Machine learning the Born-Oppenheimer potential energy surface: from molecules to materials Gábor Csányi Engineering Laboratory
Interatomic potentials for molecular dynamics Transferability biomolecular force fields (biochemistry) AMBER CHARMM AMOEBA GROMACS OPLS... Accuracy small molecules in gas phase (quantum chemistry) Bowman Szalewicz Paesani... Reactive solid state materials (physics & materials) Tersoff Brenner EAM BOP...
Ingredients Representation of atomic neighbourhood smoothness, faithfulness, continuity Interpolation of functions flexible but smooth functional form, few sensible parameters Database of configurations predictive power non-domain specific
Interpolating the solution of an eigenproblem i~ @ @t = H H el = 1 2 X i r 2 r i + X ij Z j ( 1) R j r i + X ij 1 r i r j Lowest eigenstate: E 0 V el (R 1,R 2,...) has spatial locality Weak Many-body (cluster) expansion = + + + + + + + + + higher multi-body terms... + analytic long range terms Strong E = X i "(q (i) 1,q 2, (i)...,q (i) M ) + analytic long range terms qi are descriptors of local atomic neighbourhood
Parametrisation of the H 2 O dimer Standard representation: 6 atoms 15 interatomic distances : x = {rij} Symmetrize kernel function: K(x, x 0 )= X K(p(x),x 0 ) p2s S: symmetry group of molecules Use analytic forces if available: K 0 (x, x 0 )= @ @x K(x, x0 )
H2O dimer corrections R oo
Water hexamers - relative energies - no ZPC CCSD(T) BLYP DMC Bowman BLYP+GAP 0.00 0.00 0.00 0.00 0.00 0.24-0.59 0.33 0.46 0.15 0.70-2.18 0.56 2.39 0.52 1.69-2.44 1.53 4.78 1.77 (B3LYP optimised geometries from Tschumper) [kcal / mol] GAP error < 0.03 kcal/mol / H 2 O
Ice polymorphs Relative energies [mev / H 2 O] Expt DMC±5 PBE0+vdW TS BLYP+GAP±3 Ih 0 0 0 0 II 1-4 6-5 IX 4 3-3 VIII 33 30 76 30 error < 0.15 kcal/mol / H 2 O
Ice polymorphs Absolute binding energies [mev / H 2 O] Expt DMC ±5 PBE0+vdW TS BLYP+GAP ±3 Ih 610 605 672 667 II 609 609 666 672 IX VIII 577 575 596 637 BLYP+GAP overbinds uniformly by ~ 60 mev 3-body terms!
Interpolating the solution of an eigenproblem i~ @ @t = H H el = 1 2 X i r 2 r i + X ij Z j ( 1) R j r i + X ij 1 r i r j Lowest eigenstate: E 0 V el (R 1,R 2,...) has spatial locality Weak Many-body (cluster) expansion = + + + + + + + + + higher multi-body terms... + analytic long range terms Strong E = X i "(q (i) 1,q 2, (i)...,q (i) M ) + analytic long range terms qi are descriptors of local atomic neighbourhood
Quantum mechanics is many-body Carbon: tight binding Tungsten: DFT, Embedded Atom Model 7 Tungsten, Finnis-Sinclair bcc fcc sc Energy [ev / atom] 6 5 4 3 DFT bcc fcc sc 2 1 0-1 12 14 16 18 20 22 Volume [A 3 / atom]
Quantum mechanics has some locality neighbourhood force F r far field Force errors around Si self interstitial Force errors around O in water with QM/MM r V el (R 1,R 2,...) X Is there a finite atoms range atomic energy function ij that gives the "(R 1 R i,r 2 R i,...)+ 1 2 correct total i energy and forces? ij X Finite range atomic energy function 1 ˆL i ˆLj + R ij R ij 6 ˆL i = q i + p i i +...
