Electrostatic Energy and the Madelung Potential

The calculation of the electrostatic energy is completely straightforward, in principle. For example, the electrostatic energy for the sodium chloride crystal is obtained in the following way. 

Sum the electrostatic interaction energy ( )e /rn for all other ions (indexed J), using a plus or minus depending upon whether the / th and jth ions are of the same or opposite sign. 

Divide by the number of ion pairs Np to obtain the electrostatic energy per ion pair, called the Madelung energy. 

It should be realized that taking the sum over all ions in a crystal is a nontrivial task, because the sum converges very slowly. This is a classic problem addressed early in this century by Madelung (Madelung, 1909) for a recent discussion, see Brown (1967, p. 93 ff). 

It is convenient to write the result of such a calculation in general form. Let us imagine equal numbers of ions of charge Ze and -Ze the energy for these will be proportional to Z. Since all interaction energies scale inversely with distance, and therefore with nearest-neighbor distance, d, it is convenient to write the total electrostatic energy of the crystal, divided by the number of ion pairs, in the form 

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The surface bandgap derives from bonding ionicity argtanents much in the same way as the bulk band gap. In the bulk lattice, the free space electron affinity and the ionization energy levels of the anionic (X) and cationic (K) constituents, respectively, are inverted by the Madelung potential, V, that is, by the electrostatic potential of all the ions of the lattice evaluated at the site of either the M or the X ion ... 

The anion-cation bond-breaking produced by the formation of a surface is responsible for several phenomena. Some have an electrostatic origin. This is the case for polarization which is induced by surface electric fields which are much larger than those in the bulk. This is also the case for shifts of the renormalized atomic energies of surface atoms and shifts of surface bands, which are induced by reduction of the Madelung potential at the surface. On the other hand, as far as covalent effects are concerned, a modification of the anion-cation hybridization takes place, due to the lower coordination of the surface atoms. This has important consequences on the gap width and on the electron distribution. 

The electrostatic (Madelung) part of the lattice energy (MAPLE) has been employed to define Madelung potentials of ions in crystals (Hoppe, 1975). MAPLE of an ionic solid is regarded as a sum of contributions of cations and anions the Madelung constant. A, of a crystal would then be the sum of partial Madelung constants of cation and anion subarrays. Thus,... 

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]. 

Finally, we list some interesting developments related to alternative forms of electrostatics calculations. Pask and Steme pointed out that real-space electrostatic calculations for periodic systems require no information from outside the central box. Rather, we only need the charge density within the box and the appropriate boundary conditions to obtain the electrostatic potential for the infinite system. These ideas were used earlier in an initial MG effort to compute Madelung constants in crystals. ° So long as charge balance exists inside the box, the computed potential is stable and yields an accurate total electrostatic energy. Thus, questions about conditional summation of the 1/r potential to obtain the physical electrostatic energy are unnecessary. This has also been noted in the context of Ewald methods for the simulation of liquids (See Chapter 5 in Ref. 227). 

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