glazed bodies whose hardness is usually 0.16-0.25 degrees on the Mohs scale lower than that of unglazed material. The 7-8 hardness interval for porcelain materials given by many authors is incorrect. It is specific to materials ranked under special ceramics of steatitic or mullite-corundum type (Tables 4.4.4, 5.6, 5.7). 

Correlation of nucleophilic rate data for phenyldimethylsulfonium ions with common nucleophiles, with pX e values shows that the slopes of the lines, jS[ e, correlate qualitatively with the Edwards hardness parameter for the nucleophile and not with the Swain-Scott n parameter.144 cw,cw-2,4,6-Trimethyl-l,3,5-triaminocyclohexane is weakly basic in aqueous solution, because of steric inhibition to solvation of the conjugate acid.145 The three NH2 groups are axial and the steric effect also results in reduced reactivity as a nucleophile in, S n2 reactions. Highly stereoselective syntheses of N-. and O-glycosides have been carried out by addition of anionic nucleophiles to glycosyl iodides.146 5 n2 reactions are involved, but some substrates are susceptible to E2 elimination when treated with highly basic anions. 

These differences are reflected well by the mode hardness parameters. When a pair of (EDM, MEC) diagrams exhibits a similar topology, e.g., selected modes... 

As emphasized in Section IV of this chapter, the lithiotropy is of much consequence in the reactivity of enolates, the O and C sites competing toward electrophiles. This problem has been examined recently by Meneses and coworkers202, who described a local hardness parameter that can be used as a selectivity index, in particular for a set of ketone lithium enolates. 

This is akin to the hardness parameter 77 as introduced by Pearson[34,35], thus 77 = c in the notation we use here. 

In addition we give the corresponding hardness parameters [34] in units of [V/e] and a parameter 61, which permits the determination of electronegativities as a function of the charge on an atom, to be described below. These parameters are given here already as they were obtained directly in the recalculation of the electronegativity values. 

Electronegativity Values, %, in Pauling Units and Hardness Parameters, r) in [V/e]... 

The static first and second hyperpolarizabilities of thiazole and a number of heteroaromatic ring systems were calculated by the ab initio time-dependent Flartree-Fock (TDHF) method. The computed nonlinear polarizabilities correlate well with frontier orbital energies and hardness parameters <2003CPL(376)116, 1996JA12443>. 

The first hyperpolarizabilities (/3) of the donor-acceptor (D-A) systems containing several 1,3-heteroatom 7t-bridging units (oxazole, imidazole, and thiazole) have been studied by the ab initio method (HF/6-31G) <2004JMT(677)173>. The static first and second hyperpolarizabilities of amino- and nitro-substituted chromophores containing thiazole rings have been calculated by the ab initio TDHF method. The computed nonlinear polarizabilities correlate well with frontier orbital energies and hardness parameter (77) <2003CPL(376)116>. 

Compare the results of Problems 17 and 18 with the absolute hardness parameters of Appendix B-5 for BF3, NH3, and pyridine (C5H5N). What value of t would you predict for B(CH3)3 (Compare NH3 and N(CH3)3 as a guide.)... 

All of the hard parameters will need to be taken into account in the financial model of the scheme which will evaluate the cashflows, in terms of costs and revenues to calculate the financial renim on the investment in the project. 

Two vitally important hard parameters associated with any conversion process, which have a critical effect on project viability, are efficiency and availability. These are defined as follows ... 

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