Accelerated Quantum Molecular Dynamics · occ. | i|2! ⇢ sc SCF SCF SCF SCF U(R; ) U(R; sc)...

Operated by Los Alamos National Security, LLC for the U.S. Department of Energy’s NNSA

U N C L A S S I F I E D

Accelerated Quantum Molecular Dynamics

Enrique Martinez, Christian Negre, Marc J. Cawkwell, Danny Perez, Arthur F. Voter and Anders M. N. Niklasson



Outline

§ Quantum MD

• Current approaches

• Challenges

§ Extended Lagrangian Born-Oppenheimer MD

§ Accelerated MD

§ Coupling XL-BOMD and AMD

• Vacancies in graphene



Integrate the equations of motion of classical molecular trajectories

with the forces calculated on the fly from a self-consistent quantum mechanical description of the electronic structure:

MIRI = �⇥U(R; �sc)

⇥RI

H[�]�i = ⇥i�i

⇢ =X

occ.

|�i|2! ⇢sc

SCF SCFSCF

SCF

U(R; �)

U(R; �sc)

Born-Oppenheimer MD

Goal: To develop a unique computationalcapability based on a new generationquantum based molecular dynamics thatovercomes current limitations allowingdesign and prediction of materials andprocesses on time and length scalesmultiple orders of magnitude beyondcurrent capacity.#SCF�O(N3)

Quantum based MD



New Generation Quantum Mechanical Molecular Dynamics

XL-BOMD

Car-Parrinello MD (1985)Born-Oppenheimer MD (1973)

+ Accurate and reliable+ General, any material + Long time steps- Energy drift- Expensive, SCF cost

- Parameter dependence- Material dependent- Short time steps+ Small energy drift+ Fast, no SCF required

The best of both worlds

Quantum based MD



System Size (N)

Wal

l-Clo

ck T

ime

/ MD

Tim

e St

epO(N3) DiagonalizationO(N) Regular linear scalingO(N) Low pre-factor

Significant pre-factor reduction is required for practical

large scale QMD!

Quantum Molecular Dynamics

The Challenge


U N C L A S S I F I E D Slide 6

Hi�,j⇥ [q] = hi�,j⇥ +X

l

ql�il⇥ij⇥�⇥

“Self-consistent-charge density-functional tight-binding method for simulations of complex materials properties”

M. Elstner et al. Phys. Rev. B 58, 7269 (1998) +1,400 citations

UDFTB(R; ⇥) = 2Tr[⇥h] +1

2

X

i,j

qiqj�ij + Epair[R]

qi = 2X

�

�i�,i� ⇥ = � (µI �H[q])

Maybe the simplest and most efficient approach to electronic structure theory that still covers the relevant

features of “exact” ab initio Kohn-Sham DFT.

Typically two orders of magnitude in speed up (w.r.t. DFT)!

LATTE: Self-Consistent Charge Density Functional based Tight-Binding Theory



SCFSCF

SCF

Potential energy surface at the SCF optimized electronic ground state

Born-Oppenheimer MD

Potential energy surface of a non-optimized electronic state

Car-Parrinello MDDtBO

dtCP << DtBO

Potential energy surface of a shadow H is very closeto the exact SCF optimized electronic ground state

0New Fast “SCF-free” GXL-BOMD

DtXL-BO

Niklasson & Cawkwell, JCP 141, 164123 (2014)

SCF-free Generalized XL-BOMD



SCF-free Generalized XL-BOMD

Slide 8

Niklasson & Cawkwell, JCP 141, 164123 (2014)

MIRI = � �U(R, n)

�RI

��n

n(r) = �⇥2

ZK(r, r0) (�[n](r0)� n(r0)) dr0

U(R, n) = min�

(E(1)

DFT

(R, ⇥, n)

�� ⇥ =X

occ

|�i|2, ��i|�j⇥ = �ij

)+ V

nn

(R)

E(1)DFT(R, ⇥, n) = EDFT(R, n) +

Z�EDFT(R, n)

�⇥

��=n

(⇥[n](r)� n(r)) dr

K(r, r0) ⇠✓�⇥[n](r

�n(r0)� �(r� r0

◆�1

⇥[n] = argmin�

(E(1)

DFT

(R, ⇥, n)

�� ⇥ =X

occ

|�i|2, ��i|�j⇥ = �ij

)

ETot

=1

2

X

I

MIR2

I + U(R, n)

LShadowXBO =

X

I

MIR2I

2� U(R, n) +

µ

2

Zn2dr� µ⇥2

2

Z(�� n)†K†K(�� n)dr

LShadowXBO = LKS

BO, {� BO approx. lim� ⇥ ⇤, limµ ⇥ 0, limµ� ⇥ constant}

Shadow Hamiltonian



SCF-free Generalized XL-BOMD Canonical Ensemble

Slide 9

200

250

300

350

400

T (

K)

a)

−622.4

−622.3

−622.2

−622.1

−622.0

0 2 4 6 8 10

ES +

EN

(e

V)

time (ps)

b) BO 6 diag/step

BO 9 diag/step

XLBO 1 diag/step

200

250

300

350

400

T (

K)

a)

−622.4

−622.0

−621.6

−621.2

−620.8

0 2 4 6 8 10

Energ

y (e

V)

time (ps)

b)

O(N3)

BO 6 diag/step

BO 9 diag/step

XLBO 1 diag/step

Isocyanic acid (HNCO)24

NVE ensembleNVT ensemble Nose thermostat

Is the XL-BOMD compatible with the thermostats? Does it help?



