Local Search Dmytro Antypov, Argyrios Deligkas, Vladimir Gusev, - - PowerPoint PPT Presentation
Crystal Structure Prediction via Oblivious Local Search Dmytro Antypov, Argyrios Deligkas, Vladimir Gusev, Matthew J. Rosseinsky, Paul G. Spirakis, Michail Theofilatos 18th Symposium on Experimental Algorithms June 16-18, 2020 Catania, Italy
Crystal Structure Prediction via Oblivious Local Search
Dmytro Antypov, Argyrios Deligkas, Vladimir Gusev, Matthew J. Rosseinsky, Paul G. Spirakis, Michail Theofilatos
18th Symposium on Experimental Algorithms June 16-18, 2020 Catania, Italy
Crystal = an ordered arrangement of ions, atoms or
molecules ➢ The crystal structure is periodic. ➢ Crystal lattice extends in all 3 dimensions. ➢ The unit cell is a small box containing one or more atoms in a specific spatial arrangement that form the crystal when stacked.
Crystal Structure Prediction via Oblivious Local Search Michail Theofilatos
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Crystal = composition + unit cell parameters + arrangement of atoms
▪ Chemical formula
▪ Charge neutral ▪ Atomic radius 𝜍𝑗
Composition
Crystal Structure Prediction via Oblivious Local Search Michail Theofilatos
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Crystal = composition + unit cell parameters + arrangement of atoms
▪ Chemical formula
▪ Charge neutral ▪ Atomic radius 𝜍𝑗
Composition
Crystal Structure Prediction via Oblivious Local Search Michail Theofilatos
𝑇𝑠𝑈𝑗𝑃3 2
Crystal = composition + unit cell parameters + arrangement of atoms
▪ Lengths 𝑧1, 𝑧2, 𝑧3 ▪ Angles 𝜄12, 𝜄13, 𝜄23
Unit cell parameters
Crystal Structure Prediction via Oblivious Local Search Michail Theofilatos
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Crystal = composition + unit cell parameters + arrangement of atoms
▪ Point 𝑦𝑗 = (𝑦𝑗1, 𝑦𝑗2, 𝑦𝑗3) in the unit cell for every ion 𝑗 ▪ 𝑒(𝑦𝑗, 𝑦𝑘): distance between 𝑦𝑗 and 𝑦𝑘 ▪ 𝑒(𝑦𝑗, 𝑦𝑘) ≥ 𝜍𝑗 + 𝜍𝑘
Arrangement of atoms
Crystal Structure Prediction via Oblivious Local Search Michail Theofilatos
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Crystal = composition + unit cell parameters + arrangement of atoms
▪ Orthogonal unit cell
▪ Point 𝑦𝑗 = 𝑦𝑗1, 𝑦𝑗2, 𝑦𝑗3 has “copies” in (𝑙1𝑧1 + 𝑦𝑗1, 𝑙2𝑧2 + 𝑦𝑗2, 𝑙3𝑧3 + 𝑦𝑗3) for every possible combination of integers 𝑙1, 𝑙2, 𝑙3
Example
Crystal Structure Prediction via Oblivious Local Search Michail Theofilatos
▪ Density functional theory (DFT)
expensive ▪ Interatomic forcefields
cheaper
Methods for calculating the energy
▪ Every combination of unit cell parameters and arrangement of atoms corresponds to an energy. ▪ Potential Energy Surface ▪ 6 unit cell parameters ▪ n atoms in the unit cell ▪ 3 𝑜 − 1 + 6 degrees of freedom
Crystal Structure Prediction via Oblivious Local Search Michail Theofilatos
