Jack Thomas https://jackrthomas.github.io/notes/ quarto-1.10.18 Fri, 27 Mar 2026 05:00:00 GMT Orthogonal Polynomials & Jacobi Operators Jack Thomas https://jackrthomas.github.io/notes/posts/orthogonal-polynomials/
  • Mostly from Thomas et al. (2022) paper and the references
  • Also see slides

    • References

    • Bond-order potentials (BOP): Drautz (2020), FINNIS (2007), Horsfield et al. (1996), Pettifor (1989)
    • Numerical BOP: Haydock et al. (1972), Peiponen and Vaittinen (1982), Zheludeva et al. (1985), Luchini and Nex (1987)
    • Kernel polynomial method: MAIOLI et al. (1994),
    • Quadrature-based methods: Nex (1978), Tang et al. (2013), Nagler et al. (1984)
    • Maximal entropy method: Glanville et al. (1988), Mead and Papanicolaou (1984)

    (also see Thomas et al. (2022) (section 4.3 and Appendices D, E, F))

    TO-DO: understand:

    Orthogonal polynomials

    We have some measure and we want to consider sequences of orthogonal polynomials: (polynomials of degree ) such that and

    for all .

    We have a choice of normalisation:

    • Orthonormal polynomials: ,
    • Monic orthogonal: ,
    • e.g., ,
    • ….

    Example 1 (Classical orthogonal polynomials) Suppose :

    • Jacobi: on ,
    • Gegenbauer: Jacobi with ,
    • Legendre: on (Jacobi and Gegenbauer with ),
    • Chebyshev: Jabobi with (first kind), (second kind),
    • Hermite: on ,
    • Laguerre: on .

    Why?

    • The above polynomials appear all over the place,
    • Gauss quadradure (uses the zeros of orthogonal polynomials to approximate integrals),
    • Approximation theory,
    • …..

    Example 2 (Spectral measures) For bounded self-adjoint operator, there exists a unique spectral measure such that

    When is a matrix, with eigenpairs , then .

    When is e.g. a tight binding Hamiltonian, the corresponding spectral measure is a local density of states (LDOS). We can use orthogonal polynomials (and the corresponding Jacobi operator) to approximate the LDOS.

    Jacobi operators

    Question: How to generate the sequence of orthogonal polynomials?

    Answer: Lanczos-type recursion (it turns out that orthogonal polynomials satisfy a three-term recursion relation).

    Example 3 (Chebyshev polynomials) For example, we have Chebyshev polynomials of the first kind satisfy , , and

    (Chebyshev II have the same recursion but with .)

    For simplicity, let’s suppose that and generate the sequence of orthonormal polynomials. We first define and . For to be orthogonal to , we require and we choose so that . Then, supposing we have , then we define

    Lemma 1 This recursion generates the sequence of orthonormal polynomials.

    Proof. By the choice of we have and, by the choice of , we have . We assume that are orthonormal with respect to , and note that

    and

    for all .

    Finally, we conclude by noting that, for , we have

    Notice that, by the proof of Lemma 1, we also have the following Jacobi operator:

    It turns out that and so . In other words is the spectral measure of corresponding to the -entry.

    Remark. If is the spectral measure of some Hamiltonian , then we have transformed a complicated geometry to a simple semi-infinite linear chain (with complicated coefficients).

    We can approximate with the spectral measure corresponding to a truncated version of ….

    Measures, moments, OPs, and Jacobi operators

    BOP

    See Appendix E of Thomas et al. (2022).

    Gauss quadrature

    See Appendix D of Thomas et al. (2022) for proofs of the lemmas.

    Let be orthogonal polynomials with respect to and let be the set of zeros of . (One can show that where is the trucation of the Jacobi operator i.e. ).

    Gauss quadrature integrate the polynomial interpolation of on :

    where is the polynomial interpolation of on .

    Lemma 2 is a set of distinct points. There exists weights such that and .

