Machine learning the Born-Oppenheimer potential energy surface:

from molecules to materialsGábor Csányi

Engineering Laboratory

Interatomic potentials for molecular dynamics

Transferability biomolecular force fields

(biochemistry)

Reactive solid state materials

(physics & materials)

Accuracy small molecules

in gas phase (quantum chemistry)

AMBER CHARMM AMOEBA

GROMACSOPLS

...

Tersoff Brenner

EAM BOP

...

Bowman Szalewicz

Paesani ...

Ingredients

• Representation of atomic neighbourhood

• Interpolation of functions

• Databaseof configurations

smoothness, faithfulness, continuity

flexible but smooth functional form, few sensible parameters

predictive power non-domain specific

Many-body (cluster) expansion

= + + + +

+ + + +

higher multi-body terms...+

+ analytic long range termsE =X

i

"(q(i)1 , q2,(i) . . . , q(i)M )

qi are descriptors of local atomic neighbourhood

Wea

kSt

rong

Interpolating the solution of an eigenproblem

�i~@ @t

= H Hel =1

2

X

i

r2ri +

X

ij

Zj(�1)

|Rj � ri| +X

ij

1

|ri � rj |Lowest eigenstate: E0 ⌘ Vel(R1, R2, . . .) has spatial locality

+ analytic long range terms

Parametrisation of the H2O dimer

• Standard representation: 6 atoms ⇒ 15 interatomic distances : x = {rij}

• Symmetrize kernel function: S: symmetry group of molecules

• Use analytic forces if available:

K̃(x, x0) =X

p2S

K(p(x), x0)

K

0(x, x0) =@

@x

K(x, x0)

H2O dimer corrections

Roo

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

[kcal / mol]

GAP error < 0.03 kcal/mol / H2O(B3LYP optimised geometries from

Tschumper)

Ice polymorphs

[meV / H2O] Expt DMC±5 PBE0+vdWTS BLYP+GAP±3

Ih 0 0 0 0

II 1 -4 6 -5

IX 4 3 -3

VIII 33 30 76 30

Relative energies

error < 0.15 kcal/mol / H2O

Ice polymorphs

[meV / H2O] Expt DMC ±5

PBE0+vdWTS BLYP+GAP ±3

Ih 610 605 672 667

II 609 609 666 672

IX

VIII 577 575 596 637

Absolute binding energies

BLYP+GAP overbinds uniformly by ~ 60 meV ⇒ 3-body terms!

Many-body (cluster) expansion

= + + + +

+ + + +

higher multi-body terms...+

+ analytic long range termsE =X

i

"(q(i)1 , q2,(i) . . . , q(i)M )

qi are descriptors of local atomic neighbourhood

Wea

kSt

rong

Interpolating the solution of an eigenproblem

�i~@ @t

= H Hel =1

2

X

i

r2ri +

X

ij

Zj(�1)

|Rj � ri| +X

ij

1

|ri � rj |Lowest eigenstate: E0 ⌘ Vel(R1, R2, . . .) has spatial locality

+ analytic long range terms

Quantum mechanics is many-body

-1

0

1

2

3

4

5

6

7

12 14 16 18 20 22

Ener

gy [e

V / a

tom

]

Volume [A3 / atom]

Tungsten, Finnis-Sinclairbccfccsc

bccfccsc

DFT

Tungsten: DFT, Embedded Atom ModelCarbon: tight binding

Quantum mechanics has some locality

force F

r

neighbourhood

far field

Force errors around Si self interstitial

Force errors around O in water with QM/MM

r

L̂i = qi + pi ·⇥i + . . .Vel

(R1

, R2

, . . .) ⇡atomsX

i

"(R1

�Ri, R2

�Ri, . . .) +1

2

X

ij

L̂iL̂j1

Rij+

�ij

|Rij |6

Finite range atomic energy function

Is there a finite range atomic energy function that gives the correct total energy and forces?

Traditional ideas for functional forms

• Pair potentials: Lennard-Jones, RDF-derived, etc.