Traditional ideas for functional forms Pair potentials: Lennard-Jones, RDF-derived, etc. Three-body terms: Stillinger-Weber, MEAM, etc. " i = 1 2 X j V 2 ( r ij ) Embedded Atom (no angular dependence) Bond Order Potential (BOP) Tight-binding-derived attractive term with pair-potential repulsion ReaxFF: kitchen-sink + hundreds of parameters " i = P j ( r ij ) Representation is implicit These are NOT THE CORRECT functions. Limited accuracy, not systematic given by GOAL: potentials based on quantum mechanics
Representing the atomic neighbourhood in strongly bound materials What are the arguments of the atomic energy ε? Need a representation, i.e. a coordinate transformation - Exact symmetries: Global Translation Global Rotation Reflection Permutation of atoms - Faithful: different configurations correspond to different representations - Continuous, differentiable, and smooth (i.e. slowly changing with atomic position) ( Lipschitz diffeomorphic ) r4 r3 r2 r1 Rotational invariance by itself is easy: q Rij = ri rj (Weyl) - Complete, but not invariant permutationally - Not continuous with changing number of neighbours
Matrix of interatomic distances Equivalent to Weyl matrix 2 r 1 r 1 r 1 r 2 r 1 r N r 2 r 1 r 2 r 2 r 2 r N 6....... 4..... r N r 1 r N r 2 r N r N 3 7 5 Rotationally invariant: no alignment needed Permutation problem: Weyl matrix is not permutation invariant
Drop atom ordering? Permutation: too costly for ~ 20 neighbours Unordered list of distances? Distance histograms? Not unique!
Atomic neighbour density function "(r 1 r i, r 2 r i,...) "[ i (r)] i (r) =s (r)+ j (r r ij )f cut ( r ij ) (r,, ) = n=1 l=0 l m= l c m l,ng n (r)y m l (, ) Translation Permutation Need rotationally invariant features of ρ(r)
Rotational Invariance Transformation of coefficients: Wigner matrices are unitary: Construct invariants: D 1 = D Equivalent to Steinhardt bond order Q 2, Q 4, Q 6 parameters Higher order invariants possible, (generalisation of Steinhardt W)
First few terms of power spectrum for 3 atoms Polynomials of Weyl invariants Highly oscillatory for large l : not smooth Different l channels uncoupled
Uniqueness: environment reconstruction tests 1. Select target configuration 2. Randomise neighbours 3. Try to reconstruct target by matching descriptors RMSD Error in distinguishing Number of neighbours
Construct smooth similarity kernel directly Overlap integral Z S( i, i 0)= i (r) i 0(r)dr, Integrate over all 3D rotations: Z k( i, i 0)= S( i, ˆR i 0) 2 d ˆR = Z d ˆR cutoff: compact support Z i (r) i 0( ˆRr)dr 2 After LOTS of algebra: X SOAP kernel k( i, i 0)= n,n 0,l p (i) nn 0 l p(i0 ) nn 0 l nl p nn 0 l = c nl c n 0 l
Smooth Overlap of Atomic Positions (SOAP) Error in distinguishing RMSD Average d REF 10 1 10 0 10 1 10 2 l max= 2 l max= 4 l max= 6 SOAP 4 6 8 10 12 14 16 18 20 n Number of neighbours
Stability of fit: elastic constants of Si Order of angular momentum expansion
How to generate databases? Target applications: large systems Capability of full quantum mechanics (QM): small systems QM MD on representative small systems: sheared primitive cell elasticity large unit cell phonons surface unit cells surface energy gamma surfaces screw dislocation vacancy in small cell vacancy vacancy @ gamma surface vacancy near dislocation Iterative refinement 1. QM MD Initial database 2. Model MD 3. QM Revised database What is the acceptable validation protocol? How far can the domain of validity be extended?
Existing potentials for tungsten Error DFT reference 15% 0 50% GAP BOP MEAM FS C11 C12 C44 Elastic const. Vacancy energy (100) (110) (111) (112) Surface energy
Building up databases for tungsten (W)
Improvement for tungsten (W)
Peierls barrier for screw dislocation glide
Vacancy-dislocation binding energy (~100,000 atoms in 3D simulation box)
Outstanding problems Accuracy on database accuracy in properties? Database contents region of validity? Alloys - permutational complexity? Chemical variability? Systematic treatment of long range effects Electronic temperature