Dr Jonathan Skelton Department of Chemistry, University of Manchester ([email protected]) Guided exploration of crystal potential-energy surfaces using harmonic lattice-dynamics calculations
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
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
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
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
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
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
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
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
jmskelton
gaelvaroquaux
yuyamaguchi
doradora09
ssii
dscs
nobushimi
langstat
yamasakih
icoxfog417
ishiijunpei
lynnandtonic
morganepeng
eileencodes
shpigford
orderedlist
quramy
3n
lara
philhawksworth
addyosmani
dougneiner
trallard