• Three-body terms: Stillinger-Weber, MEAM, etc.

• Embedded Atom (no angular dependence)

• Bond Order Potential (BOP) Tight-binding-derived attractive term with pair-potential repulsion

• ReaxFF: kitchen-sink + hundreds of parameters

These are NOT THE CORRECT functions. Limited accuracy, not systematic

"i =1

2

X

j

V2(|rij |)

"i = ��P

j ⇢(|rij |)�

given byGOAL: potentials based on quantum mechanics

Representation is implicit

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”)

• Rotational invariance by itself is easy: q ≡ Rij = ri ⋅ rj (Weyl)

- Complete, but not invariant permutationally

- Not continuous with changing number of neighbours

r1

r2

r3

r4

Matrix of interatomic distances

• Equivalent to Weyl matrix

• Rotationally invariant: no alignment needed

• Permutation problem: Weyl matrix is not permutation invariant

2

6664

r1 · r1 r1 · r2 · · · r1 · rNr2 · r1 r2 · r2 · · · r2 · rN

......

. . ....

rN · r1 rN · r2 · · · rN · rN

3

7775

Drop atom ordering ?

• Permutation: too costly for ~ 20 neighbours

• Unordered list of distances? Distance histograms?

Not unique!

Atomic neighbour density function

• Translation ✓

• Permutation ✓

• Need rotationally invariant features of ρ(r)

⇥i(r) = s�(r) +�

j

�(r� rij)fcut(|rij |)

"(r1 � ri, r2 � ri, . . .) ⌘ "[⇢i(r)]

⇥(r, �,⇤) =⇥�

n=1

⇥�

l=0

l�

m=�l

cml,ngn(r)Y m

l (�, ⇤)

Rotational Invariance

Transformation of coefficients:

• Wigner matrices are unitary:

• Construct invariants:

• Equivalent to Steinhardt bond order Q2 , Q4 , Q6 … parameters

• Higher order invariants possible, (generalisation of Steinhardt W)

D�1 = D†

First few terms of power spectrum for 3 atoms

• Polynomials of Weyl invariants

• Highly oscillatory for large l : not smooth

• Different l channels uncoupledf1 = Y22 + Y2−2 + Y33 + Y3−3

f2 = Y21 + Y2−1 + Y32 + Y3−2

Uniqueness: environment reconstruction tests

Number of neighbours

Error in distinguishing

1. Select target configuration

2. Randomise neighbours

3. Try to reconstruct target by matching descriptors

RM

SD

Construct smooth similarity kernel directly

• Overlap integral

k(⇢i, ⇢i0) =X

n,n0,l

p(i)nn0lp(i0)nn0l

• After LOTS of algebra: SOAP kernel pnl = c†nlcnl

pnn0l = c†nlcn0l

S(⇢i, ⇢i0) =

Z⇢i(r)⇢i0(r)dr,

k(⇢i, ⇢i0) =

Z ���S(⇢i, R̂⇢i0)���2dR̂ =

ZdR̂

����Z

⇢i(r)⇢i0(R̂r)dr

����2

cutoff: compact support• Integrate over all 3D rotations:

4 6 8 10 12 14 16 18 20

10−2

10−1

100

101

n

Average

dREF

SOAPlmax= 2

l max= 4

l max= 6

Smooth Overlap of Atomic Positions (SOAP)

Number of neighbours

Erro

r in

dis

tingu

ishi

ng

RM

SD

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 ➝ elasticitylarge unit cell ➝ phononssurface unit cells ➝ surface energygamma surfaces ➝ screw dislocationvacancy in small cell ➝ vacancy vacancy @ gamma surface ➝ vacancy near dislocation

Iterative refinement

1. QM MD ➝ Initial database2. Model MD3. QM ➝ Revised database

What is the acceptable validation protocol? How far can the domain of validity be extended?

Existing potentials for tungsten

DFT reference

Error

0

15%

C11 C12 C44

50%

Elastic const.

Vacancy

energy Surface energy

(100) (110) (111) (112)

BOPMEAM

FS

GAP

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