Fundamental Global Model for the Structures and Energetics of Nanocrystalline Ionic Solids

Abstract

This paper presents a general theory elucidating the relationships between the structures and cohesive energetics of alkali halide nanocrystals consisting of small sections of bulk rocksalt structures with m(1) and m(2) rows but infinite along the z axis. The theory introduces the electrostatic interactions between the ions treated as point charges and the short-range repulsions between the closest ion neighbors with the latter terms written in the Born form Ar(-)(n). Minimum energy structures are defined by the distances a(e) and b(e) separating the closest ions perpendicular and parallel to the z direction. The ratio a(e)/b(e), defining the crystal shape, is independent of the strength A of the short-range repulsion, greater than the bulk value of unity, and increases with decrease of the crystal cross section or n. This ratio tends toward unity in the hard sphere limit of infinite n. Both b(e)/R(6:6)(e) and a(e)/R(6:6)(e), with the bulk separation R(6:6)(e), are less than one, increase with increase of the crystal cross section or n, and are independent of A if this is independent of structure. The structural dependence of A increases its value with a decreasing crystal cross section rendering closer to unity the ratios a(e)/b(e), b(e)/R(6:6)(e), and a(e)/R(6:6)(e). Energy gains on relaxing the crystal toward equilibrium from its bulk separations decrease with increase of the crystal cross section or n, being about 60 kJ/mol for a one-dimensional chain with n = 6 but 0.5 kJ/mol for m(1) = m(2) = 4 with n = 12. The energy gained on relaxing to a structure with a(e)/b(e) constrained at unity is about 10 times greater than the further energy gains consequent on removing this constraint. The present theory neglecting the interaction between ions and the encapsulating nanotube explains the experimentally measured b(e)/R(6:6)(e) ratios. The observation that the a(e)/R(6:6)(e) values are greater than one shows that ion-wall interactions are important in determining the values of a(e).

Theoretical study of the structures and electronic properties of all-surface KI and CsI nanocrystals encapsulated in single walled carbon nanotubes

“…For nonencapsulated nanocrystalline ionic solids, a recently presented analytic theory 57 based on the Born model was shown to be capable of explaining both their structures and energetics on a global and unified basis. This approach is based on the assumption that the crystals are fully ionic and focuses only on the largest and most important terms, namely, the Coulombic interactions between the ions treated as point charges and the short-range repulsive interactions between immediately neighboring ions.…”

“…These variations in the predictions for a and b caused by changing the strength of the ion-carbon short-range repulsions V sXW 0 ͑r͒ can be understood by the following arguments based on the previous analytic model. 57 The equilibrium distance b in a nonencapsulated one dimensional single chain is less than that in the rocksalt structured bulk crystal because, on passing from the latter to the former, the attractive Madelung energy is only reduced in magnitude from 1.74756/ R to 2 ln 2 / R at constant R ͑1.74756 is the bulk Madelung constant͒, while the number of closest short-range cation-anion repulsions is reduced by a factor of 3. On passing from the one dimensional single chain to the nonencapsulated ͑2 ϫ 2 ϫϱ͒ nanocrystal, the number of closest short-range repulsions is doubled, while the additional binding resulting from the electrostatic interactions between two adjacent chains is only −0.116741/ R with the interaction between the two chains separated by the distance a ͱ 2 generating 57 a repulsion of 0.02857/ R. This explains why the equilibrium values of both a and b are larger than those in the one dimensional chain.…”

“…For a nanocrystal in which each of the ͑2 ϫ 2͒ planes is square, not inconsistent with the experimental data, the electrostatic potential energy experienced by an anion electron at distances r a less than the closest cation-anion separation is constant having the value −M T ͑x͒ / b equal to the Madelung potential determining the electrostatic contribution to the crystal cohesive energy. Here x = a / b with M T ͑x͒ being given by 57 M T ͑x͒ = 2 ln 2 + M b ͑x͒.…”

The noble gas dimers as a probe of the energetic contributions of dispersion and short-range electron correlation in weakly bound systems

“…The dipole-quadrupole dispersion χ and all the dipole-quadrupole terms subsequently derived [14,15] from the fundamental theory of [23]. Each dispersion damping parameter presented in Table 1 was calculated from the expression (4) provided by the analysis presented by Lassettre [26]. Here, I X , expressed in atomic units, is the ionization potential of the ground state of atom X, whilst ∆E X is the excitation energy (also in a.u.)…”

“…However the Hartree-Fock predictions can be refined both to eliminate this deficiency and to describe further correlation effects by techniques such as Møller-Plesset perturbation theory, the coupled cluster method or full configuration interaction. However, such post Hartree-Fock methods become so demanding of computational resources that they cease to be practical for systems much smaller than those containing around 50 or 100 atoms of interest in the modelling of materials or molecules or ionic crystals encapsulated in carbon nanotubes, topics of current interest [2][3][4]. This difficulty provided one of the main motivations for the development of density functional theory (DFT) which, although remaining in the first broad class of approach, is much less computationally demanding than the fully ab initio methods.…”