SCF-free Generalized XL-BOMD Canonical Ensemble

350

400

450

500

550

600

650

700

1 2 3 4 5 6 ∞T (

K) Born−OppenheimerXLBO

Un

sta

ble

a)(HNCO)24

Ttarget = 500 K

350

400

450

500

550

600

650

1 2 3 4 5 6 ∞ T

Diagonalizations per step

Born−OppenheimerXLBO

Un

sta

ble

b) Andersen Langevin

Nose(CH3NO2)32

260

280

300

320

340

T (

K)

Unstable

a) AndersenLangevin

Nose

(HNCO)24 Ttarget = 300 K

400

450

500

550

600

0 0.2 0.4 0.6 0.8 1.0 1.2

time step (fs)

b) nitromethane Ttarget = 500 K

Average temperature and the square-root of its second moment

XL-BOMD converges faster with the right fluctuations



XL-BOMD + Accelerated Molecular Dynamics

Slide 11

a

bg

Infrequent Events

Can we accelerate the transition without perturbing the

probabilities for the system to escape? AMD

Accelerated Molecular DynamicsHyperdynamics

Parallel Replica Dynamics

Temperature Accelerated Dynamics



Parallel Replica Dynamics

Slide 12

The properties of the exponential allow us to parallelize time, by having many processors seek the first escape event. The procedure:

Must detect every transition.

dephase allow correlated events

parallel time

AFV, Phys. Rev. B 57, 13985 (1998).

add all these times together (tsum) when first event occurs on any processor

(overhead)

tcorr

tcorr

replicate

Good parallel efficiency if trxn / nproc >> 2tcorr



Master

Replica 1 Replica 2 Replica n…

Python script

LATTE: Tight Binding Order NParallel Implementation MultiThreads

AMDF(BES program)

XL-BOMD + Accelerated Molecular Dynamics



Application: Vacancies in graphene

Slide 14

We have studied 1 and 2 vacancy diffusion

Temperature distribution of a graphene sheet containing 1V at 2500 and 3000 K in the NVT using Langevin Dynamics

run for 100 ps

a) b)

2500 3000 3500 4000 4500

Ttarget = 3000 K

2996 K3058 K

3281 K

BO−1SCFunstable

T (K)

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

2000 2500 3000 3500 4000 4500

Ttarget = 2500 K2493 K2582 K

2681 K

3631 K

frequ

ency

T (K)

BO−1SCFBO−3SCF

BO−5SCFXLBO−1SCF

We have analyzed the normal modes to pick the dephasing time ~5ps. There are some slow modes perpendicular to the sheet plane




Slide 15

1V diffusivities in NVT ensemble ParRep-LATTE, reaching on the order of hundreds of ns

10−10

10−9

10−8

10−7

10−6

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2

D = ν λ2 exp(−∆E/kBT)λ = 1.54 10−10 m

ν = 9.44 1012 s−1

∆E = 0.54 eV

ν = 1.17 1015 s−1

∆E = 1.16 eV

DV

(m2 /s

)1000/T (K−1)

Migration barriers in agreement with recent DFT calculations

Competing events at higher temperatures



2V configurations at 3500 K in NVT ensemble ParRep-LATTE

Net translation of the center of mass


4−55−7−9

5−77−5

5−8−5

555−777

5555−6−7777

0.0 100 5.0 10−9 1.0 10−8 1.5 10−8 2.0 10−8 2.5 10−8 3.0 10−8

t (s)




5-8-5 555-777 5555-6-7777 2x(57) 4-55-7-9

Experiments 50.3% 14.1% 18.8% 1.8% 0%

Simulation 20.7% 65.2% 12.7% 0.6% 0.02%

Mobility and configurational distribution compatible with

experimentsto

tal m

igra

tion

leng

th d

istri

butio

n

x m

igra

tion

leng

th d

istri

butio

n

y m

igra

tion

leng

th d

istri

butio

n

a) b)

c) d)

5-8-5 5-8-5translations

5-8-5 555-777 555-777 5555-6-7777 transitions

Bond strechingRotations

−2

0

2

4

6

8

6 7 8 9 10 11 12 13 14

T = 3500 Kt = 27.87 ns

initial

final

y (Å

)

x (Å)

00.050.1

0.150.2

0.250.3

0.350.4

0 0.5 1 1.5 2 2.5 3d (Å)

0

0.05

0.1

0.15

0.2

0.25

−3 −2 −1 0 1 2 3�x (Å)

0

0.05

0.1

0.15

0.2

0.25

−3 −2 −1 0 1 2 3�y (Å)



Conclusions

n We have developed a tool coupling XL-BOMD and AMD that reaches ~2-3 orders of magnitude speedup with respect to traditional methods

n We have calculated single vacancy mobilities in graphene reaching hundreds of ns

n We have also studied the very stable 2V-complex migration

Slide 18

Accelerated Quantum Molecular Dynamics · occ. | i|2! ⇢ sc SCF SCF SCF SCF U(R; ) U(R; sc)...

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