Locality of interatomic forces in tight binding models for insulators

Abstract

The tight binding model is a minimalistic electronic structure model for predicting properties of materials and molecules. For insulators at zero Fermi-temperature we show that the potential energy surface of this model can be decomposed into exponentially localised site energy contributions, thus providing qualitatively sharp estimates on the interatomic interaction range which justifies a range of multi-scale models. For insulators at finite Fermitemperature we obtain locality estimates that are uniform in the zero-temperature limit. A particular feature of all our results is that they depend only weakly on the point spectrum. Numerical tests confirm our analytical results.This work extends and strengthens (Chen, Ortner 2016) and (Chen, Lu, Ortner 2018) for finite temperature models.

Locality of Interatomic Interactions in Self-Consistent Tight Binding Models

“…In Theorem 3, we restrict ourselves to the grand-canonical ensemble where there is a fixed chemical potential. By following the proofs of Ortner and Thomas (2020), one can also show analogous results for the canonical ensemble where the Fermi-level arises as a Lagrange multiplier for the particle number constraint.…”

“…In addition to the partial justification for IP and multi-scale models, our results also allow the thermodynamic and zero Fermi-temperature limit results of Ortner and Thomas (2020), Chen et al (2018) to be extended to the nonlinear setting. We sketch the main ideas in the concluding remarks in Sect.…”

“…We have formulated variational problems for (possibly infinite) atomic systems and extended the zero Fermi-temperature limit results of Ortner and Thomas (2020) to the nonlinear setting. We believe that the thermodynamic limit results of Ortner and Thomas (2020) can also be extended to nonlinear tight binding models.…”

“…Following (Chen and Ortner 2016;Chen et al 2018;Ortner and Thomas 2020;, we consider finite energy displacements. For u : Λ → R d , ∈ Λ and σ ∈ Λ − , we define the finite difference D σ u( ):=u( + σ ) − u( ) and the full finite difference stencil Du( ):=(D σ u( )) σ ∈Λ− .…”

“…We emphasise here that, in order to define these problems, we require the differentiability of the site energies and so can only define these problems locally around stable configurations. We may follow the proofs of Ortner and Thomas (2020), to extend the results to the case of nonlinear tight binding models. For example, we may show the following:…”