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

Ionic crystals 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. 1 Crystal Structure Prediction via Oblivious Local Search Michail Theofilatos

Ionic crystals Crystal = composition + unit cell parameters + arrangement of atoms Composition ▪ Chemical formula • Element 𝑓 𝑗 has charge 𝑟 𝑗 • Proportions of ions ▪ Charge neutral ▪ Atomic radius 𝜍 𝑗 2 Crystal Structure Prediction via Oblivious Local Search Michail Theofilatos

Ionic crystals Crystal = composition + unit cell parameters + arrangement of atoms Composition 𝑇𝑠𝑈𝑗𝑃 3 ▪ Chemical formula • Element 𝑓 𝑗 has charge 𝑟 𝑗 • Proportions of ions ▪ Charge neutral ▪ Atomic radius 𝜍 𝑗 2 Crystal Structure Prediction via Oblivious Local Search Michail Theofilatos

Ionic crystals Crystal = composition + unit cell parameters + arrangement of atoms Unit cell parameters ▪ Lengths 𝑧 1 , 𝑧 2 , 𝑧 3 ▪ Angles 𝜄 12 , 𝜄 13 , 𝜄 23 3 Crystal Structure Prediction via Oblivious Local Search Michail Theofilatos

Ionic crystals Crystal = composition + unit cell parameters + arrangement of atoms Arrangement of atoms ▪ Point 𝑦 𝑗 = (𝑦 𝑗1 , 𝑦 𝑗2 , 𝑦 𝑗3 ) in the unit cell for every ion 𝑗 ▪ 𝑒(𝑦 𝑗 , 𝑦 𝑘 ) : distance between 𝑦 𝑗 and 𝑦 𝑘 ▪ 𝑒(𝑦 𝑗 , 𝑦 𝑘 ) ≥ 𝜍 𝑗 + 𝜍 𝑘 4 Crystal Structure Prediction via Oblivious Local Search Michail Theofilatos

Ionic crystals Crystal = composition + unit cell parameters + arrangement of atoms Example ▪ Orthogonal unit cell • 𝜄 12 = 𝜄 13 = 𝜄 23 = 90 𝑝 ▪ Point 𝑦 𝑗 = 𝑦 𝑗1 , 𝑦 𝑗2 , 𝑦 𝑗3 has “copies” in (𝑙 1 𝑧 1 + 𝑦 𝑗1 , 𝑙 2 𝑧 2 + 𝑦 𝑗2 , 𝑙 3 𝑧 3 + 𝑦 𝑗3 ) for every possible combination of integers 𝑙 1 , 𝑙 2 , 𝑙 3 Crystal Structure Prediction via Oblivious Local Search Michail Theofilatos

Energy of crystal structures

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 Methods for calculating the energy ▪ Density functional theory (DFT) • Accurate, but computationally expensive ▪ Interatomic forcefields • Less accurate, but computationally cheaper 6 Crystal Structure Prediction via Oblivious Local Search Michail Theofilatos

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 Methods for calculating the energy ▪ Density functional theory (DFT) • Accurate, but computationally expensive ▪ Interatomic forcefields • Less accurate, but computationally cheaper Buckingham-Coulomb potential 6 Crystal Structure Prediction via Oblivious Local Search Michail Theofilatos

Buckingham – Coulomb potential Buckingham ▪ Short range ▪ Depends on composition-dependent parameters • For each pair of elements 𝑓 𝑗 , 𝑓 𝑘 we have 𝐵 𝑓 𝑗 𝑓 𝑘 , 𝐶 𝑓 𝑗 𝑓 𝑘 and 𝐷 𝑓 𝑗 𝑓 𝑘 𝐷 𝑓 𝑗 𝑓 𝑘 ▪ 𝐶𝐹 𝑗,𝑘 = 𝐵 𝑓 𝑗 𝑓 𝑘 exp −𝐶 𝑓 𝑗 𝑓 𝑘 𝑒 𝑦 𝑗 , 𝑦 𝑘 − 6 𝑒 𝑦 𝑗 ,𝑦 𝑘 Coulomb ▪ Long range 𝑟 𝑗 𝑟 𝑘 ▪ 𝐷𝐹 𝑗,𝑘 = 𝑒 𝑦 𝑗 ,𝑦 𝑘 7 Crystal Structure Prediction via Oblivious Local Search Michail Theofilatos

