$30 off During Our Annual Pro Sale. View Details »

Guided exploration of crystal potential-energy surfaces using harmonic lattice-dynamics calculations

Guided exploration of crystal potential-energy surfaces using harmonic lattice-dynamics calculations

Presented at the Chemical Crystallography Group of the British Crystallographic Association Autumn Meeting on the 15th November 2023.

Jonathan Skelton

November 15, 2023
Tweet

More Decks by Jonathan Skelton

Other Decks in Science

Transcript

  1. Dr Jonathan Skelton
    Department of Chemistry, University of Manchester
    ([email protected])
    Guided exploration of crystal potential-energy surfaces
    using harmonic lattice-dynamics calculations

    View Slide

  2. Structure prediction
    Dr Jonathan Skelton
    “One of the continuing scandals in the
    physical sciences is that it remains in
    general impossible to predict the
    structure of even the simplest crystalline
    solids from a knowledge of their chemical
    composition.”
    J. Maddox writing in Nature (1988)
    M. C. Escher, “Horseman” (1946)
    J. Maddox, Nature 335, 201 (1988)
    CCG Autumn Meeting 2023, 15th Nov 2023 | Slide 2

    View Slide

  3. Structure prediction
    Dr Jonathan Skelton
    Reilly et al., Acta Cryst. B 72, 439 (2016)
    CCG Autumn Meeting 2023, 15th Nov 2023 | Slide 3

    View Slide

  4. An exponential problem...
    Dr Jonathan Skelton
    A simple mathematical argument can be made to show that the number of
    potential-energy minima is likely to rise exponentially with system size
    Stillinger, Phys. Rev. E 59 (1), 48 (1999)
    Dellago and Bolhuis in Advances in Polymer Science 221, pp. 167 (2009)
    CCG Autumn Meeting 2023, 15th Nov 2023 | Slide 4

    View Slide

  5. A continuum of approaches
    Dr Jonathan Skelton
    Random searching:
    o Random sampling (AIRSS)
    “Smart” sampling algorithms:
    o Simulated annealing
    o Evolutionary (genetic) algorithms
    o Particle-swarm optimisation
    o Basin hopping
    Some application of “chemical intuition”:
    o Data mining (e.g. MP Structure Predictor tool)
    o Modular assembly (e.g. MC-EMMA)
    Best unbiased sampling
    Lowest computational cost
    Pickard and Needs, J. Phys.: Condens. Matter 23, 053201 (2011)
    Oganov et al., Acc. Chem. Res. 44, 227 (2011)
    Wang et al., Comp. Phys. Commun. 183 (10), 2063 (2012)
    Hautier et al., Inorg. Chem. 50 (2), 656 (2011)
    Collins et al., Nature 546, 280 (2017)
    CCG Autumn Meeting 2023, 15th Nov 2023 | Slide 5

    View Slide

  6. Lattice dynamics 101
    Dr Jonathan Skelton
    We begin by considering the Taylor expansion of the crystal potential energy:
    𝜑 𝒖 = Φ0
    + ෍
    𝑘

    𝛼
    Φ𝑘
    𝛼𝑢𝑘
    𝛼 +
    1
    2

    𝑘,𝑘′

    𝛼,𝛽
    Φ
    𝑘,𝑘′
    𝛼𝛽 𝑢𝑘
    𝛼𝑢
    𝑘′
    𝛽 + 𝑂(Φ3
    ) + ⋯
    Lattice energy
    𝑈latt
    Atomic forces (vanish
    at equilibrium)
    Harmonic approx.
    The second-order force constants 𝚽𝑘,𝑘′ can be used to derive the phonon modes
    within the harmonic approximation by constructing and diagonalising the
    dynamical matrix 𝑫(𝒒):
    𝐷
    𝑘,𝑘′
    𝛼𝛽 𝒒 =
    1
    𝑚𝑘
    𝑚𝑘′

    𝑙′
    Φ
    𝑘0,𝑘′𝑙′
    𝛼𝛽 × exp 𝑖𝒒 ∙ 𝒓𝑘′𝑙′
    − 𝒓𝑘0
    CCG Autumn Meeting 2023, 15th Nov 2023 | Slide 6

    View Slide

  7. Lattice dynamics 101
    Dr Jonathan Skelton
    Kragskow et al., Chem. Soc. Rev. 52 (14), 4567 (2023)
    CCG Autumn Meeting 2023, 15th Nov 2023 | Slide 7

