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)Link
opens in a new window and (Chen, Lu, Ortner 2018)Link
opens in a new window for finite temperature models.
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 and C. Ortner, Thermodynamic limit of crystal defects with finite temperature tight binding, Arch. Ration. Mech. Anal. 230 (2018) 701–733]. We discuss the non-trivial relationship between taking zero temperature and thermodynamic limits in the finite Fermi-temperature models.
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 nonlinear setting.
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.
We study the polynomial
approximation of symmetric multivariate functions and of multi-set
functions. Specifically, we consider $f(x_1,…,x_N)$, where $x_i \in \mathbb
R^d$, and $f$ is invariant under permutations of its $N$ arguments. We
demonstrate how these symmetries can be exploited to improve the cost versus
error ratio in a polynomial approximation of the function $f$, and in
particular study the dependence of that ratio on $d,N$ and the polynomial
degree. These results are then exploited to construct approximations and
prove approximation rates for functions defined on multi-sets where $N$
becomes a parameter of the input.
Jack Thomas, William J. Baldwin, Gábor Csányi, and Christoph Ortner. Self-consistent Coulomb interactions for machine learning interatomic potentials. arXiv e-prints 2406.10915. [arXivLink opens in a new window | abstract].
A ubiquitous approach to obtain transferable machine learning-based models of potential energy surfaces for atomistic systems is to decompose the total energy into a sum of local atom-centred contributions. However, in many systems non-negligible long-range electrostatic effects must be taken into account as well. We introduce a general mathematical framework to study how such long-range effects can be included in a way that (i) allows charge equilibration and (ii) retains the locality of the learnable atom-centred contributions to ensure transferability. Our results give partial explanations for the success of existing machine learned potentials that include equilibriation and provide perspectives how to design such schemes in a systematic way. To complement the rigorous theoretical results, we describe a practical scheme for fitting the energy and electron density of water clusters.
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 - Sept 2021: PhD in Mathematics and Statistics, University of
Warwick
Supervised by Christoph Ortner
Thesis: Analysis of an ab initio Potential Energy Landscape
Prize: Faculty Thesis Prize 2022 (joint winner)
Sept 2017 - Aug 2018: MSc in Mathematics and Statistics, University of
Warwick
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: two groups of Maths & Stats
students Modules covered: Sets & Numbers, Mathematical Analysis
(Terms 1&2) and Linear Algebra.
2019/20:
First Year Supervisor: three groups of Maths & Stats
students (as above)
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)