Buckingham – Coulomb potential Buckingham ▪ Short range 𝑜 ▪ Depends on composition-dependent 𝐹 𝑧, 𝜄, 𝑦 = lim 𝜍→∞ ෍ ෍ (𝐶𝐹 𝑗,𝑘 + 𝐷𝐹 𝑗,𝑘 ) parameters 𝑗=1 𝑘 ≠𝑗,𝑘∈𝑇(𝑦 𝑗 ,𝜍) • For each pair of elements 𝑓 𝑗 , 𝑓 𝑘 we have 𝐵 𝑓 𝑗 𝑓 𝑘 , 𝐶 𝑓 𝑗 𝑓 𝑘 and 𝐷 𝑓 𝑗 𝑓 𝑘 𝐷 𝑓 𝑗 𝑓 𝑘 ▪ 𝐶𝐹 𝑗,𝑘 = 𝐵 𝑓 𝑗 𝑓 𝑘 exp −𝐶 𝑓 𝑗 𝑓 𝑘 𝑒 𝑦 𝑗 , 𝑦 𝑘 − 6 𝑒 𝑦 𝑗 ,𝑦 𝑘 Coulomb ▪ Long range 𝑟 𝑗 𝑟 𝑘 ▪ 𝐷𝐹 𝑗,𝑘 = 𝑒 𝑦 𝑗 ,𝑦 𝑘 7 Crystal Structure Prediction via Oblivious Local Search Michail Theofilatos

Buckingham – Coulomb potential Buckingham ▪ Short range 𝑜 ▪ Depends on composition-dependent 𝐹 𝑧, 𝜄, 𝑦 = lim 𝜍→∞ ෍ ෍ (𝐶𝐹 𝑗,𝑘 + 𝐷𝐹 𝑗,𝑘 ) parameters 𝑗=1 𝑘 ≠𝑗,𝑘∈𝑇(𝑦 𝑗 ,𝜍) • For each pair of elements 𝑓 𝑗 , 𝑓 𝑘 we have 𝐵 𝑓 𝑗 𝑓 𝑘 , 𝐶 𝑓 𝑗 𝑓 𝑘 and 𝐷 𝑓 𝑗 𝑓 𝑘 𝐷 𝑓 𝑗 𝑓 𝑘 ▪ 𝐶𝐹 𝑗,𝑘 = 𝐵 𝑓 𝑗 𝑓 𝑘 exp −𝐶 𝑓 𝑗 𝑓 𝑘 𝑒 𝑦 𝑗 , 𝑦 𝑘 − 6 𝑒 𝑦 𝑗 ,𝑦 𝑘 Coulomb Sphere with centre 𝑦 𝑗 and radius ρ ▪ Long range 𝑟 𝑗 𝑟 𝑘 ▪ 𝐷𝐹 𝑗,𝑘 = 𝑒 𝑦 𝑗 ,𝑦 𝑘 7 Crystal Structure Prediction via Oblivious Local Search Michail Theofilatos

Energy calculation – A simpler approach Question 1 ▪ Given a composition and Buckingham parameters for it, find a simple, combinatorial method to approximate the energy of a crystal structure 8 Crystal Structure Prediction via Oblivious Local Search Michail Theofilatos