    Lemma 3 Gaussian quadrature is exact for all polynomials of degree : for

    Proof. Write where . Notice that . Therefore, we have

    Lemma 3 leads to the following error bound for Gaussian quadrature:

    References

    Chin, Alex W., Ángel Rivas, Susana F. Huelga, and Martin B. Plenio. 2010. “Exact Mapping Between System-Reservoir Quantum Models and Semi-Infinite Discrete Chains Using Orthogonal Polynomials.” Journal of Mathematical Physics 51 (9). https://doi.org/10.1063/1.3490188.
    Drautz, Ralf. 2020. “From Electrons to Interatomic Potentials for Materials Simulations.” In Topology, Entanglement, and Strong Correlations, edited by Eva Pavarini and Erik Koch, vol. 10. Modeling and Simulation. Forschungszentrum Jülich GmbH.
    FINNIS, M. 2007. “Bond-Order Potentials Through the Ages.” Progress in Materials Science 52 (2-3): 133–53. https://doi.org/10.1016/j.pmatsci.2006.10.003.
    Glanville, S, A T Paxton, and M W Finnis. 1988. “A Comparison of Methods for Calculating Tight-Binding Bond Energies.” Journal of Physics F: Metal Physics 18 (4): 693–718. https://doi.org/10.1088/0305-4608/18/4/008.
    Haydock, R, V Heine, and M J Kelly. 1972. “Electronic Structure Based on the Local Atomic Environment for Tight-Binding Bands.” Journal of Physics C: Solid State Physics 5 (20): 2845–58. https://doi.org/10.1088/0022-3719/5/20/004.
    Horsfield, A. P., A. M. Bratkovsky, M. Fearn, D. G. Pettifor, and M. Aoki. 1996. “Bond-Order Potentials: Theory and Implementation.” Physical Review B 53 (19): 12694–712. https://doi.org/10.1103/physrevb.53.12694.
    Luchini, M U, and C M M Nex. 1987. “A New Procedure for Appending Terminators in the Recursion Method.” Journal of Physics C: Solid State Physics 20 (21): 3125–30. https://doi.org/10.1088/0022-3719/20/21/011.
    MAIOLI, CESARE, STEFANO SOLA, and FABIO VITALI. 1994. “EXTERNAL ANCHORING FOR WIDE-AREA NETWORK SUPPORT: THE RHYTHM PROJECT.” International Journal of Modern Physics C 05 (05): 769–83. https://doi.org/10.1142/s012918319400088x.
    Mead, Lawrence R., and N. Papanicolaou. 1984. “Maximum Entropy in the Problem of Moments.” Journal of Mathematical Physics 25 (8): 2404–17. https://doi.org/10.1063/1.526446.
    Nagler, S E, W J L Buyers, R L Armstrong, and R A Ritchie. 1984. “Static and Dynamic Spin Correlations in the Random One-Dimensional Antiferromagnetic CsCo<sub>x</Sub>mg<sub>1-x</Sub>cl<sub>3</Sub>.” Journal of Physics C: Solid State Physics 17 (27): 4819–35. https://doi.org/10.1088/0022-3719/17/27/012.
    Nex, C M M. 1978. “Estimation of Integrals with Respect to a Density of States.” Journal of Physics A: Mathematical and General 11 (4): 653–63. https://doi.org/10.1088/0305-4470/11/4/006.
    Peiponen, K E, and A Vaittinen. 1982. “Light-Induced Refractive Index Change in Some f Coloured Alkali Halide Crystals.” Journal of Physics C: Solid State Physics 15 (13): L415–18. https://doi.org/10.1088/0022-3719/15/13/007.
    Pettifor, D. 1989. “New Many-Body Potential for the Bond Order.” Physical Review Letters 63 (22): 2480–83. https://doi.org/10.1103/physrevlett.63.2480.
    Saberi, Hamed, Andreas Weichselbaum, and Jan von Delft. 2008. “Matrix-Product-State Comparison of the Numerical Renormalization Group and the Variational Formulation of the Density-Matrix Renormalization Group.” Physical Review B 78 (3). https://doi.org/10.1103/physrevb.78.035124.
    Tang, Shan, M. Steven Greene, and Wing Kam Liu. 2013. “Erratum on ‘Two-Scale Mechanism-Based Theory of Nonlinear Viscoelasticity’ [j. Mech. Phys. Solids 60(2) (2012) 199–226].” Journal of the Mechanics and Physics of Solids 61 (1): 91–93. https://doi.org/10.1016/j.jmps.2012.09.001.
    Thomas, Jack, Huajie Chen, and Christoph Ortner. 2022. “Body-Ordered Approximations of Atomic Properties.” Archive for Rational Mechanics and Analysis 246 (1): 1–60. https://doi.org/10.1007/s00205-022-01809-w.
    Woods, M. P., R. Groux, A. W. Chin, S. F. Huelga, and M. B. Plenio. 2014. “Mappings of Open Quantum Systems onto Chain Representations and Markovian Embeddings.” Journal of Mathematical Physics 55 (3). https://doi.org/10.1063/1.4866769.
    Zheludeva, S I, M V Kovalchuk, and V G Kohn. 1985. “The Photoelectric Voltage Excited by x-Ray Standing Waves in Semiconductors with a p-n Junction.” Journal of Physics C: Solid State Physics 18 (11): 2287–304. https://doi.org/10.1088/0022-3719/18/11/012.
    ]]>     Group Meeting Talk https://jackrthomas.github.io/notes/posts/orthogonal-polynomials/ Fri, 27 Mar 2026 05:00:00 GMT     Locality Jack Thomas https://jackrthomas.github.io/notes/posts/locality/
  • Based on Thomas et al. (2025)
  • notes