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▪ Density functional theory (DFT)
expensive ▪ Interatomic forcefields
cheaper
Methods for calculating the energy
▪ Every combination of unit cell parameters and arrangement of atoms corresponds to an energy. ▪ Potential Energy Surface ▪ 6 unit cell parameters ▪ n atoms in the unit cell ▪ 3 𝑜 − 1 + 6 degrees of freedom Buckingham-Coulomb potential
Crystal Structure Prediction via Oblivious Local Search Michail Theofilatos
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▪ Short range ▪ Depends on composition-dependent parameters
𝑘 we have
𝐵𝑓𝑗𝑓𝑘, 𝐶𝑓𝑗𝑓𝑘 and 𝐷𝑓𝑗𝑓𝑘
▪ 𝐶𝐹𝑗,𝑘 = 𝐵𝑓𝑗𝑓𝑘 exp −𝐶𝑓𝑗𝑓𝑘𝑒 𝑦𝑗, 𝑦𝑘 −
𝐷𝑓𝑗𝑓𝑘 𝑒 𝑦𝑗,𝑦𝑘
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Buckingham
▪ Long range ▪ 𝐷𝐹𝑗,𝑘 =
𝑟𝑗𝑟𝑘 𝑒 𝑦𝑗,𝑦𝑘
Coulomb
Crystal Structure Prediction via Oblivious Local Search Michail Theofilatos
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▪ Short range ▪ Depends on composition-dependent parameters
𝑘 we have
𝐵𝑓𝑗𝑓𝑘, 𝐶𝑓𝑗𝑓𝑘 and 𝐷𝑓𝑗𝑓𝑘
▪ 𝐶𝐹𝑗,𝑘 = 𝐵𝑓𝑗𝑓𝑘 exp −𝐶𝑓𝑗𝑓𝑘𝑒 𝑦𝑗, 𝑦𝑘 −
𝐷𝑓𝑗𝑓𝑘 𝑒 𝑦𝑗,𝑦𝑘
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Buckingham
▪ Long range ▪ 𝐷𝐹𝑗,𝑘 =
𝑟𝑗𝑟𝑘 𝑒 𝑦𝑗,𝑦𝑘
Coulomb
𝐹 𝑧, 𝜄, 𝑦 = lim
𝜍→∞ 𝑗=1 𝑜
𝑘 ≠𝑗,𝑘∈𝑇(𝑦𝑗,𝜍)
(𝐶𝐹𝑗,𝑘 + 𝐷𝐹𝑗,𝑘)
Crystal Structure Prediction via Oblivious Local Search Michail Theofilatos
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▪ Short range ▪ Depends on composition-dependent parameters
𝑘 we have
𝐵𝑓𝑗𝑓𝑘, 𝐶𝑓𝑗𝑓𝑘 and 𝐷𝑓𝑗𝑓𝑘
▪ 𝐶𝐹𝑗,𝑘 = 𝐵𝑓𝑗𝑓𝑘 exp −𝐶𝑓𝑗𝑓𝑘𝑒 𝑦𝑗, 𝑦𝑘 −
𝐷𝑓𝑗𝑓𝑘 𝑒 𝑦𝑗,𝑦𝑘
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Buckingham
▪ Long range ▪ 𝐷𝐹𝑗,𝑘 =
𝑟𝑗𝑟𝑘 𝑒 𝑦𝑗,𝑦𝑘
Coulomb
𝐹 𝑧, 𝜄, 𝑦 = lim
𝜍→∞ 𝑗=1 𝑜
𝑘 ≠𝑗,𝑘∈𝑇(𝑦𝑗,𝜍)
(𝐶𝐹𝑗,𝑘 + 𝐷𝐹𝑗,𝑘)
Crystal Structure Prediction via Oblivious Local Search Michail Theofilatos
Sphere with centre 𝑦𝑗 and radius ρ 7
Crystal Structure Prediction via Oblivious Local Search Michail Theofilatos
▪ Given a composition and Buckingham parameters for it, find a simple, combinatorial method to approximate the energy of a crystal structure
Question 1
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Crystal Structure Prediction via Oblivious Local Search Michail Theofilatos
▪ Given a composition and Buckingham parameters for it, find a simple, combinatorial method to approximate the energy of a crystal structure
Question 1
▪ Given a parameter 𝑙, it creates 𝑙 layers around the unit cell with copies of the structure. ▪
Depth Approach
𝐹 𝑧, 𝜄, 𝑦 = lim
𝜍→∞ 𝑗=1 𝑜
𝑘 ≠𝑗,𝑘∈𝑇(𝑦𝑗,𝜍)
(𝐶𝐹𝑗,𝑘 + 𝐷𝐹𝑗,𝑘)
8
Crystal Structure Prediction via Oblivious Local Search Michail Theofilatos
▪ Given a composition and Buckingham parameters for it, find a simple, combinatorial method to approximate the energy of a crystal structure
Question 1
▪ Given a parameter 𝑙, it creates 𝑙 layers around the unit cell with copies of the structure. ▪
Depth Approach
𝐹 𝑧, 𝜄, 𝑦 = lim
