Evolutionary Crystal Structure Predictionhan.ess.sunysb.edu/~aoganov/...StructurePrediction.pdf ·...

Evolutionary Crystal Structure Prediction

Artem R. Oganov (ARO)

(1) Department of Geosciences(2) Department of Physics and Astronomy(3) New York Center for Computational Sciences

State University of New York, Stony Brook, NY 11794-2100

(from http://nobelprize.org)

Structure is the basis for understanding materials and their properties

Zincblende ZnS.One of the first structures solved by Braggs in 1913.

Structure Diffraction

With time, incredibly complex structures were discovered

Host-guest elements(Rb-IV phase, U.Schwarz’99)

QuasicrystalsNew state of matter discovered in lab in 1984. In nature found only in 2009!

Proteins

When the structure is known, many properties can be computed reliably

• State of art: DFT.• Molar volumes: within 1-3% (LDA,GGA).• Transition pressures: 10% underestimated (LDA), 5 GPa (GGA). • Band gaps: ~30% underestimated (LDA, GGA), 10% (GW).• Unsatisfactory – for van der Waals crystals, systems with localised d-

and f-electrons.

VolumesLDA: Mujica‘03

Transition pressuresLDA: Mujica‘03

We simulate materials using density functional theory (DFT)

• Hohenberg & Kohn (1964), Kohn & Sham (1965): exact theory, recipe for approximate practical calculations. Set of coupled one-electron equations:

• Approximate exchange-correlation functionals (LDA, GGA):

• “Difficult” wavefunctions: rapid variation close to the nuclei, slow variation between the atoms.

E.Schrödinger

W. Kohn

• PAW: accurate and extremely efficient method (Blöchl’94; Kresse’99).

Experimentalists also like DFT

When the structure is known, many properties can be computed reliably

• State of art: DFT.• Molar volumes: within 1-3% (LDA,GGA).• Transition pressures: 10% underestimated (LDA), 5 GPa (GGA). • Band gaps: ~30% underestimated (LDA, GGA), 10% (GW).• Unsatisfactory – for van der Waals crystals, systems with localised d-

and f-electrons.

VolumesLDA: Mujica‘03

Transition pressuresLDA: Mujica‘03

(Free) energy landscape: key to thermodynamics and kinetics

Transition paths, rate constants etc. Global minimum & low-energy local minima

EH=E+PVF=E-TSG=E+PV-TS

Need to find GLOBAL energy minimum.Trying all structures is impossible:

1031 yrs.1039301017 yrs.102520103 yrs.1011101 sec.11CPU timeVariantsNatoms

Overview of USPEX(Oganov & Glass, J.Chem.Phys. 2006)

J. Maddox(Nature, 1988)

New developments in crystal structure prediction extend the range of problems that can be solved

3. Predicting new materials

1. Predicting crystal structures by evolution

2. Why does it work?

1. Predicting structures by evolution

Oganov A.R., Lyakhov A.O., Valle M. (2011). How evolutionary crystal structure prediction works - and why. Acc. Chem. Res. 44, 227-237.

Evolutionary simulations learn & explore the most promising regions of search space

USPEX (Universal Structure Predictor: Evolutionary Xtallography)

• (Random) initial population• Evaluate structures by relaxed (free) energy• Select lowest-energy structures as parents for new

generation• New developments – ARO & Valle (2009), Chua et al.

(2010), Lyakhov & ARO (2010).

(1) Heredity

(2) Lattice mutation (4) Permutation(3) Softmode mutation

Selection of “good” fragments• Use local order parameter (ARO& Valle, 2009; Lyakhov et al., 2010)

Ordered regions – redDefective regions – blue/green

Single defects in 256-atom fcc cell

Selection of “good” fragments• Use local order parameter (ARO& Valle, 2009; Lyakhov et al., 2010).• Can deal efficiently with very large systems (sometimes, up to 500-

1000 atoms).


