Madelung correction

Constant k is usually assigned a value of 1.75 to 2.00. Normally, the valence state ionization energies are corrected for charge effects and the calculations cycled to self-consistency (75). Madelung correction terms have also been included by some workers (109). 

Based upon the method of calculation adopted, a complete computer programme consisting of three main parts can easily be written for support of such calculations. The three parts are as follows (a) the CNDO part or EHMO part with Madelung correction for calculation of the localized electron orbitals in the anionic group (b) the transition matrix element calculation part and (c) the second-order susceptibility part for the calculation of the microscopic susceptibility of the anionic group followed by the calculation of the macroscopic SHG coefficients of the crystal. 

Here F denotes the Fock matrix, F ° the Madelung-corrected Fock matrix, P the density matrix, H is the core Hamiltonian matrix, Etot the total energy of the system and Vmn the nuclear repulsion term. The terms VMAD(P) and denote the... 

Madelung correction of the Fock matrix and of the nuclear repulsion, respectively. Due to the application of the ZDO approximation in MSINDO, V (P) is a diagonal matrix,... 

The principle that embodies this relationship can be stated as the products will be those in which the smaller ions will combine with oppositely charged ions of higher charge. In these cases, the Madelung constants may be different, so other factors may be involved. However, the principle correctly predicts reactions such as the following ... 

A finite array of charges is built taking into account the symmetry elements of the crystal. The charges of the outermost ions are adjusted in order to provide the correct value of the Madelung potential on each cluster site as well as the electrical neutrality... 

The covalent curve of Figure 3-5 has been drawn in the same way as for the HX curves. The ionic curve represents a Coulomb attractive potential and a repulsive potential b/R9f in which the constant b is given values which lead (with use of the Madelung constant and corresponding constant in the repulsive potential) to the correct interatomic distance in the corresponding crystal (Chap. 13). Polarization is neglected. 

It was noted in Chapter 4 that the Madelung constant of a structure may be expressed in various ways. The way that is conceptually simplest in terms of the Bom-Lande equation is the simple geometric factor. A, such that when combined with the true ionic charges, Z and Z, the correct electrostatic energy is formulated. It was noted that some workers have favored using another constant, A. combined with the highest common factor of Z and Z. Z -... 

The enthalpy of formation of an ionic compound can be calculated with an accuracy of a few percent by means of the Born-Land6 equation (Eq. 4.13) and the Bcrn-Haber cycle. Consider NaCI. for example. We have seen that by using the predicted internuclear distance of 283 pm (or the experimental value of 281.4 pm), the Madelung constant of 1.748, the Rorn exponent, n, and various constants, a value of — 755 kJ mol-1 could be calculated for the lattice energy. The heat capacity correction is 2.1 kJ mol-1, which yields U 9i = —757 kJ mol-1. The Bom-Haber summation is then... 

The theorem has the important implication that intramolecular interactions can be calculated by the methods of classical electrostatics if the electronic wave function (or charge distribution) is correctly known. The one instance where it can be applied immediately is in the calculation of cohesive energies in ionic crystals. Taking NaCl as an example, the assumed complete ionization that defines a (Na+Cl-) crystal, also defines the charge distribution and the correct cohesive energy is calculated directly by the Madelung procedure. 

The results of our band structure calculations for GaN crystals are based on the local-density approximation (LDA) treatment of electronic exchange and correlation [17-19] and on the augmented spherical wave (ASW) formalism [20] for the solution of the effective single-particle equations. For the calculations, the atomic sphere approximation (ASA) with a correction term is adopted. For valence electrons, we employ outermost s and p orbitals for each atom. The Madelung energy, which reflects the long-range electrostatic interactions in the system, is assumed to be restricted to a sum over monopoles. 

Table 1. The ionization energy I of the loosest bound orbitals of the cation M and anion X corrected for the charging effect S, and the values 7Calc derived from the monatomic ion and the full Madelung potential in the cubic crystals, all in eV...

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