Lattice Madelung energy

For simple monovalent metals, the pseudopotential interaction between ion cores and electrons is weak, leading to a uniform density for the conduction electrons in the interior, as would obtain if there were no point ions, but rather a uniform positive background. The arrangement of ions is determined by the ion-electron and interionic forces, but the former have no effect if the electrons are uniformly distributed. As the interionic forces are mainly coulombic, it is not surprising that the alkali metals crystallize in a body-centered cubic lattice, which is the lattice with the smallest Madelung energy for a given density.46 Diffraction measurements... 

The experimental trends in bonding and structure which we have discussed in the previous chapter cannot be understood within a classical framework. None of the elements and only very few of the thousand or more binary AB compounds are ionic in the sense that the electrostatic Madelung energy controls their bonding. And even for ionic systems, it is a quantum mechanical concept that stops the lattice from collapsing under the resultant attractive electrostatic forces the strong repulsion that arises as the ion cores start to overlap is direct evidence that Pauli s exclusion principle is alive and well and hard at work ... 

Fig. 8.15 The Madelung energy in ionic compounds as a function of the radius ratio for CsCl, NaCl and cubic ZnS lattices (assuming the anion radius, / , is held constant).

In order for an ionic compound to dissolve, the Madelung energy or electrostatic attraction between the ions in the lattice must be overcome. In a solution in which the ions are separated by molecules of a solvent with a high dielectric constant ( H 0 81.7 ) the attractive force will be considerably less. The process of solution of an ionic compound in water may be considered by a Bom-Haber type of cycle. The overall enthalpy of the process is the sum of two terms, the enthalpy of dissociating the ions from the lattice (the lattice energy) and the enthalpy of introducing the dissociated ions into the solvent (the solvation energy) ... 

The neutral-ionic transition (NIT) at t = 0 occurs abruptly[94] when the Madelung energy M of the ionic lattice exceeds the energy I — A to transfer an electron form D to A. Long-range Coulomb interactions are treated self-consistently as part of A in the modified Hubbard model[95],... 

The Madelung constant a is defined as the ratio (a = EM/H) of the Madelung energy to the Coulomb attractive energy H between the nearest-neighbor anion and cation. Thus, for the monoatomic one-dimensional lattice of equidistant cations and anions of Eq. (8.10.3), a = 1.385294361. 

The absolute values of the Madelung energies Em for naphthalene TCNE and hexamethylbenzene p-chloranil are smaller than the cost of ionizing the lattice Iu Aa, so they are predicted by Eq. (12.2.8) to be complexes of almost neutral constituents, in agreement with experiment Em for TMPD TCNQ and TMPD chloranil are larger than fD — Aa, so they are predicted by Eq. (12.2.7) to be complexes of almost fully ionic constituents, in agreement with experiment these results are for mixed-stack crystals, where the D and A species are stacked atop each other, with large intermo-lecular overlap [13]. 

The estimation of lattice energies is based on the calculation of the coulombic (Madelung) energy, which comprises most of the lattice energy, to which an additional bonding energy due to metal-metal attractive interaction—for example, as in some rutile type oxides (26)— is added. The latter may be obtained empirically or by use of ligand field theory (11). 

The materials may be in a quasi-ionic phase when 1 > p > 0.5, or in a quasi-neutral phase when 0.5 > p > 0. In the simplest theoretical approach, the value of p depends on only three parameters D) the ionization potential of D Aa, the electron affinity of A and M, the electrostatic Madelung energy of the crystal lattice. A fully ionic lattice (p = 1) is then realized when /D - Aa > M, and a fully neutral lattice (p = 0) when /D — Aa < M. This result is, however, greatly obscured by the neglect of transfer integral t and of other relevant parameters [44]. 

Neutral DA charge transfer complexes D . A ( 0) have a lower melting point than those of ionic DA complexes (8 > 0.5), since the Madelung energy in the neutral DA makes a much smaller contribution to the lattice energy. For the... 

This energy is termed the Madelung energy since it represents a lattice energy internal to the molecule with a Madelung constant, of course, equal to 1.00. It is a maximum in a purely ionic bond (5 = Z" ") and decreases to whatever extent the charges on X and Y decrease. 

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