Single defects in 256-atom fcc cell



Crystalline islands in amorphous matrix - 256-atoms/cell



Crystalline islands in amorphous matrix - 512-atoms/cell

Graphite, correctly predicted to be the stable phase at 1 atm

Test1: „Who would guess that graphite is the stableallotrope of carbon at ordinary pressure?“ (Maddox, 1988)

Metastable sp2 forms, harder than diamond?First proposed by R.Hoffmann (1983)

Low-energy structures reveal chemistry sp-hybridisation

(carbyne)sp2-hybridisation sp3-hybridisation

[ARO & Glass, J.Chem.Phys. (2006)]

Test1: High-pressure phases of carbon are also successfully reproduced

2000 GPa: bc8 phase, potentiallyimportant in astrophysics

100 GPa: diamond is stable

Metastable bc8 form of SiIs known (Kasper, 1964)

[ARO & Glass, J.Chem.Phys. (2006)]

+found metastable form that matches „superhard graphite“ of W.Mao(Li, ARO, Ma, et al., PRL 2009)

Test 2: USPEX is self-learning, self-improving

MgSiO3, 40 atoms/cell

40 atoms/cell. Ground state not found among 120,000 random structures, but takes 600-950 structures with USPEX

MgSiO3, 80 atoms/cell

80 atoms/cell. Evolutionary runs take only ~3200 structures to find the ground state

[ARO & Glass, J. Phys. C (2008)]

Test 2: USPEX vs random sampling (random sampling - Freeman’1993, Schmidt’1996)

Random structures, all locally optimisedDid NOT find PPV after 120000 steps

Search with USPEXFound PPV after 600-950 steps

Test case: 40-atom cell of MgSiO3 with fixed lattice parameters of post-perovskite

[Martonak, ARO & Glass, Phase Transitions (2007)]

Test 2: USPEX is self-learning, self-improving

Best structure obtained after 120000 steps Is not PPV

Search with USPEXFound PPV after 600-950 steps

Test case: 40-atom cell of MgSiO3 with fixed lattice parameters of post-perovskite

[Martonak, ARO & Glass, Phase Transitions (2007)]

Applications of this method proved its great utlilty:

Random sampling (Freeman & Catlow, 1993; Schmidt et al., 1996; van Eijck & Kroon, 2000;

Pickard & Needs, 2006)• No „learning“. Works well only for small problems (<30 degrees of freedom – e.g.

10 atoms).

Minima hopping (Gödecker 2004)• Keep history of visited minima. Escape minima with MD, using feedback to

control temperature

Metadynamics (Martonak, Laio, Parrinello 2003)•Taboo search with reduced dimensionality.

Simulated annealing (Pannetier 1990; Schön & Jansen 1996)• Random walk. Ever decreasing probability to accept step to worse solution• Difficult to control parameters.• No „learning“ - only current position as source of information!

Alternative methods:

Genetic and evolutionary algorithms• Bush (1995), Woodley (1999) – works only for small systems, inefficient.• Deaven & Ho (1995) – developed only for clusters. Efficient.• ARO& Glass (2006), Abraham (2006), Fadda (2010), Wang (2010), Lunie (2011)

Blind test (2010): USPEX is superior to random sampling and simulated annealing

• Blind test (ARO, Schon, Hennig, 2010) – on extremely difficult cases:

4071-978# of structure relaxations before ground state

-70.37--68.82Minimum energy, eV

1 (1)-1 (1)Number of runs (runs producing lowest E)

Test #4, Mg13Al8P3 with variable cell (ab initio)

461068513029# of structure relaxations before ground state

-1950.53-1949.10-1943.46Minimum energy, eV

1 (1)9 (1)1 (1)Number of runs (runs producing lowest E)

Test #3, Mg10Al4Ge2Si8O36 with variable cell (with forcefields)




Test #2, Ba2Mg2Al8Si8O32 with fixed cubic cell (with forcefields)




Test #1, BaMgAl4Si4O16 with fixed cubic cell (with forcefields)

USPEXSim. annealingRandom sampling

Random sampling failed to give lowest-enthalpy structures for 2 phases (out of 3 predicted) of SiH4 (Pickard, PRL 2006), 1 for Nitrogen (Pickard PRL 2009), 1 for SnH4 (Pickard, 2010)

Benchmarking the power of the methoda b

cde

Test #2 (Ba2Mg2Al8Si8O32, with fixed cell): (a) Variation of the lowest energy during the evolutionary USPEX run, (b) Summary of simulated annealing runs, (c-e) Lowest-energy structures obtained by random sampling, simulated annealing and USPEX, respectively. Thin horizontal line in (a) shows the lowest energy found in 14102 random sampling attempts.