𝜍→∞ 𝑗=1 𝑜
𝑘 ≠𝑗,𝑘∈𝑇(𝑦𝑗,𝜍)
(𝐶𝐹𝑗,𝑘 + 𝐷𝐹𝑗,𝑘)
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Crystal Structure Prediction via Oblivious Local Search Michail Theofilatos
▪ Given a composition and Buckingham parameters for it, find a simple, combinatorial method to approximate the energy of a crystal structure
Question 1
▪ Given a parameter 𝑙, it creates 𝑙 layers around the unit cell with copies of the structure. ▪
Depth Approach
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𝐹 𝑧, 𝜄, 𝑦 = lim
𝜍→∞ 𝑗=1 𝑜
𝑘 ≠𝑗,𝑘∈𝐸(𝑙)
(𝐶𝐹𝑗,𝑘 + 𝐷𝐹𝑗,𝑘)
Crystal Structure Prediction via Oblivious Local Search Michail Theofilatos
▪ Given a composition and Buckingham parameters for it, find a simple, combinatorial method to approximate the energy of a crystal structure
Question 1 Depth Approach
𝑙 = 1
▪ Given a parameter 𝑙, it creates 𝑙 layers around the unit cell with copies of the structure. ▪
Depth Approach
𝐹 𝑧, 𝜄, 𝑦 = lim
𝜍→∞ 𝑗=1 𝑜
𝑘 ≠𝑗,𝑘∈𝐸(𝑙)
(𝐶𝐹𝑗,𝑘 + 𝐷𝐹𝑗,𝑘)
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Crystal Structure Prediction via Oblivious Local Search Michail Theofilatos
▪ Given a composition and Buckingham parameters for it, find a simple, combinatorial method to approximate the energy of a crystal structure
Question 1
𝑙 = 2 8
▪ Given a parameter 𝑙, it creates 𝑙 layers around the unit cell with copies of the structure. ▪
Depth Approach
𝐹 𝑧, 𝜄, 𝑦 = lim
𝜍→∞ 𝑗=1 𝑜
𝑘 ≠𝑗,𝑘∈𝐸(𝑙)
(𝐶𝐹𝑗,𝑘 + 𝐷𝐹𝑗,𝑘)
Crystal Structure Prediction via Oblivious Local Search Michail Theofilatos
▪ Given a parameter 𝑙, it creates 𝑙 layers around the unit cell with copies of the structure. ▪
Depth Approach
𝐹 𝑧, 𝜄, 𝑦 = lim
𝜍→∞ 𝑗=1 𝑜
𝑘 ≠𝑗,𝑘∈𝐸(𝑙)
(𝐶𝐹𝑗,𝑘 + 𝐷𝐹𝑗,𝑘) ▪ Comparison between depth approach and GULP for SrTiO3.
Experimental results
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Crystal Structure Prediction via Oblivious Local Search Michail Theofilatos
▪ Comparison between depth approach and GULP for SrTiO3. ▪ Fast convergence
Experimental results
9
▪ Given a parameter 𝑙, it creates 𝑙 layers around the unit cell with copies of the structure. ▪
Depth Approach
𝐹 𝑧, 𝜄, 𝑦 = lim
𝜍→∞ 𝑗=1 𝑜
𝑘 ≠𝑗,𝑘∈𝐸(𝑙)
(𝐶𝐹𝑗,𝑘 + 𝐷𝐹𝑗,𝑘)
Crystal Structure Prediction via Oblivious Local Search Michail Theofilatos
▪ Comparison between depth approach and GULP for SrTiO3. ▪ Fast convergence ▪ “Monotonicity”
between any two feasible arrangements remain almost always the same.
Experimental results
9
▪ Given a parameter 𝑙, it creates 𝑙 layers around the unit cell with copies of the structure. ▪
Depth Approach
𝐹 𝑧, 𝜄, 𝑦 = lim
𝜍→∞ 𝑗=1 𝑜
𝑘 ≠𝑗,𝑘∈𝐸(𝑙)
(𝐶𝐹𝑗,𝑘 + 𝐷𝐹𝑗,𝑘)
Crystal Structure Prediction via Oblivious Local Search Michail Theofilatos
▪ Comparison between depth approach and GULP for SrTiO3. ▪ Fast convergence ▪ “Monotonicity”
between any two feasible arrangements remain almost always the same.