    • References mentioned:

    • Chen and Ortner (2016) - Combes-Thomas estimates
    • Chen et al. (2018) - thermodynamic limit
    • Ortner et al. (2020) - improved results for insulators
    • Thomas (2020) - self-consistent tight-binding models
    • Thomas et al. (2022) - body-ordered approximations
    • Xie et al. (2020) - “XPS”

    References

    Chen, Huajie, Jianfeng Lu, and Christoph Ortner. 2018. “Thermodynamic Limit of Crystal Defects with Finite Temperature Tight Binding.” Archive for Rational Mechanics and Analysis 230 (2): 701–33. https://doi.org/10.1007/s00205-018-1256-y.
    Chen, Huajie, and Christoph Ortner. 2016. “QM/MM Methods for Crystalline Defects. Part 1: Locality of the Tight Binding Model.” Multiscale Modeling &Amp; Simulation 14 (1): 232–64. https://doi.org/10.1137/15m1022628.
    Ortner, Christoph, Jack Thomas, and Huajie Chen. 2020. “Locality of Interatomic Forces in Tight Binding Models for Insulators.” ESAIM: Mathematical Modelling and Numerical Analysis 54 (6): 2295–318. https://doi.org/10.1051/m2an/2020020.
    Thomas, Jack. 2020. “Locality of Interatomic Interactions in Self-Consistent Tight Binding Models.” Journal of Nonlinear Science 30 (6): 3293–319. https://doi.org/10.1007/s00332-020-09651-8.
    Thomas, Jack, Will Baldwin, Gabor Csanyi, and Christoph Ortner. 2025. “Self-Consistent Coulomb Interactions for Machine Learning Interatomic Potentials.” Nonlinearity 38 (9): 095024. https://doi.org/10.1088/1361-6544/ae0402.
    Thomas, Jack, Huajie Chen, and Christoph Ortner. 2022. “Body-Ordered Approximations of Atomic Properties.” Archive for Rational Mechanics and Analysis 246 (1): 1–60. https://doi.org/10.1007/s00205-022-01809-w.
    Xie, Xiaowei, Kristin A. Persson, and David W. Small. 2020. “Incorporating Electronic Information into Machine Learning Potential Energy Surfaces via Approaching the Ground-State Electronic Energy as a Function of Atom-Based Electronic Populations.” Journal of Chemical Theory and Computation 16 (7): 4256–70. https://doi.org/10.1021/acs.jctc.0c00217.
    ]]>     Group Meeting Talk https://jackrthomas.github.io/notes/posts/locality/ Thu, 23 Oct 2025 05:00:00 GMT     Decay of interatomic force constants in the reduced Hartree-Fock model Jack Thomas https://jackrthomas.github.io/notes/posts/rHF/
  • Based on Cancès et al. (2025),
  • notes

    • References mentioned:

    • Lions et al. (2001)
    • Cancès et al. (2008)
    • Cancès and Lewin (2009)
    • Levitt (2020)