For more information…

2. Why does it work?

ARO & Valle, J.Chem.Phys. 130, 104504 (2009)

(Free) energy landscape: key to thermodynamics and kinetics

Transition paths, rate constants etc. Global minimum & low-energy local minima

EH=E+PVF=E-TSG=E+PV-TS

Crystal structure prediction: What are we facing?

• Find lowest free energy structure from chemical composition

• High-dimensional problem. Dimensionality d = 3N + 3.

• Very sensitive to small changes

• Thus: HUGE and ‘noisy’ search space

• Don‘t have to search the whole configuration space

• Global minimum surrounded by many very good local minima

• Can assume some overall shape to which we can tune an approach

• Can easily calculate the energies from first principles

Example of a (very) simple landscape

After local optimization intrinsic dimensionality of landscape is reduced:

d* = 3N + 3 – κd*=10.9 (d=39) for Au8Pd4, d*=11.6 (d=99) for Mg16O16, d*=32.5 (d=39) for Mg4N4H4.

Complexity C ~ exp(βd*)

Fingerprint theory is the basis of our analysisFingerprint function is a 1D-descriptor of the structure:

diffraction spectrum, PCF, ...

Difference between 2 structures is given by „distance“, e.g.:

[ARO & Valle, J. Chem. Phys. 130, 104504 (2009)]

Pedagogical cartoonReal system (GaAs): correlation of energy and the distance from the ground-state structure

Fingerprint is related to the pair correlation function

Feature of evolution: emergence of order from chaos

Increase of order during evolutionary simulation of GaAs


Fingerprinting allows to monitor diversity and emergence of order from chaos in simulations

Similarity matrix:Measure of structural diversity Symmetric, values in [0;1]Colour coding: 0-blue, 1-red.

Diversity, measured by collectivequasi-entropy Scoll, should not decrease too fast


Statistical confirmation of Pauling‘s 5th rule: „The number of essential structural elements of stable structures tends to be small“

Some of the (many) remarkable silicate frameworks:

Correlation plot for 6900 structures of SiO2 with 24 atoms/cell

Increasing energy

Grouping structures into similarity classes: quest for more insight in complex systems

Distance-preserving mapping of crystal structures of H2O(darker – lowest E, lighter – higher E).

DNA grouping in Europe


From USPEX-61.957 eV-61.960 eV

Visualizing energy landscapes (and seeing energy funnels!)Au8Pd4 - simple L4J8 - complex

-99.12 -99.05Cluster expansion Binary Lennard-Jones crystal (RL:RJ=1:2)


Visualizing energy landscapes (and seeing energy funnels!)


Energy landscape of MgO

Karplus‘s „tree“ representation of the landscapes

38-atom Lennard-Jones cluster(Wales, 2010)

SrO (Schoen & Jansen, 2010)

bct4-carbon

M-carbon

diamond

Finding actual transition mechanism and barriers in solids is an extremely complex task!

graphite

graphite

graphite

The problem is toaccount for nucleation[Boulfelfel, 2007]

[Boulfelfel, 2011]

3. Predicting new materials

-Matter under pressure-Materials with target properties

Units: 100 GPa = 1 Mbar = 200x

Matter under pressure: new phenomena and ubiquity in nature

P.W. Bridgman1946 Nobel Prize for Physics

Sodium is an alkali metal, at normal conditions well described by the nearly free electron model

Sodium shows unexpectedly complex behavior under pressure

H. DavyH. Davy

1807: Discovered by Sir Humphrey Davy.

2002: Hanfland, Syassen, et al. – firstevidence of remarkably complex structures above 1 Mbar. Further evidence – Gregoryanz et al. (2008).

2005: Gregoryanz et al. find melting curve minimum at ~1.2 MbarDoes sodium become a d-metal?

Na

Theory predicts a new structure that is insulating and …transparent!

Sodium becomes transparent at ~200 GPa(Ma, Eremets, ARO et al., Nature 2009)

Localized interestitial electron pairs make Na insulating. Structure – close packing of interstitial electron pairs!

Sodium becomes transparent at ~200 GPa(Ma, Eremets, ARO et al., Nature 2009)

Localized interestitial electron pairs make Na insulating. Structure – close packing of interstitial electron pairs!

Theory predicts a new structure that is insulating and …transparent!