Experimental results
𝐹1(𝑦): energy of arrangement 𝑦 for 𝑙 = 1. 𝐹𝐻(𝑦): energy of arrangement 𝑦 computed by GULP. ✓ If for two random (feasible) configurations 𝑦𝑗 and 𝑦𝑘 it holds that 𝐹1(𝑦𝑗) < 𝐹1(𝑦𝑘), then 𝐹𝐻(𝑦𝑗) < 𝐹𝐻(𝑦𝑘) for almost all pairs 𝑦𝑗 and 𝑦𝑘.
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▪ Given a parameter 𝑙, it creates 𝑙 layers around the unit cell with copies of the structure. ▪
Depth Approach
𝐹 𝑧, 𝜄, 𝑦 = lim
𝜍→∞ 𝑗=1 𝑜
𝑘 ≠𝑗,𝑘∈𝐸(𝑙)
(𝐶𝐹𝑗,𝑘 + 𝐷𝐹𝑗,𝑘)
Crystal Structure Prediction via Oblivious Local Search Michail Theofilatos
▪ Comparison between depth approach and GULP for SrTiO3. ▪ Fast convergence ▪ “Monotonicity”
between any two feasible arrangements remain almost always the same.
Experimental results
Formal proof?
𝐹1(𝑦): energy of arrangement 𝑦 for 𝑙 = 1. 𝐹𝐻(𝑦): energy of arrangement 𝑦 computed by GULP. ✓ If for two random (feasible) configurations 𝑦𝑗 and 𝑦𝑘 it holds that 𝐹1(𝑦𝑗) < 𝐹1(𝑦𝑘), then 𝐹𝐻(𝑦𝑗) < 𝐹𝐻(𝑦𝑘) for almost all pairs 𝑦𝑗 and 𝑦𝑘.
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▪ Given a parameter 𝑙, it creates 𝑙 layers around the unit cell with copies of the structure. ▪
Depth Approach
𝐹 𝑧, 𝜄, 𝑦 = lim
𝜍→∞ 𝑗=1 𝑜
𝑘 ≠𝑗,𝑘∈𝐸(𝑙)
(𝐶𝐹𝑗,𝑘 + 𝐷𝐹𝑗,𝑘)
▪ Crystal structure prediction (CSP) is the calculation of the crystal structures of solids. ▪ The most stable structure corresponds to the global minimum of the potential energy surface. ▪ Computational methods employed include:
Crystal Structure Prediction via Oblivious Local Search Michail Theofilatos
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▪ Move from ad-hoc Edisonian approach to a systematic theoretical approach
by the aimed properties, and then guide the experimentalist to synthesize them in the laboratory ▪ Search for materials with desired properties
Crystal Structure Prediction via Oblivious Local Search Michail Theofilatos
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▪ Obtain structural information of materials under any external conditions
Crystal Structure Prediction via Oblivious Local Search Michail Theofilatos
Nanosecond freezing of water at high pressures: nucleation and growth near the metastability limit, Philip C. Myint et al.
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Crystal Structure Prediction via Oblivious Local Search Michail Theofilatos
▪ Input: A composition with its corresponding Buckingham constants, a positive 𝑜, and a rational 𝐹. ▪ Question: Is there a crystal structure (𝑧, 𝜄, 𝑦) for the composition with 𝑜 ions that is neutrally charged and achieves Buckingham-Coulomb energy 𝐹 𝑧, 𝜄, 𝑦 < 𝐹?
Question 1 - MinEnergy
▪ Input: A composition with its corresponding Buckingham constants and a positive 𝑜. ▪ Task: Find a crystal structure (𝑧, 𝜄, 𝑦) for the composition with 𝑜 ions that is neutrally charged and the Buckingham- Coulomb energy 𝐹 𝑧, 𝜄, 𝑦 is minimized.
Question 2 - MinStructure
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Crystal Structure Prediction via Oblivious Local Search Michail Theofilatos
▪ Input: A composition with its corresponding Buckingham constants and a rational 𝐹. ▪ Question: Is there a crystal structure for the composition that is neutrally charged and
𝐹 𝑧,𝜄,𝑦 𝑜
< 𝐹?
Question 3 - AvgEnergy
▪ Input: A composition with its corresponding Buckingham constants. ▪ Task: Find a crystal structure for the composition that is neutrally charged and the average Buckingham-Coulomb energy per ion in the unit cell,
𝐹 𝑧,𝜄,𝑦 𝑜
, is minimized.