    References

    Cancès, Éric, Amélie Deleurence, and Mathieu Lewin. 2008. “A New Approach to the Modeling of Local Defects in Crystals: The Reduced Hartree-Fock Case.” Communications in Mathematical Physics 281 (1): 129–77. https://doi.org/10.1007/s00220-008-0481-x.
    Cancès, Éric, Antoine Levitt, and Jack Thomas. 2025. Decay of Interatomic Force Constants in the Reduced Hartree-Fock Model. https://arxiv.org/abs/2506.09595.
    Cancès, Éric, and Mathieu Lewin. 2009. “The Dielectric Permittivity of Crystals in the Reduced Hartree–Fock Approximation.” Archive for Rational Mechanics and Analysis 197 (1): 139–77. https://doi.org/10.1007/s00205-009-0275-0.
    Levitt, Antoine. 2020. “Screening in the Finite-Temperature Reduced Hartree–Fock Model.” Archive for Rational Mechanics and Analysis 238 (2): 901–27. https://doi.org/10.1007/s00205-020-01560-0.
    Lions, P.-L, I Catto, and C Le Bris. 2001. “On the Thermodynamic Limit for Hartree–Fock Type Models.” Annales de l’Institut Henri Poincaré C, Analyse Non Linéaire 18 (6): 687–760. https://doi.org/10.1016/s0294-1449(00)00059-7.
    ]]>     Group Meeting Talk https://jackrthomas.github.io/notes/posts/rHF/ Fri, 10 Oct 2025 05:00:00 GMT     Combes-Thomas estimate for banded matrices Jack Thomas https://jackrthomas.github.io/notes/posts/CT/ The following argument is contained in Benzi et al. (2013).

    Proposition 1 (Combes–Thomas estimate for banded matrices) Suppose is banded: for all . Then, for ,

    Proof. Since for all , we have

    for all .

    Therefore, for all , we have

    where .

    Example 1 Here, we let be the polynomial interpolation of on Chebyshev nodes and plot the error on for different choices of (this gives an upper bound on the locality of the resolvent as above):

    This is decay with
    1.0 0.880853
    0.5 0.479982
    0.25 0.24404
    0.125 0.116845
    0.0625 0.0505203

    i.e. it roughly halves when the distance to the “spectrum” is halved.

    References

    Benzi, Michele, Paola Boito, and Nader Razouk. 2013. “Decay Properties of Spectral Projectors with Applications to Electronic Structure.” SIAM Review 55 (1): 3–64. https://doi.org/10.1137/100814019.
    ]]>     Notes https://jackrthomas.github.io/notes/posts/CT/ Wed, 10 Sep 2025 05:00:00 GMT     Screening in the reduced Hartree-Fock model Jack Thomas https://jackrthomas.github.io/notes/posts/screening-rhf/

    Placing a point charge at the origin in a vacuum produces a long-range Coulomb potential , with being the distance to the charge and the dielectric permittivity of the vacuum. However, in a material, the material reorganises itself; electrons move towards positive charges (or away from negative charges) and the total potential (including the Coulomb interactions, and the response from the electrons) is screened. Metals and insulators display fundamentally different screening behaviours. In metals at finite temperature, electrons are mobile and are able to move long distances to fully screen the defect; a simple empirical model being given by the Yukawa potential . On the other hand, electrons in insulators are tightly bound to the nuclei and are thus unable to move too far from their periodic arrangement. A simple model for the total potential is given by for some ; one observes partial screening. This heuristic description has been rigorously proved for the reduced Hartree-Fock model at finite temperature (Levitt 2020), and for insulators at zero temperature (Cancès et al. 2008; Cancès and Lewin 2009). In metals at zero temperature, the sharp cut-off in the Fermi-Dirac distribution at the Fermi surface (the surface in reciprocal space separating occupied and unoccupied electron states) causes singularities in Fourier space. As a result, one expects to observe Friedel oscillations; the total potential decays with an algebraic rate with oscillatory tails. The rate of decay and frequency of oscillation depends on the Fermi surface, and thus on the metal, in a non-trivial way. This dependence is only known for the simplest of cases when the external potential is zero, and thus the Fermi surface is spherical (i.e. the Free electron gas). For real metals, the Fermi surface can significantly deviate from this spherical picture. We will discuss ongoing work in explaining this complicated screening behaviour.

    Joint work with Éric Cancès and Antoine Levitt.