Why do electrons concentrate in the interstitial space?

corecore

core core

e

ee

e

e

corecore

corecore

Pressure

Similar model was first proposed for Li by Neaton & Ashcroft (1999),see also Rousseau & Ashcroft (2008)

Dense transparent sodium can be represented as an „electride“, a compound made of ionic cores and strongly localized interstitial electrons.

Can expect new phenomena for such materials.

s-electrons

p,d-electrons

CaLi2: another illustration of the importance of core electrons

• Feng (2007), Debessai (2008), Tse (2009) gave mutually inconsistent results.

• Our study (Xie et al., 2010) reconciled theory and experiment and found unique new structures with Li-Li pairing.

Located between metals and non-metals, boron is a frustrated element. As a compromise, it forms complex and unusual crystal structures.

B

alpha-B beta-B T-192

Boron is the most complex element. Even its discovery was full of troubles

[ARO & Solozhenko, J. Superhard Mat. 2009]

J.L. GayJ.L. Gay--LussacLussac H. DavyH. Davy

H. Moissan

F. Wöhler

1808: J.L.Gay-Lussac and H.Davyreport discovery of a new chemical element - boron.

1895:1895: H. Moissan proves that the discovered substance was not H. Moissan proves that the discovered substance was not elemental boron and contained at most 60% of boron. elemental boron and contained at most 60% of boron. MoissanMoissan’’s material was later found to be less than 90% boron. s material was later found to be less than 90% boron.

1858:1858: F. F. WWööhlerhler concludesconcludes that boron exists inthat boron exists in two forms two forms --"diamond"diamond--like" and "graphitelike" and "graphite--like"; both were later found to be like"; both were later found to be compounds compounds -- AlBAlB1212 and Band B4848CC22Al, respectively. Al, respectively.

2007:2007: ~16 crystalline forms of boron are known (most believed ~16 crystalline forms of boron are known (most believed to be compounds!). Stable form is still unknown.to be compounds!). Stable form is still unknown.

B

Boron forms a charge-transfer phase under pressure

2004: Chen & Solozhenko: synthesised new phase, structure could not be solved.

2006: ARO: found the structure, demonstrated that it is stable.

2008: Solozhenko, Kurakevych & ARO – its hardness is 50 GPa.

B

Structure of ionic boron: (B2)δ+(B12)δ- (ARO et al., Nature 2009)δ=+0.5 (Bader partitioning), +2.2 (Born dynamical charges).

The concept of electron-deficiency was devised to explain boron chemistry. ”it is the ideas that are deficient, not the electrons” (J.Burdett).

Red - theory, black - experiment

BCharge transfer is clear from the electronic structure

Unlike other forms of boron, γ-B does not metallize even at very high pressure

Infrared spectra are entirely due to large dynamical charges on atoms

DOS

Phase diagram of boron (ARO et al., Nature 2009)

BFirst phase diagram of boron – after 200 years

Superconducting -Ga-type phaseis purely theoretical and has yet to be synthesized

γ-B28 was probably observed in a largely discarded 1965 paper by R. Wentorf

How many phases of boron? B

CH4

•Uranus and Neptune: H2O:CH4:NH3 = 59:33:8.•Neptune has internal heat source (Hubbard’99).•Ross’81 (and Benedetti’99):

CH4=C(diamond) + 2H2. Sinking of diamond – main source of heat in Neptune?•Theory (Ancilotto’97; Gao‘2010) confirms this.

Planet Neptune has an internal source of heat. What is it?

[Gao, ARO et al., J. Chem. Phys. 133, 144508 (2010) ]

diamond

methane

hydrocarbons

Looking for the densest possible material: carbon allotrope(s) denser than diamond

Diamond has the highest bulk modulus and lowest atomic volume among all elements (and compounds)From Brazhkin (2009).

diamond structure

new structure, 3.7% denserthan diamond!(Zhu, ARO, et al., PRB 2011)

Superdense carbon allotropes [Zhu, ARO, et al., 2011]

(1) sp3 hybridization(2) superhard(3) huge refractive indices (up to 2.8!)(4) strong dispersion of light(6) tP12 is the widest-gap form of

carbon (7.0-7.3 eV)

Special Issue “Theory of superhard materials” (editor – A.R. Oganov)

Theory of hardness? Yes!

kj

ki

kj

ki

k CNCNX

3330.831.8SiO2 stishovite

2123.423.4β-Si3N4

8-1012.312.4TiO2 rutile

0.140.1757.4Graphite

9089.791.2Diamond

Exp.Lyakhov& ARO(2011)

Model of Liet al. (2009)

Material

Lyakhov & ARO (2011) – augmented model of Li (2009) by bond valence model and graph theory.

