First-principles calculatiton combined with multicanonical ... · First-principles calculatiton combined with multicanonical ensemble method Yoshihide YOSHIMOTO Materials Design and

First-principles calculatiton combined with multicanonical ensemble method

Yoshihide YOSHIMOTOMaterials Design and Characterization Laboratory

Institute for Solid State PhysicsUniversity of Tokyo

Yoshimoto, JCP 125, 184103 (2006)

Multi-order Multi-thermal ensembleand

Thermodynamic downfolding

Contents

• Introduction

• multi-order multi-thermal ensemble by structure factor

• thermodynamic downfolding

• results for Xtal↔Liq transition of Si

• results for Xtal↔Liq transition of MgO

Yoshimoto, JCP 125, 184103 (2006)

DSC Cycle

developing materials

DSC CycleDesign, Synthesis, Characterization (MDCL, ISSP)



D

S C



D

S Cmajor part of

first-principles works

minor part offirst-principles works

? ? ?



D

S Cmajor part of

first-principles works

minor part offirst-principles works

? ? ?

crystallization asa basic synthesis developing

materials

Purpose of development

✦ Capture the transition directly

➡ By multicanonical method

• Without simulations of co-existing state

✓ keep simulated system small

• Without ladder ofthermodynamic integration/adiabatic switching

✓ No need for a reference system

✦ Optimized model potential based on multicanonical ensemble and ab-initio calc.

multicanonical ensembleA generalized ensemble

multicanonical ensembleBerg and Neuhaus, PRL 68, 9 (1992) Lee, PRL 71, 211(1993) e!!E !W (E)!1 = e!S(E)

multicanonical ensemble

• Wide range of E is visited with equal prob.

➡overcome free energy barriers rapidly

➡capture the transition in principle

• produces entropy, S(E) [as weight]

• Re-weighting of the simulation run produces any thermal average at wide range of T

➡Simul. run represents the total thermodynamics of the system

Berg and Neuhaus, PRL 68, 9 (1992) Lee, PRL 71, 211(1993)

Wang and LandauPRL, 86, 2050 (2001)

e!!E !W (E)!1 = e!S(E)










e!!E !W (E)!1 = e!S(E)

P (E) !W (E)e!!E










e!!E !W (E)!1 = e!S(E)

P (E) !W (E)W (E)!1 = 1










e!!E !W (E)!1 = e!S(E)










e!!E !W (E)!1 = e!S(E)

!A" =!

i Ai!i 1










e!!E !W (E)!1 = e!S(E)

!A" =!

m AmW (Em)e!!Em

!m W (Em)e!!Em










Ferrenberg and SwendsenPRL61, 2635 (1988); 63 1658 (1989)

e!!E !W (E)!1 = e!S(E)

Limitation of multicanonical

0 5

10 15 20 25 30

0 0.5 1 1.5 2

O

t [108 a.u.]

multicanonical

Simulation by Wang-Landau algorithm to determine S(U)

no tunneling from Liq. to Xtal. of Si

Xtal.

Liq.

O: An order parameter of crystallization

Wang and Landau, PRL, 86 (2001) 2050

• sampling probability in MD

• entropy [mult.-ca. weight]

Order param. as 2nd deg. of freedom

P (U,O)

Multi-Order Multi-Thermal: MOMT

explicitly generate random walk between ordered and disordered states

keep tunneling time in MD short

S(U,O)

two-component multicanonical: Higo et al., 1997 multibaric multithermal: Okumura et al., 2004

Wang and Landau, 2001

Crystalline order Structure factorcf. Bragg spots in diffraction experiments

potential energy

order param.