Energy calculation – A simpler approach Question 1 ▪ Given a composition and Buckingham parameters for it, find a simple, combinatorial method to approximate the energy of a crystal structure Depth Approach ▪ Given a parameter 𝑙 , it creates 𝑙 layers around the unit cell with copies of the structure. 𝑜 ▪ 𝐹 𝑧, 𝜄, 𝑦 = lim 𝜍→∞ ෍ ෍ (𝐶𝐹 𝑗,𝑘 + 𝐷𝐹 𝑗,𝑘 ) 𝑗=1 𝑘 ≠𝑗,𝑘∈𝑇(𝑦 𝑗 ,𝜍) 8 Crystal Structure Prediction via Oblivious Local Search Michail Theofilatos

Energy calculation – A simpler approach Question 1 ▪ Given a composition and Buckingham parameters for it, find a simple, combinatorial method to approximate the energy of a crystal structure Depth Approach ▪ Given a parameter 𝑙 , it creates 𝑙 layers around the unit cell with copies of the structure. 𝑜 ▪ 𝐹 𝑧, 𝜄, 𝑦 = lim 𝜍→∞ ෍ ෍ (𝐶𝐹 𝑗,𝑘 + 𝐷𝐹 𝑗,𝑘 ) 𝑗=1 𝑘 ≠𝑗,𝑘∈𝑇(𝑦 𝑗 ,𝜍) 8 Crystal Structure Prediction via Oblivious Local Search Michail Theofilatos

Energy calculation – A simpler approach Question 1 ▪ Given a composition and Buckingham parameters for it, find a simple, combinatorial method to approximate the energy of a crystal structure Depth Approach ▪ Given a parameter 𝑙 , it creates 𝑙 layers around the unit cell with copies of the structure. 𝑜 ▪ 𝐹 𝑧, 𝜄, 𝑦 = lim 𝜍→∞ ෍ ෍ (𝐶𝐹 𝑗,𝑘 + 𝐷𝐹 𝑗,𝑘 ) 𝑗=1 𝑘 ≠𝑗,𝑘∈𝐸(𝑙) 8 Crystal Structure Prediction via Oblivious Local Search Michail Theofilatos

Energy calculation – A simpler approach Question 1 ▪ Given a composition and Buckingham parameters for it, find a simple, combinatorial method to approximate the energy of a crystal structure Depth Approach Depth Approach ▪ Given a parameter 𝑙 , it creates 𝑙 layers around the unit cell with copies of the structure. 𝑜 ▪ 𝐹 𝑧, 𝜄, 𝑦 = lim 𝜍→∞ ෍ ෍ (𝐶𝐹 𝑗,𝑘 + 𝐷𝐹 𝑗,𝑘 ) 𝑗=1 𝑘 ≠𝑗,𝑘∈𝐸(𝑙) 𝑙 = 1 8 Crystal Structure Prediction via Oblivious Local Search Michail Theofilatos

Energy calculation – A simpler approach Question 1 ▪ Given a composition and Buckingham parameters for it, find a simple, combinatorial method to approximate the energy of a crystal structure Depth Approach ▪ Given a parameter 𝑙 , it creates 𝑙 layers around the unit cell with copies of the structure. 𝑜 ▪ 𝐹 𝑧, 𝜄, 𝑦 = lim 𝜍→∞ ෍ ෍ (𝐶𝐹 𝑗,𝑘 + 𝐷𝐹 𝑗,𝑘 ) 𝑗=1 𝑘 ≠𝑗,𝑘∈𝐸(𝑙) 𝑙 = 2 8 Crystal Structure Prediction via Oblivious Local Search Michail Theofilatos

Energy calculation – A simpler approach Depth Approach Experimental results ▪ ▪ Given a parameter 𝑙 , it creates 𝑙 layers around the unit Comparison between depth approach cell with copies of the structure. and GULP for SrTiO 3 . 𝑜 ▪ 𝐹 𝑧, 𝜄, 𝑦 = lim 𝜍→∞ ෍ ෍ (𝐶𝐹 𝑗,𝑘 + 𝐷𝐹 𝑗,𝑘 ) 𝑗=1 𝑘 ≠𝑗,𝑘∈𝐸(𝑙) 9 Crystal Structure Prediction via Oblivious Local Search Michail Theofilatos