Linear Relationship between Weighted-Average Madelung Constants and Density Functional Theory Energies for MgO Nanotubes

Abstract

For systems containing large numbers of ions, calculations using Density Functional Theory (DFT) are often impractical because of the amount of time needed to perform the computations. In this paper, we show that weighted-average Madelung constants of MgO nanotubes correlate in an essentially perfectly linear way with cohesive energies determined by DFT. We discuss this correlation in terms of the relationship between lattice energies and cohesive energies. Through this linear correlation, Madelung constants are used to predict cohesive energies and average ion charges of nanostructures containing up to 3940 ions. Cohesive energies of MgO nanotubes are shown to converge to a value lower than those of bulk MgO. Using the slopes of the DFT versus Madelung constant plots, the average charges on the ions in the nanotubes are determined. For nanotubes containing the same number of ions, the relative stability of longer tubes versus disc-like structures is discussed.

Bonding in Sodium Chloride Nanotubes: A New Analysis via Madelung Constants and Cohesive Energies

“…In this paper we present DFT and MC(wa) determinations for NaCl nanotubes, which are used as an archetype to convey a new method for assessing partial ionic charges associated with charge transfer (CT) effects in alkali halides and related structures. This is achieved via the linear correlation between E coh and (MC(wa)) 11 (see Tables 1−3 and Figure 3). Recall that the cohesive energy is relative to the energy of the groundstate atoms (see eq 2 above) and will not scale linearly with the Coulombic attractive energies which are by definition relative to the energies of the gas-phase ions.…”

“…Optimized geometries are slightly distorted as we have reported previously for MgO nanotubes. 11 Examples are shown in Figure 2 for three L6 nanotubes. In each case the distortion is small, where the average bond length decreased from 2.8 Å (unoptimized geometry) to 2.7 Å (also see Table 2).…”

“…Ionic charges were determined from the slopes of the E coh vs MC(wa) plots as we have described for MgO nanotubes in a previous publication. 11 It is important to note that no specific charges or distances are assumed in the Madelung constant derivation. Although the charges and distances are for convenience set to unity in the computer code, they can be any value as ultimately they cancel because the dimensionless MC is a ratio of the average interionic Coulombic energy in the tube of interest divided by the Coulombic energy of a single ion pair having the same average charges and distances.…”

“…Coulombic attractive energies (E coul : eq 3) were computed using average near-neighbor distances and charges determined from the E coh vs MC(wa) plots. 11 This is described below (see also eq 3). The average coordination number (ACN, vide infra, discussion of Figure 4) was calculated by considering that in any nanotube the ions in the two end layers have a coordination number of 3, while all ions in interior layers have a coordination number of 4.…”

“…MC(wa) values were computed via Coulomb sum algorithms for each nanotube. , It is appropriate at this point to revisit this method since there can be confusion between terms. Madelung constants are strictly defined as single-ion quantities that are given by the sum of the Coulombic interactions (attractive and repulsive) of the ion in question with every other ion in the lattice.…”