Efficiency of MOMT

0 5

10 15 20 25 30

0 0.5 1 1.5 2

O

t [108 a.u.]

multicanonical

0 5

10 15 20 25 30

0 0.5 1 1.5 2

O

t [108 a.u.]

multi-order multi-thermal

-11

-10.5

-10

-9.5

-9

-8.5

0 0.5 1 1.5 2

U [H

t.]

t [108 a.u.]

multi-order multi-thermal

-11

-10.5

-10

-9.5

-9

-8.5

0 0.5 1 1.5 2

U [H

t.]

t [108 a.u.]

multicanonical

by Wang-Landau algorithm

short tunneling time between Liq. and Xtal. of Si

Xtal.

Liq.

Xtal.

Liq.

multi-order multi-thermal MD run

Repeating transition between liquid and crystal

variable cell simulation

impuritiesin a crystal

amorphous

Still too much steps for first-principles MD

0 5

10 15 20 25 30

0 0.5 1 1.5 2

O

t [108 a.u.]

multicanonical

0 5

10 15 20 25 30

0 0.5 1 1.5 2

O

t [108 a.u.]

multi-order multi-thermalXtal.

Liq.

Xtal.

Liq.


• > 106 steps required (1 step ~ 100 a.u.)

0 5

10 15 20 25 30

0 0.5 1 1.5 2

O

t [108 a.u.]

multicanonical

0 5

10 15 20 25 30

0 0.5 1 1.5 2

O

t [108 a.u.]


Liq.

Xtal.

Liq.



←available steps for first-principles MD

~ 104 steps

0 5

10 15 20 25 30

0 0.5 1 1.5 2

O

t [108 a.u.]

multicanonical

0 5

10 15 20 25 30

0 0.5 1 1.5 2

O

t [108 a.u.]


Liq.

Xtal.

Liq.




~ 104 steps

• even more light weight method is required

0 5

10 15 20 25 30

0 0.5 1 1.5 2

O

t [108 a.u.]

multicanonical

0 5

10 15 20 25 30

0 0.5 1 1.5 2

O

t [108 a.u.]


Liq.

Xtal.

Liq.




~ 104 steps

• even more light weight method is required

downfolding of inter-atomic potential from a first-principles one to a model one

Thermodynamic downfolding of U

• Multicanonical simul. : typical config.

• Re-weighting of produces any thermodynamic quantities at any T

➡ : a measure of a potential

• Optimize model potential via function L

• Self-consistency for

➡ depends on UM

!U(X) = UM (X)! UA(X)

Q = {Xi}Q

Q

QQ

model accurate

L =!

Xi!Q[!U(Xi) ! "!U#]2

Iterative optimize of UM

UM

Q

opt UM by L(UM; Q, UA on Q)

MD/MCFPC

UA on Q

feed

back

embarrassingly parallel

downfolded model potential of Si

Original EDIP, Justo et al. PRB 58 (1998) 2539

Stillinger Weber

optimized EDIP by DForiginal EDIP

Stillinger-Weber, PRB 31 (1985) 5262

UA: first-principles

UM: model

-0.5

0

0.5

1

1.5

2

2.5

UM

! U

A [H

t.]

0

2

4

6

8

10

12

0 0.5 1 1.5 2

!F

[10

-2 H

t. a

.u.-1

]

UA [Ht.]

Erro

r in

For

ce

per

atom

diff.

in p

ot. e

nerg

y

each point: sampled config.

clearly superior potential was obtained

Applications

Targets ofThermodynamic Downfolding

+ Multiorder Multithermal

Classification by type of bonding Instances

Covalent Silicon, Diamond

Ionic MgO, SiO2

Molecular H2O

Metallic Fe, Al

Silicon

Why crystal⇔liquid transition of Silicon

• Large ΔS ~ 3kB/mol for the transition

‣ comparable to that of water

‣ typical elemental substances: ΔS ~ 1kB/mol

• Overcooled as much as 300K (m.p. 1685K)

It is hard to observe the transition without over-cooling/heating by direct MD reversibly