    View Slide

  8. Lattice dynamics 101
    Dr Jonathan Skelton CCG Autumn Meeting 2023, 15th Nov 2023 | Slide 8

    View Slide

  9. Lattice dynamics 101
    Dr Jonathan Skelton
    Skelton and Tang, J. Phys.: Condens. Matter 33 (16), 164002 (2021)
    CCG Autumn Meeting 2023, 15th Nov 2023 | Slide 9

    View Slide

  10. Phonons and phase transitions
    Dr Jonathan Skelton
    Skelton et al., Phys. Rev. Lett 117, 075502 (2016)
    Pnma
    Cmcm
    CCG Autumn Meeting 2023, 15th Nov 2023 | Slide 10

    View Slide

  11. Phonons and phase transitions
    Dr Jonathan Skelton
    U(Q)
    Q
    Real PES HA
    U(Q)
    Q
    Real PES HA
    𝑈 𝑄 =
    1
    2
    𝜇𝜔2𝑄2
    CCG Autumn Meeting 2023, 15th Nov 2023 | Slide 11

    View Slide

  12. Phonons and phase transitions
    Dr Jonathan Skelton
    Skelton et al., Phys. Rev. Lett 117, 075502 (2016)
    CCG Autumn Meeting 2023, 15th Nov 2023 | Slide 12

    View Slide

  13. Phonons and phase transitions
    Dr Jonathan Skelton
    Whalley et al., Phys. Rev. B 94, 220301(R) (2016)
    Cubic (𝑇 > 327 K)
    CCG Autumn Meeting 2023, 15th Nov 2023 | Slide 13

    View Slide

  14. Lattice dynamics 101
    Dr Jonathan Skelton
    Kragskow et al., Chem. Soc. Rev. 52 (14), 4567 (2023)
    CCG Autumn Meeting 2023, 15th Nov 2023 | Slide 14

    View Slide

  15. Phonons and phase transitions
    Dr Jonathan Skelton
    Skelton et al., Phys. Rev. Lett 117, 075502 (2016)
    Orthorhombic
    (𝑇 < 165 K)
    Tetragonal
    (𝑇 =165-327 K)
    Cubic
    (𝑇 > 327 K)
    CCG Autumn Meeting 2023, 15th Nov 2023 | Slide 15

    View Slide

  16. Phonons and phase transitions
    Dr Jonathan Skelton
    Skelton et al., Phys. Rev. Lett 117, 075502 (2016)
    CCG Autumn Meeting 2023, 15th Nov 2023 | Slide 16

    View Slide

  17. Back to structure prediction...
    Dr Jonathan Skelton
    The harmonic phonon modes of a crystal are (orthogonal) combinations of the
    atomic degrees of freedom with an associated energy, and which can describe
    the natural thermal motion of a crystal
    The presence of imaginary modes in the harmonic phonon spectrum identifies the
    structure as a maximum on the potential-energy surface
    The atomic displacements associated with the imaginary modes tell us how to
    distort the structure to lower the energy (i.e. to go “downhill”)
    If we know of (or can guess) a high-energy structure (usually high-temperature,
    often also high-symmetry), we can “follow” the imaginary modes to map out the
    PES and locate the connected minima
    If we start from a high-temperature structure and progressively “freeze in” the
    imaginary modes, we are likely to be mimicking how the crystal explores its PES
    during cooling
    CCG Autumn Meeting 2023, 15th Nov 2023 | Slide 17

    View Slide

  18. Example 1: Simple metals
    Dr Jonathan Skelton
    Togo and Tanaka., Phys. Rev. B 87, 184104 (2013)
    CCG Autumn Meeting 2023, 15th Nov 2023 | Slide 18

    View Slide

  19. Example 1: Simple metals
    Dr Jonathan Skelton
    Togo and Tanaka., Phys. Rev. B 87, 184104 (2013)
    CCG Autumn Meeting 2023, 15th Nov 2023 | Slide 19

    View Slide

  20. Example 1: Simple metals
    Dr Jonathan Skelton
    Togo and Tanaka., Phys. Rev. B 87, 184104 (2013)
    CCG Autumn Meeting 2023, 15th Nov 2023 | Slide 20