Is diamond the hardest structure for carbon? Yes [Lyakhov & ARO, 2011].

Simulation for carbon, 16 atoms/cell

0.11181.3P6522

0.22482.0Cmcm

0.32282.2Fmmm

0.78482.9I212121

0.16683.4P2/m

0.28283.5Cmcm

0.19884.0I4/mmm

0.16384.3C2/m

0.02689.1Lonsdaleite

0.00089.7Diamond

Enthalpy,eV/atom

Knoophardness,

GPaStructure

All of the hardest structures are sp3-hybridized

Looking for the hardest materials… What is the hardest oxide?

• Leger (Nature 1996) – SiO2 stishovite (33 GPa). • Dubrovinsky (Nature 2001) – TiO2-cotunnite (38 GPa).• He (Appl.Phys.Lett. 2002) – B6O (45 GPa).

Suggestion of TiO2 is clearly incorrect – (1) it is unstable at 1 atm (!), (2) Experiments of Dubrovinsky were low-quality

(bulk modulus is 43% overestimated – Al-Khatatbeh (2009), Hamane-Nishio (2010))(3) No phase of TiO2 can be harder than ~15 GPa [ARO & Lyakhov, 2010; Lyakhov & ARO, 2011].

Pseudo-hard TiO2: Dubrovinsky et al. (Nature 2001) overestimated bulk modulus by 43%, hardness by 140%

Lyakhov & ARO (2011)

Nishio-Hamane (2010): bulk modulus is ~300 GPa,not 431 GPa.

New developments in crystal structure prediction extend the range of problems that can be solved

Predicted new materials and phenomena

Powerful methods forcrystal structure prediction

Fingerprints - new language for

crystallography

Colleagues around the world:• A. Bergara (U. Basque Country, Spain)• G. Gao (Jilin University, China)• C. Gatti (U. Milano, Italy)• I. Errea (U. Basque Country, Spain)• C. Hu (Guilin, China)• M. Martinez-Canales (UCL, U.K.)• M. Salvado & P.Pertierra (Oviedo, Spain)• H. Stokes (Brigham Young U., USA)• F. Zhang (Perth, Australia)• USPEX Users and Developers Community (~500 people)USPEX code is freely available at: http://han.ess.sunysb.edu/~USPEX[2 tutorials – Poitier, France (26 June – 1 July) and Xi‘an, China (2-6 August 2011)]

A. Lyakhov Q. ZhuY. Ma

Acknowledgments:

S.E. Boulfelfel Y. Xie

My postdocs and students:

C.W. Glass

Algorithm:• Evolutionary optimization (USPEX algrorithm). Options to use random sampling,

minima-hopping-like, particle-swarm optimization, metadynamics-like algorithms.• Initialization using fully random, symmetric random structures, or user-fed structures

(seeding).• Fingerprint niching technique, local order parameter (ARO&Valle, 2009). Graph theory

and Brown‘s model for analyzing topology. New variation operators (Lyakhov & ARO, 2010).

Types of runs:• Global optimization of either the energy or properties (density, hardness, band gap, etc.)• Fixed-cell or variable-cell, fixed-composition or variable-composition runs are possible. • For molecular crystals, can operate with ready-made molecules.Software aspects:• Interfaced with VASP, SIESTA, CP2k, QuantumEspresso, DMACRYS, GULP.• Excellent parallel scaling on up to >104 CPUs.Analysis:• Automatic detection of space groups. • Benefits from powerful analysis and visualization code STM4.Distribution:• USPEX code is freely available at: http://han.ess.sunysb.edu/~USPEX• Code distributed with a 50-page manual and ~12 tutorials/tests. • USPEX Users and Developers Community (>500 people)

Forthcoming tutorials: Poitiers, France (26 June – 1 July) ; Xi‘an, China (2-6 August 2011)]

Features of the USPEX code:

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