Atomic hardness parameter

Atomic hardness parameters can be derived for every hardness matrix if only the Fukui function indices dNj/dN are known, (cf. Sect. 3.2). 

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 ... 

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]. 

The resultant pair potentials for sodium, magnesium, and aluminium are illustrated in Fig. 6.9 using Ashcroft empty-core pseudopotentials. We see that all three metals are characterized by a repulsive hard-core contribution, Q>i(R) (short-dashed curve), an attractive nearest-neighbour contribution, 2( ) (long-dashed curve), and an oscillatory long-range contribution, 3(R) (dotted curve). The appropriate values of the inter-atomic potential parameters A , oc , k , and k are listed in Table 6.4. We observe that the total pair potentials reflect the characteristic behaviour of the more accurate ab initio pair potentials in Fig. 6.7 that were evaluated using non-local pseudopotentials. We should note, however, that the values taken for the Ashcroft empty-core radii for Na, Mg, and Al, namely Rc = 1.66, 1.39, and... 

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. 

An item about which there is a divergence in viewpoint at present concerns the interactions between non-identical atoms. If we use the Hill function as a vehicle for the discussion (not all workers use this function, but their functions can be recast in this form), there are just two variables to consider a distance parameter (r ) and a hardness parameter (e). Most workers take the sum of the radii of the individual atoms interacting as r, but this is not universally done. The parameter e was taken originally by Hill to be the product of the square roots of the e-values for the interacting atoms, which is what simple physical considerations seem to dictate (Hirschfelder et al.,... 

When the DF theory provided chemists with an unanimous and easy tractable definition of hardness, the older concepts of polarizability (I.l) or atomic dimensions (1.2) measuring hardness were put in the shade. However, the expertise gained by chemists in using polarizabilities, especially refractions, led to a formal unification of this measure of hardness with the new one, rigorously based on atomic energies. Refractions are the key to experimental hardness parameters for bonded atoms. 

The chemical approximation also explained the question of units for hardness, 1 V/e = 6.2418 X 10 F" , and opened a way to evaluation of atomic hardness from available chemical data of polarizability and atomic radii for free as well as for bonded atoms and ions t) = (47tCo) Rn - The versatile properties of hardness parameters derived from atomic refractions are shown in Table 2. Chemical approximation made possible the transformation of the... 

Hardness parameters for free atoms became an easy target on the basis of DF theory any formalism correctly reproducing first ionization energies and electron affinities could be used as a reliable source of q = 1/2(1 — A). A number of studies devoted to that topic presented hardness parameters with various degree of refinement [32, 33, 34]. Studies concentrated on bonded atoms were not as abundant and considerably less conclusive in describing properties of the atom in the molecule. 

Table 7. Charge Transfer AlTmity calculated by various approaches for selected pairs of atoms in diatomic molecules. Absolute electronegativity and hardness parameters as in Ref. [4]...

R, the sum of atomic radii plays important role in this formalism. As demonstrated in Sect 1.5, equivalence between the absolute atomic hardness and atomic radius is best for van der Waals or ionic radii. Thus, the R distance may be much larger than the normal bond in the AB molecule. For the purpose of this analysis, however, R/2 = Rab as a working approximation might be used, which is strictly valid for van der Waals radii and homonuclear diatomic molecules. Another choice is to follow the ehemieal approximation and use some available hardness parameters to estimate R as R = (rA + Tb) = (r A )-... 

The standard bond lengths from Sutton, L.E. Tables of Interatomic Distances and ConAguration in Molecules and Ions, The Chemical Society, London 1958, were assumed, together with the experimental neutral atom electronegativity and hardness parameters [11]... 

NAin the test configuration. Cx are the standard deviations for parameter type X Ij and Lj are the measured and target jth bond lengths, respectively. Similar definitions are given for bond angle roj and Qj, dihedrals 9j and roj, coordinates xj and Xj, and the distance djk between non-bonded atoms j and k with the atomic hard-sphere radii ri and ij, respectively (Hellinga Richards, 199 1). 

Concerning the calibration of electronegativities and hardnesses, we must refer to some known partial charge distributions and see what values should be used for the %s and atomic radius parameters to reproduce... 

The first term in Eqs. (7) and (8) increases steeply with diminishing distance r between the atom pair, reflecting the strong repulsion between the atoms at a short distance. In the Buckingham potential [Eq. (7)], the curvature of the potential in this region depends on the hardness parameter p. The second term in Eqs. (7) and (8) represents several types of long-range attraction between atoms, sometimes... 

In the same spirit, considering the transfer of an electron from an atom X to another identical atom X, the associated energy quantity is the energy difference, (7x - Ax), which should be a measure of the ease or difficulty of charge fluctuation between two identical atoms. This quantity, therefore, is interpreted as the measure of hardness/softness of an atomic species. The quantitative definition of the chemical hardness parameter q has thus been made fl = (7 - A)/2, which is interpreted " to represent (again, within a finite difference approximation) the second derivative of the energy quantity with respect to the electron number N, viz., q = (l/2)(3 W3N ). 

Equations 6.26 provide a set of linear equations for each of the atoms, a = 1,..., N, which are to be solved to obtain the atomic charges and atomic dipoles. However, one has to define the parameters involved. Here, the quantity J.o( is the same for all the atoms and is equal to the chemical potential of the cluster which is unknown and hence is to be determined during the solution process. One has N atomic charges, 3N atomic dipole components and the quantity a as the unknowns and there are N scalar equations, N vector equations and one equation = 0, for the charge conservation for the neutral cluster. Among the other parameters, (ta is an atomic chemical potential parameter, and q(a, a) = r ° is the atomic self-hardness term. Among the other quantities, r (a,P) is the mutual atom-atom hardness which can be approximated... 

---