Challenging test case for new methods

calculation conditions for Si

✓ isobaric-isothermal MD by HA Stern

• 64 atoms in a cubic cell

✓Order param. from 8 shortest reciprocal lattice vectors

✓Model potential: EDIP type by Justo et al

✓Accurate potential: First-principles calc. by TAPP

• density functional: PBE, pseudo-potential

• plane wave basis set: Gcut ~ 3.8

• k point sampling: 2x2x2

✓Thermodynamic downfolding: |Q| = 1000, 2 iteration

PRB 58 (1998) 2539

J. Compt. Chem. 25 (2004) 749

Bernasconi and Parrinello et al.J. Phys. Chem. Solids 56 (1995) 501

results for Si

Present Landman Kaczmarski Sugino

CarAlfe

Gillanexp.

potential opt-EDIPStillinger Weber

EDIP LDA GGA

Tm[K] 1572 1665 1572 1350 1492 1685

ΔH(ΔU)[kJ mol-1]

43.3 31.4 36.0 34 43 48.3

ΔV/Vs 0.101 0.070 -0.0095 0.1 0.106 0.1

Sugino and Car, PRL 74 (1995) 1823

Alfe and Gillan, PRB 68 (2003) 205212

Landman et al., PRB 37 (1988) 4637

Kaczmarski et al., PRB 69 (2004) 214105

results for Si

Present Landman Kaczmarski Sugino

CarAlfe

Gillanexp.

potential opt-EDIPStillinger Weber

EDIP LDA GGA

Tm[K] 1572 1665 1572 1350 1492 1685

ΔH(ΔU)[kJ mol-1]

43.3 31.4 36.0 34 43 48.3

ΔV/Vs 0.101 0.070 -0.0095 0.1 0.106 0.1

Sugino and Car, PRL 74 (1995) 1823

Alfe and Gillan, PRB 68 (2003) 205212

Landman et al., PRB 37 (1988) 4637

Kaczmarski et al., PRB 69 (2004) 214105

accurate potential was by GGA

downfolded model potential of Si

Original EDIP, Justo et al. PRB 58 (1998) 2539

Stillinger Weber

optimized EDIP by DForiginal EDIP

Stillinger-Weber, PRB 31 (1985) 5262

UA: first-principles

UM: model

-0.5

0

0.5

1

1.5

2

2.5

UM

! U

A [H

t.]

0

2

4

6

8

10

12

0 0.5 1 1.5 2

!F

[10

-2 H

t. a

.u.-1

]

UA [Ht.]

Erro

r in

For

ce

per

atom

diff.

in p

ot. e

nerg

y

each point: sampled config.

clearly superior potential was obtained

liquid(1800 K) crystal(1200 K)

0

0.5

1

1.5

2

2.5

3

3.5

0 2 4 6 8 10

g(r

)

r [a.u.]

DFFP

orig

0 0.5

1 1.5

2 2.5

3 3.5

4 4.5

5

0 2 4 6 8 10

g(r

)

r [a.u.]

DFFP

orig

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 0.5 1 1.5 2 2.5 3

g3(!

)

! [rad.]

rm = 5.85 a.u.

DFFP

orig

0

0.5

1

1.5

2

2.5

0 0.5 1 1.5 2 2.5 3

g3(!

)

! [rad.]

rm = 5.85 a.u.

DFFP

orig

bondangle

pair corr.

bondangle

pair corr.

Summary (Si)• Crystal⇔Liquid transition of Silicon was

successfully simulated

• Downfolded potential gives good agreement with first-principles results

• simultaneously for both phase

‣ pair correlation function

‣ bond-angle distribution function

Conclusion• multi-order multi-thermal ensemble by

structure factor

• thermodynamic downfolding

• a method to determine optimized interatomic model potentials

• results for Xtal↔Liq transition of Si

• results for Xtal↔Liq transition of MgO

Documents

First-principles calculatiton combined with multicanonical ... · First-principles calculatiton combined with multicanonical ensemble method Yoshihide YOSHIMOTO Materials Design and