Geometry equilibration of crystalline defects in quantum and atomistic descriptions

Abstract

We develop a rigorous framework for modelling the geometry equilibration of crystalline defects. We formulate the equilibration of crystal defects as a variational problem on a discrete energy space and establish qualitatively sharp far-field decay estimates for the equilibrium configuration.This work extends [12] by admitting infinite-range interaction which in particular includes some quantum chemistry based interatomic interactions.2000 Mathematics Subject Classification. 65L20, 65L70, 70C20, 74G40, 74G65.

Adaptive QM/MM coupling for crystalline defects

“…A rigorous framework for modelling the geometry equilibration of crystalline defects has been developed in [6,11], which formulates the equilibration of crystal defects as a variational problem in a discrete energy space, and establishes qualitatively sharp far-field decay estimates for the corresponding equilibrium configuration. We emphasize that these results rely heavily on a "locality" assumption of the models, which has been shown for tight binding model in Lemma 2.2.…”

“…For sake of simplicity, we only present results on point defects here. All analysis and algorithms can be generated to straight dislocations (see [6,12]). Given d ∈ {1, 2, 3}, A ∈ R d×d non-singular, Λ hom := AZ d is the homogeneous reference lattice which represents a perfect single lattice crystal formed by identical atoms and possessing no defects.…”

“…The following result from [6] gives the decay estimates for the equilibrium state for point defects.…”

Locality of interatomic forces in tight binding models for insulators

“…This grand potential difference functional is well defined if µ ∈ σ(H(y)). This can be shown by applying results of [6] together with the locality estimates of this paper. We can then consider the meaningful problem: for fixed δ > 0, where "arg min" denotes the set of local minimisers.…”

“…All of the semi-norms D· 2 Υ for Υ > 0 are equivalent [6] and so we fix an exponent Υ > 0 for the remainder of this paper and define the following function space of finite energy displacements:…”