    View Slide

  21. Example 1: Simple metals
    Dr Jonathan Skelton
    Togo and Tanaka., Phys. Rev. B 87, 184104 (2013)
    Mg @ 0 GPa Mg @ 10 GPa
    CCG Autumn Meeting 2023, 15th Nov 2023 | Slide 21

    View Slide

  22. Example 2: Bi2
    Sn2
    O7
    Dr Jonathan Skelton
    Rahim et al., Chem. Sci. 11, 7904 (2020)
    CCG Autumn Meeting 2023, 15th Nov 2023 | Slide 22
    𝛾-Bi2
    Sn2
    O7
    (𝐹𝑑ത
    3𝑚, 𝑇 > 900 K)
    𝛽-Bi2
    Sn2
    O7
    (𝐴𝑏𝑎2, 𝑇 ≈ 400 K)
    𝛼-Bi2
    Sn2
    O7
    (𝑃𝑐 or 𝐶𝑐, RT)

    View Slide

  23. Dr Jonathan Skelton
    Lewis et al., J. Am. Chem. Soc. 138 (25), 8031 (2016)
    Example 2: Bi2
    Sn2
    O7
    CCG Autumn Meeting 2023, 15th Nov 2023 | Slide 23

    View Slide

  24. Dr Jonathan Skelton
    Lewis et al., J. Am. Chem. Soc. 138 (25), 8031 (2016)
    Example 2: Bi2
    Sn2
    O7
    CCG Autumn Meeting 2023, 15th Nov 2023 | Slide 24

    View Slide

  25. Dr Jonathan Skelton
    Rahim et al., Chem. Sci. 11, 7904 (2020)
    Example 2: Bi2
    Sn2
    O7
    CCG Autumn Meeting 2023, 15th Nov 2023 | Slide 25

    View Slide

  26. Dr Jonathan Skelton
    Rahim et al., Chem. Sci. 11, 7904 (2020)
    Example 2: Bi2
    Sn2
    O7
    CCG Autumn Meeting 2023, 15th Nov 2023 | Slide 26

    View Slide

  27. Dr Jonathan Skelton
    Rahim et al., Chem. Sci. 11, 7904 (2020)
    Example 2: Bi2
    Sn2
    O7
    CCG Autumn Meeting 2023, 15th Nov 2023 | Slide 27

    View Slide

  28. Dr Jonathan Skelton
    Rahim et al., Chem. Sci. 11, 7904 (2020)
    𝛾-Bi2
    Sn2O7
    𝛼-Bi2
    Sn2O7
    𝛽-Bi2
    Sn2O7
    𝜁-Bi2
    Sn2O7
    𝛿-Bi2
    Sn2O7
    𝜀-Bi2
    Sn2O7
    Example 2: Bi2
    Sn2
    O7
    CCG Autumn Meeting 2023, 15th Nov 2023 | Slide 28

    View Slide

  29. Dr Jonathan Skelton
    Morris et al., in prep.
    Example 3: FA-SNP
    CCG Autumn Meeting 2023, 15th Nov 2023 | Slide 29

    View Slide

  30. Dr Jonathan Skelton
    Example 3: FA-SNP
    Morris et al., in prep.
    CCG Autumn Meeting 2023, 15th Nov 2023 | Slide 30

    View Slide

  31. Summary
    Dr Jonathan Skelton
    Structure prediction is an inherently complex problem: the number of minima on
    the PES likely grows exponentially with system size
    The presence or absence of imaginary harmonic modes in the phonon spectrum
    can be used to identify a structure as a minimum or maximum on the PES
    In cases where a high-temperature phase with imaginary modes is known, or can
    be obtained e.g. as an “educated guess”, iteratively “following” the imaginary
    modes allows the PES to be mapped out to locate connected minima
    Where applicable (it isn’t always...!), this approach may have some advantages:
    o The 3𝑛𝑎
    degrees of freedom are effectively reduced a smaller number of
    imaginary modes
    o Off-Γ imaginary modes indicate when a distortion requires an expansion of
    the cell volume
    o Possibly mimics how the system explores the PES during cooling
    CCG Autumn Meeting 2023, 15th Nov 2023 | Slide 31

    View Slide

  32. Acknowledgements
    Dr Jonathan M. Skelton
    J. Morris
    CCG Autumn Meeting 2023, 15th Nov 2023 | Slide 32

    View Slide

  33. https://bit.ly/40DAzbn
    These slides are on Speaker Deck:

    View Slide