    Given at:

    • February 2025 – Mathematical Challenges in Quantum Mechanics, GSSI, L’Aquila
    • December 2024 – Mathematical Physics and PDEs, Herrsching

    Mathematical Challenges in Quantum Mechanics, L’Aquila (February 2025)

    [open fullscreen]

    Mathematical Physics and PDEs, Herrsching (December 2024)

    [open fullscreen]

    References

    Cancès, Éric, Amélie Deleurence, and Mathieu Lewin. 2008. “A New Approach to the Modeling of Local Defects in Crystals: The Reduced Hartree-Fock Case.” Communications in Mathematical Physics 281 (1): 129–77. https://doi.org/10.1007/s00220-008-0481-x.
    Cancès, Éric, and Mathieu Lewin. 2009. “The Dielectric Permittivity of Crystals in the Reduced Hartree–Fock Approximation.” Archive for Rational Mechanics and Analysis 197 (1): 139–77. https://doi.org/10.1007/s00205-009-0275-0.
    Levitt, Antoine. 2020. “Screening in the Finite-Temperature Reduced Hartree–Fock Model.” Archive for Rational Mechanics and Analysis 238 (2): 901–27. https://doi.org/10.1007/s00205-020-01560-0.
    ]]>     Talk Conference https://jackrthomas.github.io/notes/posts/screening-rhf/ Sun, 01 Dec 2024 06:00:00 GMT     Friedel Oscillations Jack Thomas https://jackrthomas.github.io/notes/posts/friedel-oscillations/ Presented at:

    [open fullscreen]

    Abstract

    When a defect potential is placed in a material, the material rearranges and the total potential at long-range is screened by the electrons. In the finite temperature reduced Hartree-Fock model, small defects are completely screened [2]; the total change in potential decays exponentially. On the other hand, in metals at zero temperature, the presence of the Fermi-surface introduces non-analytic behaviour into the independent-particle susceptibility , leading to what are known as Friedel oscillations; the total potential oscillates and decays algebraically, with exponent depending on the dimensionality.

    Here, you can find links to the references that are cited in the poster:

    References

    1. Michael Greenblatt. Resolution of singularities, asymptotic expansions of integrals and related phenomena. Journal d’Analyse Mathematique 111.1 (2010), pp. 221–245. [doi] [arXiv]
    2. Antoine Levitt. Screening in the Finite-Temperature Reduced Hartree–Fock Model. Archive for Rational Mechanics and Analysis 238.2 (2020), pp. 901–927. [doi] [arXiv]
    3. Antoine Levitt and Jack Thomas. Locality in the reduced Hartree–Fock model. Unpublished manuscript.
    4. Bogdan Mihaila. Lindhard function of a d-dimensional Fermi gas (2011). [link]
    5. George E. Simion and Gabriele F. Giuliani. Friedel oscillations in a Fermi liquid. Physical Review B 72.4 (2005). [doi]
    6. Elias M. Stein. Harmonic Analysis. Princeton University Press, Dec. 1993. [link]
    7. Jens Lindhard. On the properties of a gas of charged particles. Dan. Mat. Fys. Medd. 28, no.8 (1954).
    8. Table of Integrals, Series, and Products. Elsevier, 2015. [link]

    Here are some additional details not on the poster, including a section on the results for the free electron gas:

    Introduction

    Suppose we have a lattice and an associated unit cell . Define . For a fixed periodic potential , associated Fermi level , consider the response to an effective potential : where is the Fermi-Dirac distribution with temperature and is the independent particle susceptibility operator, describing the linear response to the density of a non-interacting system of electrons.

    In the linear model the interaction between the electrons is neglected and we consider , whereas the reduced Hartree-Fock model takes the form

    Finite temperature

    In the finite temperature case (), the response decays “as quickly as” : That is, we have where . Moreover, the asymptotic behaviour of is given by the first term of the expansion: for all . One is therefore able to apply a fixed point argument to to conclude that small defects are completely screened: decays exponentially [2].

    Zero temperature

    At zero temperature, the Fermi surface leads to fundamentally different behaviour: all terms in the expansion oscillate and decay algebraically with rate depending on the Fermi surface. Here, we consider the behaviour of for the free electron gas. This behaviour more generally follows from the corresponding behaviour of the Green’s function (see poster).

    Free electron gas

    We may compute explicitly in Fourier space: The expression leads to [4] is the Heaviside function, and , , . In this case, is known as the Lindhard function [7]. In particular, is smooth away from points that connects two points of the Fermi surface where the tangent planes are parallel. For the free electron gas, this occurs on the sphere . For , the logarithmic divergence is responsible for the Peierls instability, whereas has inverse square root and logarithmic singularities in dimensions and , respectively.

    Lindhard function. Plot of the Lindhard function for (blue), (red), and (green).