Question 4 - AvgStructure
Unbounded number of atoms
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Crystal Structure Prediction via Oblivious Local Search Michail Theofilatos
Given a current solution 𝑦, choose a new potential solution 𝑦′. Perform gradient descent on the potential energy surface starting from 𝑦′. This is known as relaxation. Decide whether to keep 𝑦 as the current solution, or to update it to the solution found after relaxing 𝑦′.
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Crystal Structure Prediction via Oblivious Local Search Michail Theofilatos
Given a current solution 𝑦, choose a new potential solution 𝑦′. Perform gradient descent on the potential energy surface starting from 𝑦′. This is known as relaxation. Decide whether to keep 𝑦 as the current solution, or to update it to the solution found after relaxing 𝑦′. Genetic algorithms Simulated annealing Basin hopping
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Crystal Structure Prediction via Oblivious Local Search Michail Theofilatos
Given a current solution 𝑦, choose a new potential solution 𝑦′. Perform gradient descent on the potential energy surface starting from 𝑦′. This is known as relaxation. Decide whether to keep 𝑦 as the current solution, or to update it to the solution found after relaxing 𝑦′. Genetic algorithms Simulated annealing Basin hopping and Local Search
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▪ Discretize unit cell ▪ Ions are placed on the nodes of the grid ▪ Local search neighbourhoods
Crystal Structure Prediction via Oblivious Local Search Michail Theofilatos
2-ion swap
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▪ Discretize unit cell ▪ Ions are placed on the nodes of the grid ▪ Local search neighbourhoods
Crystal Structure Prediction via Oblivious Local Search Michail Theofilatos
2 swap
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▪ Discretize unit cell ▪ Ions are placed on the nodes of the grid ▪ Local search neighbourhoods
Crystal Structure Prediction via Oblivious Local Search Michail Theofilatos
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𝑇𝑠𝑈𝑗𝑃3 with 15 atoms per unit cell and discretization parameter 𝜀 = 1Å (375 grid points). Energy is in electronvolts (𝑓𝑊). Results averaged over 1000 initial configurations.
Crystal Structure Prediction via Oblivious Local Search Michail Theofilatos
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𝑇𝑠𝑈𝑗𝑃3 with 15 atoms per unit cell and discretization parameter 𝜀 = 1Å (375 grid points). Energy is in electronvolts (𝑓𝑊). Results averaged over 1000 initial configurations.
Crystal Structure Prediction via Oblivious Local Search Michail Theofilatos
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𝑇𝑠𝑈𝑗𝑃3 with 15 atoms per unit cell and discretization parameter 𝜀 = 1Å (375 grid points). Energy is in electronvolts (𝑓𝑊). Results averaged over 1000 initial configurations.
Crystal Structure Prediction via Oblivious Local Search Michail Theofilatos
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𝑇𝑠𝑈𝑗𝑃3 with 15 atoms per unit cell and discretization parameter 𝜀 = 1Å (375 grid points). Energy is in electronvolts (𝑓𝑊). Results averaged over 1000 initial configurations.
Crystal Structure Prediction via Oblivious Local Search Michail Theofilatos
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𝑇𝑠𝑈𝑗𝑃3 with 15 and 20 atoms per unit cell and discretization parameter 𝜀 = 1Å (375 grid points). Energy is in electronvolts (𝑓𝑊). Results averaged over 200 runs.
Crystal Structure Prediction via Oblivious Local Search Michail Theofilatos
energy.
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𝑇𝑠𝑈𝑗𝑃3 with 15 and 20 atoms per unit cell and discretization parameter 𝜀 = 1Å (375 grid points). Energy is in electronvolts (𝑓𝑊). Results averaged over 200 runs.
Crystal Structure Prediction via Oblivious Local Search Michail Theofilatos
energy.
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𝑇𝑠𝑈𝑗𝑃3 with 15 and 20 atoms per unit cell and discretization parameter 𝜀 = 1Å (375 grid points). Energy is in electronvolts (𝑓𝑊). Results averaged over 200 runs.
Crystal Structure Prediction via Oblivious Local Search Michail Theofilatos
energy.
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Crystal Structure Prediction via Oblivious Local Search Michail Theofilatos
➢ We introduced and studied Crystal Structure Prediction through the lens of computer science. ➢ Identified several open questions whose solution would have significant impact to the discovery
depth 𝑙 matches the arrangement that minimizes the energy when it is computed by GULP
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theofila@liverpool.ac.uk