Force Fields

Outline1. Force fields

1.1 Definition

1.2 Force field types

2. Potential models

2.1. Bond potential

2.2. Angle potential

2.3. Dihedral potential

2.4. Improper potential

2.5. Van der Waals potential

2.6. Electrostatic potential

2.7. Examples – Hexane, Benzene

3. Frequently used force fields

1.1. Force field definiton

Force Field

Force field

F⃗ ( r⃗ )=−∇⋅V ( r⃗ )

V ( r⃗ )=∑ V intra( r⃗ )+∑V inter ( r⃗ )

● Force field is a set of parameters and functional form describing potential energy of system V(r)

● Functional form of potential energy includes bonded (intramolecular) terms and nonbonded (intermolecular) terms

1.2. Force field types

Force Field Types• Class 1

• Harmonic potentials with no cross terms

• Class 2• Potentials includes cross terms and anharmonic terms, can

be used to model reactive events (Reactive force fields)

• Reactive FF – allows continuous bond forming/breaking for simulating chemical reactions

• Class 3• Accounts for electronegativity or polarizability (Polarizable

force fields)

• Polarizable FF – FF with explicit representation of polarizability for system with variable charge distribution

Class 1 Force Field (Intraparticle Potential)

V inter=∑bonds

V bond+ ∑angles

V angle+ ∑dihedrals

V dihedral+ ∑impropers

V improper

Class 2 Force Field (Intraparticle Potential)

2.1. Bond potential

ri r

j

rij

Harmonic potential

V bond(r ij)=12kb(r ij−r0)

2

● Describes harmonic vibrational motion between (i,j)-pair of covalently bonded atoms

kb = bond strength

rij = bond length between (i,j)-pair

r0 = equilibrium distance

ri r

j

rij

FENE bond potential

V bond(r ij)=−0.5 Kr02 ln [1−(

r ijr0

)2

]+4 ε[( σr ij

)12

−( σr ij

)6

]+ε

rij = bond length between (i,j)-pair

r0 = equilibrium distance

ε = repulsive strength

σ = repulsive interaction distance

K = bond strength

● Finite extensible nonlinear elastic potential, used for bead-spring polymer models

WCAFENE

Morse bond potential

● Accounts for anharmonicty and can simulate bond-breaking effects

V bond(r ij)=D [1−e−α(rij−r0)]2

rij = bond length between (i,j)-pair

r0 = equilibrium bond distance

α = potential well width (the smaller α is, the larger the well)

D = potential well depth

rij

Vbo

nd (r

ij )

D

r0

Other bond potentials

V bond (r ij)=ε(r ij−r0)

2

λ2−(r ij−r0)2

rij = bond length between (i,j)-pair

r0 = equilibrium distance

λ = finite extension

ε = energy constant

● Nonlinear bond potential

● Quartic bond potential

– Mimics FENE bond potential for coarse-grained polymer chains

V bond (rij)=K (r ij−Rc)2(r−Rc−B1)(r−Rc−B2)+V 0+4 ϵ[( σ

r ij)

12

−( σr ij

)6

]+ε

ε = energy constant of LJ potential

σ = distance with zero LJ potential

Rc = cut-off distance

rij = bond length between (I,j)-pair

V0 = energy

B1, B

2 = distance parameters

Bond potentials comparison

2.2. Angle potential

ri

rj

rkθ

ijk

θijk=acosr⃗ ij⋅r⃗ jk

r ijr jk

Harmonic potential

● Describes angular vibrational motion between (i,j,k)-triplet of covalently bonded atoms

● Has a discontinuity in the force and should be used with caution

V angle(θijk)=12kθ(θijk−θ0)

2

kθ = angle bond strength

θijk = angle between (i,j,k) atoms

θ0 = equilibrium angle

Harmonic cosine potential

● Describes angular vibrational motion between (i,j,k)-triplet of covalently bonded atoms

● Periodic and smooth for all angle θijk

kθ = angle bond strength

θijk = angle between (i,j,k) atoms

θ0 = equilibrium angle

V angle(θijk)=kθ

2[1−cos(θijk−θ0)]

Squared cosine potential

● Describes angular vibrational motion between (i,j,k)-triplet of covalently bonded atoms

