Interatomic Potentials and Force Fields

Judith A. Harrison and Guangtu Gao

Chemistry DepartmentUnited States Naval Academy

Annapolis, MD 21402jah@usna.edu

Lennard-Jones Potential

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛=

612

rσ

rσ4εU(r) r

ε and σ are energy and length parameters, respectively.r is the intermolecular distance.For inert monatomic molecules.

Rare Gas Parameters:

Ne σ = 2.74 Å ε/kB = 36.2KAr σ = 3.405 Å ε/kB = 121KKr σ = 3.65 Å ε/kB = 163KXe σ = 3.98 Å ε/kB = 232K

Combining Rules for Unlike Pairs:

jiij σσσ = jiij εεε =

Global Minima of LJN Clusters – The Magic Number Cluster Structures(Data are from The Cambridge Cluster Database, http://www-wales.ch.cam.ac.uk/)

N = 13 E = -44.33ε

N = 19E = -72.66ε

N = 25E = -102.37ε

N = 55E = -279.25ε

Solvation of Halogens in Rare-Gas ClustersSolvated (12K)

Implanted (0K)Fragmented (37K)

Ar55Cl2 7 K 42 K

Ar20Cl2

J. A. Harrison & M. G. PrisantDuke University (1990)

F. G. Amar and B. J. Berne, J. Phys. Chem. 88 (1984) 6720

Br2ArN Clusters

Potentials:

Br-Ar σ = 2.74 Å ε/kB = 143 K LJ potentialAr-Ar σ = 3.405 Å ε/kB = 119.8 K LJ potential

Br-Br: Morse Potential

F. G. Amar and B. J. Berne, J. Phys. Chem. 88 (1984) 6720

( ) ( )[ ]{ } o2

e Errαexp1 DrV −−−−=

Ground state: X

( ) ( ) ( )[ ]2ooo rrβrrαexpVrV −−−−=

Excited State: u11 Π

Vo = 11012.95 Kro = 2.3 Åα= 4.637 Å-1

β = 0.879 Å-2

F. G. Amar and B. J. Berne, J. Phys. Chem. 88 (1984) 6720

Br2ArN Clusters

Interaction Potential Bead-Spring ModelEach polymer contains N spherical beads

All interact with Lennard-Jones potentialVLJ(r) = 4ε[(σ/r)12 − (σ/r)6] for r < rc

Vb (r) = VLJ(r) + k r4[(r-r1)(r-r2)] + Vc r < rbrVb (r) = 0 r > rbr

k, r1, r2, Vc = obtained from fitting FENE near mean equilibrium bond length. rbr = length at which a bond is considered broken, kSemiflexible chains → add bond-bending terms

→ less flexible, smaller Ne

Backbone – FENE potential

Standard Finite Extensible Nonliner Elastic potential

α

Courtesy of Mark. O. Robbins, The Johns Hopkins UniversityK. Kremer and G. S. Grest, J. Chem. Phys. 92, 5057 (1990)

S. W. Sides, G. S. Grest, and M. J. Stevens, Phys. Rev. E. 64, 050802 (2001)

Adhesion of surface-tethered chains entangled in a polymer melt

• tensile pull simulation• 2 x 105 particles (Nt = 250, chain length)• move bottom wall at constant velocity• Chains attached to bottom wall only• wall-chain interaction LJ• T << Tg

• Red = tethered chains• Blue = represent the polymer melt• Green = chains were tethered but

have broken

S. W. Sides, G. S. Grest, and M. J. Stevens, Phys. Rev. E. 64, 050802 (2001)

Quantifying the amount of chain scission

∑=tn

i t

fit

t NN

nF ,1

<F > = Ave fractional chain length

<F> = 1 Pure chain pull out<F> = 0 Chain scission at wall

Nt,i = length of ith chain at end ofsimulation

<F> decreases as chain length increases!

T > Tg

T < Tg

United Atom Potentials

How to build:Molecule are represented by different functional groups, ex., CHn groups.Functional groups are represented as spherical pseudo-atoms.There are bonded and non-bonded interactions among the pseudo-atoms.Potential is the sum the these bonded and non-bounded interactions terms.Potential parameters are obtained from fitting the experimental and quantummechanical data.

Dihedral term

θ

Bonded interactions:Bond term , Morse function, or rigid.

Angle term

P. van der Ploeg et al. Mol. Phys. 49, 233 (1983)

( )∑ += ωω nsVE n cos1

Non-bonded interactions: Lennard-Jones potential and Coulomb potential.

J. Hautman and M.L. Klein, J. Chem. Phys. 91, 4994 (1989)J. Hautman and M. L Klein, J. Chem. Phys. 93, 7483 (1990)

J. Ryckaert & A. Bellemans. Faraday Discuss. Chem. Soc. 66, 95 (1978)

( )∑ −= 0llkE ll

( )∑ −= 0θθϑθ kE

Modeling Alkylthiol Chains S(CH2)nCH3

CH3

CH2

CH2

CH2

CH2

CH2

S

dSC

dcc

θCC

Functional groups: S, CH2, and CH3.Bonds are rigid: dCC=1.53Å and dSC=1.82Å.

Intramolecular potential:

(1) Bond angle term

(2) Dihedral angle term

(3) LJ interaction between atoms greater than fourth nearest neighbors

Intermolecular (chain-chain) potential:LJ potential between all sites on different Chains.

( )202

1 θθθ −= kVb

( ) ( ) ( )( ) ( ) ( )φφφ

φφφ5

54

43

3

2210

coscoscos

coscos

aaa

aaaVt

+++

++=

J. Hautman and M.L. Klein, J. Chem. Phys. 91, 4994 (1989)J. Hautman and M. L Klein, J. Chem. Phys. 93, 7483 (1990)

Self-Assembled Monolayers: Alkanethiol Chains on Au (Au--S(CH2)nCH3 )

Au

CH3

CH2

CH2

CH2

CH2

CH2

S

dSC

dcc

Surface Potential: Au--S

( )( ) ( )3

o

312

o

12

zzC

zzCzV

−−

−=

Polarizability of Kr about the same as CH2/CH3Use Kr on Ag(111) experimental data to fix parameters(model z

Courtesy of Scott Perry, University of Houston

Improving the Model - All-Atom Representation(J.P. Bareman and M.L. Klein, J. Phys. Chem. 94, 5202 (1990))

CH

HCH2 Group: C

HH

HCH3 Group:

Using C and H atoms to represent the CHn groups.

dCC=1.53Å , dCH = 1.040Å, and H-C-H bond angles are all tetrahedral.

Interactions between atoms on different chains (used for studies of bulk n-alkanes)

6Br

rCAeV(r) −= − A, B, C = parameters

D. E. Williams, J. Chem. Phys. 47, 4680 (1967)

Intramolecular and molecule-surface potential = same as united atom

J.P. Bareman and M.L. Klein, J. Phys. Chem. 94, 5202 (1990)

Improvement over the united-atom model:

(1) Better representation of the collective tilt of the chains in LB films.(2) Better estimation of the effective diameters of long chains.(3) United atom cannot reproduce tilt angles

ExperimentUnited AtomAll Atom

W. Mar & M. L. Klein, Langmuir 10, 188 (1994)

All-Atom Representation yields herringbone structure

A. Ulman, Chem. Rev. 96, 1533 (1996)

Can MD simulations predict the experimentally determined structure?

L. Zhang, W. A. Goddard III, & S. Jiang, J. Chem. Phys. 117, 7342 (2002)

Need an accurate surface interaction potential.

( )

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−=

−=

1RR

2S-expχ

2χχDE

e

2e

Re = equilibrium position determined to fit the ab initio interatomicS-Au distance

De = Dissociation energyS = Scale factor (binding energy of thiolate with Au and energy diff

for a thiol adsorbed on different sites (fcc, hcp, atop, bridge)Parameter fits with POLYGRAF (DMOL, Molecular Simulations, Inc.)

III & IV = c(4x2) superlattice(4 chains / unit cell)

II = herringbone (2 chains/unit cell)

I = one chain/unit cellV = 4 chains / unit cell

= Au

= S

= hydrocarbonbackbone

oR30 33 ×

L. Zhang, W. A. Goddard III, & S. Jiang, J. Chem. Phys. 117, 7342 (2002)

J. A. Harrison, et al., “The Friction of Model Self-Assembled Monolayers”, inEncyclopedia of Nanoscience and Nanotechnology, Vols. 3, ed. H. S. Nalwa, American Scientific Publishers, Los Angeles (2004), pp. 511-527.

Reviews of SAMS

A. Ulman, Chem. Rev. 96, 1533 (1996)

Some Force Fields:MM3 Allinger at University of Georgia Organic MoleculesAMBER Kollman at UC San Francisco Proteins and Nucleic AcidsCHARMM Karplus at Harvard University Proteins and Nucleic AcidsGROMOS Van Gunsteren et al. at ETH Zurich Proteins and Nucleic AcidsLAMMPS Plimpton et al. at SNL General Parallel MD CodeDISCOVERECEPP Empirical Computational Energy Program for Peptides

A Typical Form of Force Fields:

∑ ∑

∑∑∑

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

+⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟

⎟⎠

⎞⎜⎜⎝

⎛+

−++−+−=→→→→

nonbond jipairs ij

ji

ij

ij

ij

ijij

itorsionsiiii

ianglesiii

ibondsii

biN

rqq

rr

nkkrrkrrrrU

),(,

612

,,

20

,

20321

21

)]cos(1[)(21)(

21)...,,,(

εσσ

ε

δφϑθ θθ

Strength: Contains more atomic types;Can be used for many systemsfrom Proteins to simple monatomic molecules.

Weakness: cannot simulate chemical reactions.

Force Field References:

MM3N.L. Allinger, et al., J. Am. Chem. Soc. 111, 8551 (1989) ; J. Am. Chem. Soc. 111, 8566 (1989);J. Am. Chem. Soc. 111, 8576 (1989); Theochem: J. Molecular Structure 428, 203 (1998).

AMBERP.A. Kollman, et al., J. Am. Chem. Soc. 117, 5179 (1995);AMBER 4.0, University of California, San Francisco (1991).

CHARMM. Karplus, et al., J. Comp. Chem. 4, 187 (1983).

GROMOSW.F. van Gunsteren, et al., Biopolymers 23, 1 (1984);"The GROMOS force field" in Encyclopaedia of Computational Chemistry.

LAMMPShttp://www.cs.sandia.gov/~sjplimp/lammps.htmlS. Plimpton, J. Comp. Physiol. 117, 1 (1995)

J. A. Harrison, et al., “The Friction of Model Self-Assembled Monolayers”, inEncyclopedia of Nanoscience and Nanotechnology, Vols. 3, ed. H. S. Nalwa, American Scientific Publishers, Los Angeles (2004), pp. 511-527.

Modeling Biological MoleculesCan we predict Protein Structure from amino acid sequence ? HARD!

primary structuresecondary structuretertiary structuresolvent effects

Amino Acid sequence

α-helix

Brown, LeMay, & Bursten, “Chemistry: The Central Science”, Prentice Hall

Acetylcholineesterase (enzyme)with bound inhibitor

Prion-Pnas

Adams, Pannu, Read and Brünger PNAS 94, 5018-5023, 1997.Garcia-Moreno, E. B. and Fitch, C. Methods Enzymology 308: 20-51, 2004.

Force-fields to do energy minimizations to refine protein (and nucleicAcid) structural models from NMR and X-Ray diffraction data.

Reactive Empirical Bond-Order (REBO) Potential*

∑≠

=ji

REBOijEE

21

)()( ijA

ijijijR

ijREBOij rVbrVE −=

(1) VR and VA are repulsive and attractive terms, respectively.

ijij rij

ij

ijijijij

R eArQ

rfrV α−

⎟⎟⎠

⎞⎜⎜⎝

⎛+= 1)()(

∑=

−=3

1

)( )(

)()(n

rnijijCij

A ijn

ijeBrfrV β

Switching function fC(rij) restricts the pair potential to nearest neighbors.

*D.W. Brenner, Phys. Rev. B 42, 9458 (1990)*D. W. Brenner, J. A. Harrison, S. B. Sinnott et al., J. Phys. Condens. Matter 14, 783 (2002)D. W. Brenner, Phys. Stat. Sol. (b) 217, 23 (2000);

(2) Bond-order term bij depends on the local environment.

( ) dhij

RCijjiijij ppb ππσπσπ +++=

21

5.0

,),()(cos)(1

−

≠⎟⎟⎠

⎞⎜⎜⎝

⎛++= ∑

jik

Hi

CiijijkiikCij NNPegrfp ijkλσπ θ

θijk

i

jk

Gi a sixth-order polynomial spline of cos(θ) Pij a bicubic spline.Ni

C: the number of carbon neighbors of atom i.Ni

H the number of hydrogen neighbors of atom i.

a) The pijθπ term (pij

σπ ≠ pjiσπ)

b) The πijRC term

),,( conjij

tj

tiij

RCij NNNF=π πij

RC a tricubic splineNi

t: atom i’s coordination numberNj

t: atom j’s coordination numberNij

conj: local conjugation

c) The πijdh term

⎥⎦

⎤⎢⎣

⎡−= ∑ ∑

≠ ≠jik jiljlCikCijkl

conjij

tj

tiij

dhij rfrfNNNT

, ,

2,

)()()(cos1(),( θπ

Θijkl is the dihedral angle over the atoms i and j, from k and lA barrier for rotation around the carbon-carbon double bonds.

i j

kl

REBO are bond order potentials. The bond order term is a function of local atomic hybridization and conjugation. They Allow for covalent bond breaking and forming associated with the change of atomic hybridization. Therefore they can model chemical reactions.

Harrison & Brenner J. Am. Chem. Soc. 116, 10399 (1994).

Reactive Empirical Bond-Order (REBO) PotentialModeling Chemical Reactions

Computational Study of Fluorocarbon Modification of Polymersand Composites

Susan B. Sinnott’s group atUniversity of Florida, Department of Materials Science and Engineering

• Classical molecular dynamics • Reactive empirical bond order (REBO) potentials for

short-range interactions developed by Tersoff and parameterized by Brenner et al. for C-C, C-H, H-H, Graves et al. for C-F, F-F, Sinnott et al. for H-F, C-O, O-H, O-O

• Lennard-Jones potential for long-range interactions

)]()()([ ijvdwijaijiji ji

r rVrVBrVU +−= ∑∑<

Brenner et al., J. Phys: Condens. Matt. 14, 783 (2002)

Polystyrene Surface Details

56 Å

30 Å

20 Å

52 Å

10 Å

5 Å

Active CarbonActive Hydrogen

Heat-bath CarbonHeat-bath Hydrogen

Front View Top View

Polystyrene

Deposition of C3F5+ at 50

eV/ion• Incoming ions are

embedded in, scattered from, or react chemically with the PS surface

• PS surface is chemically modified, becomes disordered, and swells as a result of the deposition

CF2 Fragments Important for Bonding of FC to PS

• Produced from incident ions by dissociation

• Cross-link PS chains

• Nucleate FC polymer within the PS

∑∑ ∑ ∑≠ ≠ ≠

⎥⎦

⎤⎢⎣

⎡++=

i ij jik kjii

torsijkl

LJij

REBOij EEEE

, ,,21

Adaptive Intermolecular REBO (AIREBO) Potential*(next generation REBO)

a) EijLJ is the LJ potential, only used for the non-bonded pairs of atoms.

b) The torsion term is extended to all dihedral angles in the system.

⎥⎦⎤

⎢⎣⎡ −= 1.0)

2(cos

405256)()()( 10 φεjlCikCijC

torsijkl rfrfrfE

*S.J. Stuart, A.B. Tutein, and J.A. Harrison, J. Chem. Phys. 112, 6472 (2000)

Stuart, Tutein, & Harrison, J. Chem Phys. 112, 6472 (2000).

Fij = FREBO + Ftors + FLJ

K. D. Krantzman, Z. Postawa, B. J. Garrison, S. J. Stuart, and J. A. HarrisonNuclear Instruments and Methods in Physics Research B, 180 (2001) 159-163.

Barbara Garrison’s Group at Penn. StateMD Simulations of Sputtering in Molecular Solids

http://galilei.chem.psu.edu/bjgpubs.htm

0 fs 30 fs

110 fs 3000 fs

C60 vs Ga bombardment

Z. Postawa, B. Czerwinski, M. Szewczyk, E. J. Smiley, N. Winograd and B. J. Garrison, Analytical Chemistry, 75, 4402 (2003).

Barbara Garrison’s Group at Penn. Statehttp://galilei.chem.psu.edu/bjgpubs.htm

C60 in a Nanotube(Ni, Sinnott, Mikulski, & Harrison,

Phys. Rev. Lett. 88 (2002) 205505-1 )

Luzzi et al. (Univ. of Penn.)

Lieber et al. (Harvard)

Ni, Sinnott, Mikulski, & Harrison, Phys. Rev. Lett. 88 (2002) 205505-1

Effect of filling on carbon nanotube stiffness

Short Tube300 K

Ni, Sinnott, Mikulski, & Harrison, Phys. Rev. Lett. 88 (2002) 205505-1

Effect of temperature on carbon nanotube stiffness

Short Tubeempty

Ni, Sinnott, Mikulski, & Harrison, Phys. Rev. Lett. 88 (2002) 205505-1

Effect of temperature on filled carbon nanotube stiffness

Short TubeC60

Ni, Sinnott, Mikulski, & Harrison, Phys. Rev. Lett. 88 (2002) 205505-1

Ni, Sinnott, Mikulski, & Harrison

REBO FORM

J. Tersoff, Phys. Rev. B 37, 6991 (1988)J. Tersoff, Phys. Rev. B 39, 5566 (1989)J. Tersoff, Phys. Rev. Lett. 61, 2879 (1988)J. Tersoff, “New Empirical Model for the Structural Properties of Silicon”, Phys. Rev. Lett. 56, 632 (1986)

A. J. Dyson and P. V. Smith, “Extension of the Brenner Empirical Interatomic Potential to C-Si-H Systems”, Surf. Sci. 355, 140 (1996).A. J. Dyson and P. V. Smith, “A Molecular Dynamics Study of the Chemisorption of C2H2 and CH3on the Si(001)-(2x1) Surface”, Surf. Sci. 375, 45 (1997); Ibid., “Empirical Potential Study of the Chemisorption of C2H2 and CH3 on the -SiC(001) Surface”, Surf. Sci. 396 (1998) 24-39.

Similar to AIREBO (switching differs)

J. Che, T. Cagin, W. A. Goddard, III, Theor. Chem. Acc. 102, 346 (1999).K. Bearmore

Other Reactive Potentials References:

Embedded-atom method (EAM) potentials*

∑∑<

+=ji

ijiji

iiEAM rFE )(][ φρ

Based on density-functional theory and quasi-atom approach

Fi[ρi] is the energy required to embed atom i in a local electron density ρi.

∑≠

=ij

iji r )(ρρ

ρi is the superposition of the atomic density functions fro all other atoms.

)( ijij rφ is the residual pair interaction between atoms I and j.

Commonly accepted and widely used for modeling metallic bonding.

*M.S. Daw and M.I. Baskes, Phys. Rev. B 29, 6443 (1984).

i

iiii AF ξ

ρρ −=][

Electrostatic potential for metal-oxide surfaces and interfaces

F. H. Streitz & J. W. Mintmire, Phys. Rev. B 50, 11996 (1994)

+− += ESEAMoxidemetal EEE

( ) ( )∑∑≠

+ +=ji

jiijiji

iiES qqVqEE ;21 r

EES+ = electrostatic energy, qi = atomic charges, Ei = atomic energiesVij = Coulomb pair interaction