    Computing the Fourier transform explicitly, one can show that oscillates and has decay: as . Here, is the sine integral and are Bessel functions of the first and second kind, respectively. Using the asymptotic behaviour of these special functions [8] we obtain as .

    ]]>     Poster Conference https://jackrthomas.github.io/notes/posts/friedel-oscillations/ Mon, 01 Jan 2024 06:00:00 GMT     Body-ordered Approximations of Atomic Properties Jack Thomas https://jackrthomas.github.io/notes/posts/body-ordered/ Given at:

    • September 2025 – Group meeting
    • October 2023 – Journée de rentrée de l’équipe, IHES

    Group meeting (September 2025)

    Equispaced nodes

    Chebyshev nodes

     plt3 = plot( 2:N, err[2:N], marker=:circle, yaxis=:log, title="absolute error: Chebyshev", legend=false)

    What about approximation on ?

    Leja points:

    plot( 1:N, err, marker=:circle, yaxis=:log, title="absolute error: Leja", legend=false)

    Journée de rentrée de l’équipe, IHES (October 2023)

    [open fullscreen]

    References

    Thomas, Jack, Huajie Chen, and Christoph Ortner. 2022. “Body-Ordered Approximations of Atomic Properties.” Archive for Rational Mechanics and Analysis 246 (1): 1–60. https://doi.org/10.1007/s00205-022-01809-w.
    ]]>     Talk Seminar Group Meeting https://jackrthomas.github.io/notes/posts/body-ordered/ Sun, 01 Oct 2023 05:00:00 GMT     Locality of Interatomic Interactions Jack Thomas https://jackrthomas.github.io/notes/posts/locality-interatomic/ Given at:

    • December 2025 – PDE Seminar, University of Minnesota
    • June 2023 – Problèmes Spectraux en Physique Mathématique, Institut Henri Poincaré

    PDE Seminar, University of Minnesota (December 2025)

    Computational methods, such as density functional theory, have been used successfully to model electronic structure and thus allowed the investigation and prediction of properties of molecules and materials. However, even now, the high computational cost of these methods severely limits their applicability in material modelling to thousands of atoms for static and hundreds of atoms for long-time dynamic simulations.

    On the other hand, in practice, one may bypass the electronic structure model entirely and replace it with a surrogate interatomic potential, a functional form which is designed to remain computationally tractable but also capture a minimal subset of desired properties of the system of interest. Empirical interatomic potentials are computationally inexpensive and can thus be used for large-scale, long-time dynamic simulations, for example. However, the simplicity of these parametric models limits their accuracy and transferability.

    A continuous increase in the complexity of parameterisations has naturally led to machine-learned interatomic potentials (MLIPs), where one learns the parameters in the model via a fitting procedure. In this talk, we will survey some recent results on the sparsity of the potential energy landscape aimed to justify and extend the theory of MLIPs.

    Joint work with Huajie Chen and Christoph Ortner.

    • Mostly based on Thomas et al. (2022)

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    Resolvent estimates: Proof by polynomial approximation

    The following argument is contained in Benzi et al. (2013):




    Suppose that is a nearest neighbour (NN) Hamiltonian:

    Since

    etc., we have for all .

    Therefore, we can bound the off-diagonal decay of functions of the Hamiltonian: e.g. if , then

    where .

    Example 1 Here, we let be the polynomial interpolation of on Chebyshev nodes and plot the error on for different choices of (this gives an upper bound on the locality of the resolvent as above):

    This is decay with
    1.0 0.880853
    0.5 0.479982
    0.25 0.24404
    0.125 0.116845
    0.0625 0.0505203

    i.e. it roughly halves when the distance to the “spectrum” is halved.

    Polynomials are body-ordered

    Approximating on

    This is with
    10.0 4.20986
    1.0 1.83189
    0.5 1.23431
    0.25 0.720808
    0.125 0.382578

    Leja points

    Using these points to approximate on , we get:

    Problèmes Spectraux en Physique Mathématique, IHP (June 2023)

    [open fullscreen]