● Very flat at θ0 V angle(θijk)=

kθ

2[cos(θijk)−cos(θ0)]

2

kθ = angle bond strength

θijk = angle between (i,j,k) atoms

θ0 = equilibrium angle

Angle potentials comparison

2.3. Dihedral potential

Φijkl

ri

rj

rk

rl

ϕijk=acos( r⃗ij×r⃗ jk)⋅(r⃗ jk×r⃗kl)

|r⃗ ij×r⃗ jk||r⃗ jk×r⃗kl|

Harmonic cosine potential

● Describes angular spring motion between planes formed by first three and last three atoms of (i,j,k,l)-quartet of atoms

kΦ = dihedral bond strength

Φijkl

= angle between (i,j,k,l) atoms

δ = phase factor

N = {0,1,2,3,4,5,6}Number of potential minima

V dihedral(ϕijkl)=kϕ

2(1+cos(nϕijkl−δ))

2.4. Improper potential

Ψijkl

ri

rj

rk

rl

ψijkl=acos (r⃗ ij⋅w⃗kl

rijwkl

)

w⃗kl=( r⃗ij⋅^⃗ukl)

^⃗ukl+(r⃗ij⋅^⃗v kl)

^⃗vkl

^⃗ukl=^⃗r ik+

^⃗r il| ^⃗r ik+

^⃗r il|

^⃗vkl=^⃗r ik−

^⃗r il| ^⃗rik+

^⃗r il|

Harmonic potential

● Potential for restricting geometry of molecule (maintains chirality/planarity of molecule)

kΨ = improper bond strength

Ψijkl

= improper bond strength

Ψ0 = equilibrium angle

V improper (ψijkl)=kψ

2(ψijkl−ψ0)

2

Harmonic cosine potential

● Potential for restricting geometry of molecule (maintains chirality/planarity of molecule)

kΨ = improper bond strength

Ψijkl

= improper bond strength

Ψ0 = equilibrium angle

V improper (ψijkl)=kψ

2[1+cos(nψijkl−δ)]

d = {-1,+1}

N = {0,1,2,3,4,5,6}Number of potential minima

δ = phase factor

2.5. Van der Waals potential

ri r

j

rij

rij

Lennard-Jones potential

● Describes weak dipole attraction between distant atoms and hard-core repulsion between close atoms

V VDW (r ij)=4 ε[( σr ij

)12

−( σrij

)6

]

ε = depth of potential well

σ = distance with zero potential

r = distance between (i,j)-pair

Combination rules for Lennard-Jones potential

εij=√εiε j σ ij=√σ iσ j

● Equations that provides interaction energy between two dissimilar non-bonded particles

● Geometric combination rules

● Arithmetic (Lorentz-Berthelot) combination rules

● Sixth power (Waldman-Hagler) combination rules

εij=√εiε j σ ij=σ i+σ j

2

εij=2√εiε jσ i

3σ j3

σ i6+σ j

6 σ ij=(σ i

6+σ j6

2)

16

Buckingham potential

● Provides better description of strong repulsion due to electron shell overlap then Lennard-Jones potential

● Problem in charged systems at very small distances

V VDW (r ij)=A exp(−Br ij)−C

r−6

A,B,C = potential parameters

Soft potential

● Potential that allows particles to overlap

● Useful for pushing particles apart

● The A prefactor can change over time

V VDW (r ij)=A [1+cos(πr ijrc

)]

A = prefactor of potential

rij = distance between (i,j)-pair

rc = cutoff distance

repulsion

2.6. Electrostatic potential

Coloumbic potential

● Repulsive potential for atomic charges with same sign and attractive for atomic charges with opposite sign

V ES(r ij)=qiq j

4 πε0εr r ij

qi,q

j = atomic charges

ε0 = vacuum permitivity

rij = distance between (i,j)-pair

εr = vacuum permitivity

opositesigns

2.7. Classical force field examples

Force field example: Hexane model

Vvdw

Force field example: Benzen model

Vimproper

Vbond

Vangle

Vdihedral

Vvdw

3. Frequently used force fields

Frequently used force fields

● Class1 FF: CHARMM, AMBER, OPLS, GROMOS

● Class2 FF: UFF, COMPASS, MM2, MM3, MM4, CFF, PCFF, MMFF94

● