1

Dishing out the dirt on ReaxFF

Force field subgroup meeting 29/9/2003

2

Contents

- ReaxFF: general principles and potential functions

-All-carbon compounds: Training set

- Sample simulation: Ethylene+O2 reactive NVE

3

ReaxFF: general principles and potential functions

Tim

e

DistanceÅngstrom Kilometres

10-15

years

QC

ab initio,DFT,HF

ElectronsBond formation

MD

Empiricalforce fields

AtomsMolecular

conformations

MESO

FEA

Design

Grains

Grids

Hierarchy of computational chemical methods

ReaxFF Simulate bond formationin larger molecular systems

Empirical methods:- Allow large systems- Rigid connectivity

QC methods:- Allow reactions- Expensive, only small systems

4

Current status of ReaxFF program and force fields

Program:

- 18,000 lines of fortran-77 code; currently being integrated into CMDF- MD-engine (NVT/NVE/limited NPT), MM-engine- Force field optimization methods: single parameter search, anneal- Can handle periodic and non-periodic systems- User manual (under development) available online

Force fields

Published: hydrocarbons, nitramines, Si/SiO/SiH, Al/AlOAdvanced: proteins/CH/CN/CO/NO/NN/NH/OH/OO, MoOx, all-carbon, Mg/MgHIn development: SiN/SiC, Pt/PtO/PtN/PtC/PtCo/PtCl, Ni/NiAl/NiC, Co/CoC, Cu/CuC, Zr/ZrO, Y/YO, Ba/BaO, Y-BaZrOH, BH/BB/BN/BC, Fe/FeO

Method seems universally available; has been tested now for covalent, ceramic, metallic and ionic materials.

5

MM or MD routine

Connection table

1: 2 3 42: 1 5 63: 14: 15: 26: 2

Non-reactive force field

Atom positions

1: x1 y1 z1

2: x2 y2 z2

3: x3 y3 z3

4: x4 y4 z4

5: x5 y5 z5

6: x6 y6 z6

Fix

ed

MM or MD routine

Determineconnections

Reactive force field

Atom positions

1: x1 y1 z1

2: x2 y2 z2

3: x3 y3 z3

4: x4 y4 z4

5: x5 y5 z5

6: x6 y6 z6

Program structureB

ond

orde

r

Interatomic distance (Angstrom)

)( ijij rfBO =

1 2

3

4

5

6

6

ReaxFF: General rules

- MD-force field; no discontinuities in energy or forces

- User should not have to pre-define reactive sites or reactionpathways; potential functions should be able to automatically handlecoordination changes associated with reactions

- Each element is represented by only 1 atom type in force field;force field should be able to determine equilibrium bond lengths,valence angles etc. from chemical environment

7

underover

torsvalCoulombvdWaalsbondsystem

EE

EEEEEE

++

++++=

conjpen EE ++ ReaxFFCH

ReaxFFSiO: System energy description

2-body

multibody

3-body 4-body

8

Sigma bond

Pi bond

Double pi bond

Bond energy 1. Calculation of bond orders from interatomic distances

0

1

2

3

1 1.5 2 2.5 3

Interatomic distance (Å)

Bond order

Bond order (uncorrected)

Sigma bond

Pi bond

Double pi bond

9

Bond energy 2. Bond order correction for 1-3 bond orders

H

HH

HH

H

0.950.

97

0.97

0.97

0.97

0.97

0.97

0.10

0.10 0.10

0.10

0.10

0.10

BOC=4.16

BOH=1.17

Uncorrected bond orders

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1

3.5

3.7

4.1

4.5

5

6

BO

Uncorrected bond order

Cor

rect

ed b

ond

orde

r

- Unphysical- Puts strain on angle and overcoordination potentials

BOC=3.88

BOH=0.98

H

HH

HH

H

0.94

0.97

0.97

0.97

0.97

0.97

0.97

0.01

0.01 0.01

0.01

0.01

0.01

Corrected bond orders

- Correction removes unrealistic weak bonds but leaves strong bonds intact- Increases computational expense as bond orders become multibody interactions

10

Bond energy 3. Calculate bond energy from corrected bond orders

€

Ebond = −Deσ ⋅BOij

σ ⋅exp pbe,1 1− BOijσ

( )pbe,2

( ){ }

− Deπ ⋅BOij

π − Deππ ⋅BOij

ππ

0

1

2

3

1 1.5 2 2.5 3

Interatomic distance (Å)

Bond order

Bond order (uncorrected)

Sigma bond

Pi bond

Double pi bond

-500

-400

-300

-200

-100

0

100

1 1.5 2 2.5 3

Interatomic distance (Å)

Bond energy (kcal/mol)

Sigma energy

Pi energy

Double pi energy

Total bond energy

11

Nonbonded interactions

- Nonbonded interactions are calculated between every atom pair, including bonded atoms; this avoids having to switch off interactions due to changes in connectivity- To avoid excessive repulsive/attractive nonbonded interactions at short distances both Coulomb and van der Waals interactions are shielded

( ){ } 3/133 /1 ijij

jiCoulomb

r

qqCE

γ+

⋅⋅=

Shielded Coulomb potential

0

250

500

750

1000

0 0.5 1 1.5 2 2.5 3

Unshielded Coulomb

Shielded Coulomb

Unshielded vdWaals

Shielded vdWaals

+0.5 +0.5

Interatomic distance (Å)

Ene

rgy

(kca

l/mol

)

vdWaals: Shielded Morse potential

12

Charge calculation method

- ReaxFF uses the EEM-method to calculate geometry-dependent, polarizable point charges

- 1 point charge for each atom, no separation between electron and nucleus

- Long-range Coulomb interactions are handled using a 7th-order polynomal (Taper function), fitted to reproduce continuous energy derivatives. Taper function converges to Ewald sum much faster than simple spline cutoff.

NaCl-crystal (33.84x33.84x33.84 Å)

-200000

-190000

-180000

-170000

-160000

0 5 10 15 20 25 30 35

Outer cutoff

Coulomb energy

Taper 7

Ewald (12.5 Å innerspace cutoff)Spline 3

13

Interatomic distance (Å)

Ene

rgy

(kca

l/mol

)- Summation of the nonbonded and the bonded interactions gives the two-body interactions- Bond energies overcome van der Waals-repulsions to form stable bonds

-500

-250

0

250

500

750

0 1 2 3

Bond energy

Shielded vdWaals energy

Total pair energy

Total two-body interaction

14

Valence angle energy1. General shape

i

j

k

ba

( )( )ππbaoijkbaval BOBOfBOfBOfE ,)()( Θ−Θ⋅⋅=General shape:

Ensures valenceangle energy contributiondisappears when bond a

or bond b dissociates

Modifies equilibriumangle o according

to π-bond order in bond a andbond b

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0 0.5 1 1.5 2

Valence angle energy2. Bond order/valence angle energy

i

j

k

ba

( )( )ππbaoijkbaval BOBOfBOfBOfE ,)()( Θ−Θ⋅⋅=

( )21exp1)( λλ aa BOBOf ⋅−−=

Bond order bond a

Eval,max

Eva

l

0

16

Valence angle energy3. π-Bond order/equilibrium angle

i

j

k

ba

( )( )ππbaoijkbaval BOBOfBOfBOfE ,)()( Θ−Θ⋅⋅=

( )21exp1)( λλ aa BOBOf ⋅−−=

⎟⎟⎠

⎞⎜⎜⎝

⎛

⎭⎬⎫

⎩⎨⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛−⋅−−⋅−= ∑

=

)(

13, 2exp1180),(

jneighbours

njnoobao BOBOBO πππ λ

∑=

)(

1

jneighbours

njnBOπ

Equ

ilibr

ium

ang

le(d

egre

es)

17

Torsion angle energy1. General shape

( ) ( ) ( )⎭⎬⎫

⎩⎨⎧ +⋅+⋅−⋅⋅⋅⋅= ijklbijklcbators VBOfVBOfBOfBOfE ωω π 3cos1

2

12cos1

2

1)()()( 32

General shape:

Ensures torsion angle energy contribution

disappears when bond a, b or cdissociates

(similar to valence angle)

i

j

k

bac

l

Controls V2-contributionas a function of the

π-bond order in bond b

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Torsion angle energy2. π-bond order influence on V2-term

( ) ( ) ( )⎭⎬⎫

⎩⎨⎧ +⋅+⋅−⋅⋅⋅⋅= ijklbijklcbators VBOfVBOfBOfBOfE ωω π 3cos1

2

12cos1

2

1)()()( 32

i

j

k

bac

l

( ) ( )( )24 1exp ππ λ bb BOBOf −−=

0 0.25 0.5 0.75 1

πbBO

eff

V2

V2,max

0

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Avoid unrealistically high amounts of bond orders on atoms

3 3.5 4 4.5A

tom

ene

rgy

BOi ji

nbonds

,

1

BO (C)=4i ji

nbonds

,

1BO (C)=3i j

i

nbonds

,

1BO (C)=5i j

i

nbonds

,

1

∑=

−=Δ

Δ⋅+⋅Δ⋅=

neighbours

jijii

iiijover

BOValency

BOfE

1

)exp(1

1)(

λ

Overcoordination energy

20

0.01

0.1

1

10

100

1000

10000

100000

1000000

0 100 200 300 400

ReaxFF

QM (DFT)

Nr. of atoms

Tim

e/ite

ratio

n (s

econ

ds)

Computational expense

x 1000,000

21

All-carbon compounds: training set

Strategy for parameterizing reactive force fields

- Pick an appropriate QC-method - Determine a set of cluster/crystal cases; perform QC- Fit ReaxFF-parameters to QC-data

Interatomic distance (Angstroms)

Bonds

Valence/Torsions

Nonbonded

Non-reactive force field

Interatomic distance (Angstroms)

Bonds

Valence/Torsions

Nonbonded

Overcoordination

ReaxFF

Complications

22

Binding energies in all-carbon compounds relative to Graphite

0

20

40

60

80

100

120

Acyclic C2Acyclic C3Cyclic C3Acyclic C4Cyclic C4C4 pyramidAcyclic C5Cyclic C5Acyclic C62_C3Cyclic C6Acyclic C7Cyclic C7Acyclic C8C8 3ringC8 3ringIIC8 cubeCyclic C8Acyclic C9Cyclic C9Acyclic C10Cyclic C10Tricyclic C10Acyclic C12Acyclic C13Cyclic C13Tricyclic C13Acyclic C14Cyclic C15Cyclic C17Bicyclic C17Acyclic C20Hexacyclic C20C20-dodecaC60-buckyballDiamond

Relative binding energy (kcal/atom)

Reax

QC

- Even-carbon acyclic compounds are more stable in the triplet state; odd-carbon, mono and polycyclic compounds are singlet states- Small acyclic rings have low symmetry ground states (both QC and ReaxFF)- ReaxFF reproduces the relative energies well for the larger (>C6) compounds; bigger deviations (but right trends) for smaller compounds- Also tested for the entire hydrocarbon training set (van Duin et al. JPC-A, 2001); ReaxFF can describe both hydro- and all-carbon compounds

23

0

50

100

1.5 2 2.5

DFTReaxFF

0

50

100

1.5 2 2.5

DFTReaxFF

C-C distance (Å)

Ene

rgy

(kca

l/m

ol)

Ene

rgy

(kca

l/m

ol)

Bond formation between two C20-dodecahedrons

- ReaxFF properly describes the coalescence reactions between C20-dodecahedrons

24

Angle bending in C9

- ReaxFF properly describes angle bending, all the way towards the cyclization limit

25

C6+C5 to C11 reaction

- ReaxFF properly predicts the dissociation energy but shows a significantly reduced reaction barrier compared to QC

26

3-ring formation in tricyclic C13

-ReaxFF describes the right overall behaviour but deviates for both the barrier height and the relative stabilities of the tetra- and tricyclic compounds

27

0

0.05

0.1

0.15

0.2

10 15 20

c-axis (Å)

ΔE (

eV/a

tom

)

diamond

graphite

Diamond to graphite conversionCalculated by expanding a 144 diamond supercell in the c-direction and relaxing

the a- and c axes

QC-data: barrier 0.165 eV/atom(LDA-DFT, Fahy et al., PRB 1986, Vol. 34, 1191)

-ReaxFF gives a good description of the diamond-to-graphite reaction path

28

Relative stabilities of graphite, diamond, buckyball and nanotubes

Compound ERef (kcal/atom) EReaxFF

Graphite 0.00a 0.00

Diamond 0.8a 0.52

Graphene 1.3a 1.56

10_10 nanotube 2.8b 2.83

17_0 nanotube 2.84b 2.83

12_8 nanotube 2.78b 2.81

16_2 nanotube 2.82b 2.82

C60-buckyball 11.5a 11.3

a: Experimental data; b: data generated using graphite force field (Guo et al. Nature 1991)

- ReaxFF gives a good description of the relative stabilities of these structures

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Ongoing all-carbon projects

- Nanotube failure, buckyball collision (Claudio)

- Si-tip/nanotube interactions (Santiago)

- Nanotube growth, buckyball polymerization (Weiqiao)

- Buckyball/nanotube nucleation (Kevin)

- Buckyball/nanotube oscillator (Haibin)

- Diamond surface interactions (Sue Melnik)

30

Sample simulation: Ethylene+O2 reactive NVE

-12 Ethylene, 36 O2

- Pre-equilibrated at 4000K. Switched off C-O and H-O bonds during equilibration to avoid reactions

- Time-step: 0.025 fs.; cannot go much higher due to high temperature + reactive potential

- Should react; main expected products H2O, CO2 and CO

31

QuickTime™ and aGIF decompressorare needed to see this picture.

32MD-iteration

- Fast reaction after initiation

- Exothermic; temperature rises to 7000K

- Energy is not perfectly conserved at elevated temperatures.

- Future work: investigate potential; see if energy conservation can be improved.

33

- ReaxFF gets pretty reasonable product distribution; probably slightly too much CO; may need to check CO+0.5O2 to CO2 reaction energy