    References

    Benzi, Michele, Paola Boito, and Nader Razouk. 2013. “Decay Properties of Spectral Projectors with Applications to Electronic Structure.” SIAM Review 55 (1): 3–64. https://doi.org/10.1137/100814019.
    Thomas, Jack, Huajie Chen, and Christoph Ortner. 2022. “Body-Ordered Approximations of Atomic Properties.” Archive for Rational Mechanics and Analysis 246 (1): 1–60. https://doi.org/10.1007/s00205-022-01809-w.
    ]]>     Talk Seminar https://jackrthomas.github.io/notes/posts/locality-interatomic/ Thu, 01 Jun 2023 05:00:00 GMT     Analysis of an ab initio Potential Energy Landscape Jack Thomas https://jackrthomas.github.io/notes/posts/phd-thesis/
  • PhD in Mathematics and Statistics, Mathematics Institute, University of Warwick (September 2021)
  • Supervised by Christoph Ortner
  • Faculty Thesis Prize 2022 (joint winner)
  • Permanent record: WRAP

    • A wide range of electronic, optical and magnetic properties of solids are determined by electronic structure. Computational methods, such as density functional theory, have been used successfully to model electronic structure and thus allowed the investigation and prediction of properties of materials. However, even now, the high computational cost of these ab initio methods severely limits their applicability in material modelling to thousands of atoms for static and hundreds of atoms for long-time dynamic simulations.

    In modelling tasks involving only the mechanics of the atoms, one may bypass the electronic structure model entirely and replace it with a surrogate interatomic potential (IP), a functional form that may be fitted to theoretical data from a high fidelity ab initio model. The aim of this thesis is to provide a rigorous basis for IP models by justifying some of the underlying assumptions, in particular with an eye towards physics-informed machine learning approaches.

    More specifically, we consider the tight binding model, a quantum mechanical model sharing many similarities with the more complex Kohn-Sham density functional theories, and prove that the potential energy landscape (PEL) can be decomposed into atom-centred site energy contributions depending only on atoms within a small cut-off radius of the central atom. Moreover, we obtain error estimates for body-ordered approximations, which allows one to reduce the dimensionality of the site energy contributions. In particular, we are able to decompose the PEL into low dimensional components which are easier to analyse, manipulate, and fit.

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    ]]>     Thesis https://jackrthomas.github.io/notes/posts/phd-thesis/ Wed, 01 Sep 2021 05:00:00 GMT     Locality of Interatomic Interactions in Tight Binding Models for Insulators Jack Thomas https://jackrthomas.github.io/notes/posts/locality-poster/ Presented at:

    Based on Ortner et al. (2020).

    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 and Ortner (2016) and Chen et al. (2018) for finite temperature models.

    [open fullscreen]

    References

    Chen, Huajie, Jianfeng Lu, and Christoph Ortner. 2018. “Thermodynamic Limit of Crystal Defects with Finite Temperature Tight Binding.” Archive for Rational Mechanics and Analysis 230 (2): 701–33. https://doi.org/10.1007/s00205-018-1256-y.
    Chen, Huajie, and Christoph Ortner. 2016. “QM/MM Methods for Crystalline Defects. Part 1: Locality of the Tight Binding Model.” Multiscale Modeling &Amp; Simulation 14 (1): 232–64. https://doi.org/10.1137/15m1022628.
    Ortner, Christoph, Jack Thomas, and Huajie Chen. 2020. “Locality of Interatomic Forces in Tight Binding Models for Insulators.” ESAIM: Mathematical Modelling and Numerical Analysis 54 (6): 2295–318. https://doi.org/10.1051/m2an/2020020.
    ]]>     Poster Conference https://jackrthomas.github.io/notes/posts/locality-poster/ Wed, 01 Jul 2020 05:00:00 GMT     Analysis of the Tight Binding Model Jack Thomas https://jackrthomas.github.io/notes/posts/msc-thesis/
  • MSc in Mathematics and Statistics (Summer Research Project), Mathematics and Statistics Doctoral Training Centre, University of Warwick (August 2018)
  • Supervised by Christoph Ortner

    • We apply the super-cell method to define Fermi levels on finite sub-lattices within the framework of the canonical ensemble and tight binding model. These Fermi levels of course depend on the size of the sub-lattice and the model, but also on temperature via the Fermi-Dirac distribution. We show that the zero temperature and thermodynamic limits of the Fermi levels exist and commute. We then perturb the system by considering more general configurations of the lattice and find that, in the case of metals, the aforementioned results still hold. In the more complex case of insulators, we note that the results break down due to the introduction of new eigenstates within the band gap.

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    ]]>     Thesis https://jackrthomas.github.io/notes/posts/msc-thesis/ Wed, 01 Aug 2018 05:00:00 GMT