Christoph Ortner, Jack Thomas, and Huajie Chen. Locality of interatomic forces in tight binding models for insulators. ESAIM: Math. Model. Num. Anal., 54(6): 2295-2318 (2020). [doi | arXiv | abstract | poster]

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 Fermi-temperature 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.

Christoph Ortner and Jack Thomas. Point defects in tight binding models for insulators. Math. Model. Methods Appl. Sci., 30(14): 2753-2797 (2020). [doi | arXiv | abstract]

We consider atomistic geometry relaxation in the context of linear tight binding models for point defects. A limiting model as Fermi-temperature is sent to zero is formulated, and an exponential rate of convergence for the nuclei configuration is established. We also formulate the thermodynamic limit model at zero Fermi-temperature, extending the results of [H. Chen, J. Lu, C. Ortner. Arch. Ration. Mech. Anal., 2018]. We discuss the non-trivial relationship between taking zero temperature and thermodynamic limits in the finite Fermi-temperature models.

Jack Thomas. Locality of interatomic interactions in self-consistent tight binding models. J. Nonlinear Sci., 30(6): 3293-3319 (2020). [doi | arXiv | abstract]

A key starting assumption in many classical interatomic potential models for materials is a site energy decomposition of the potential energy surface into contributions that only depend on a small neighbourhood. Under a natural stability condition, we construct such a spatial decomposition for self-consistent tight binding models, extending recent results for linear tight binding models to the non-linear setting.

Jack Thomas, Huajie Chen, and Christoph Ortner. Rigorous body-order approximations of an electronic structure potential energy landscape. arXiv:2106.12572 (2021). [arXiv | abstract]

We show that the local density of states (LDOS) of a wide class of tight-binding models has a weak body-order expansion. Specifically, we prove that the resulting body-order expansion for analytic observables such as the electron density or the energy has an exponential rate of convergence both at finite Fermi-temperature as well as for insulators at zero Fermi-temperature. We discuss potential consequences of this observation for modelling the potential energy landscape, as well as for solving the electronic structure problem.

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 Fermi-temperature 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.

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Education:

Sept 2018 - present: PhD in Mathematics and Statistics, University of Warwick

Sept 2017 - Aug 2018: MSc in Mathematics and Statistics, University of Warwick

Sept 2013 - Jun 2017: MMath in Mathematics, University of Warwick

Past Projects:

Aug 2018. MSc thesis: Analysis of the Tight Binding Model (Supervised by Prof. Christoph Ortner)

July 2021. 16th U.S. National Congress on Computational Mechanics Contributed Talk in "Materials Modelling Across Scales: From First Principles Calculations to Mesoscale Physics": Sparsity of the Tight Binding Potential Energy Landscape

April 2021. British Applied Mathematics Colloquium (BAMC), University of Glasgow Contributed Talk: Tight Binding Models for Insulators: Locality of interatomic forces & geometry optimisation

June 2019. Visited Huajie Chen, Beijing Normal University, China.

May 2019. MASDOC CDT Summer Retreat, Borth, Wales. Contributed Talk: Relaxation of a Crystalline Defect in the Tight Binding Model

April 2019. British Applied Mathematics Colloquium, University of Bath. Contributed Talk: Zero Temperature Limit of the Tight Binding Model for Point Defects

March 2019. Solid Mechanics Working Group Meeting, University of Warwick. Seminar Talk: Zero Temperature Limit of the Tight Binding Model for Point Defects

First Year Supervisor: three groups of Maths & Stats students Modules covered: Sets & Numbers, Mathematical Analysis (Terms 1&2) and Linear Algebra.

This year I also helped out marking Mathematical Analysis (first year module for external maths students)

2018/19:

First Year Supervisor: one group of Discrete Mathematics students (as above)

Second Year Supervisor: two groups of Mathematics students Modules covered: Analysis III, Algebra I: Advanced Linear Algebra, Multivariable Calculus (Term 1) & Algebra II: Groups and Rings, Norms Metrics & Topologies (Term 2).

2017/18:

First Year Supervisor: one group of MORSE, Data Science and Maths & Stats students (as above)

This year I also helped out marking Mathematical Analysis (first year module for external maths students)

2016/17:

First Year Supervisior: one group of MORSE and Maths & Stats students (as above)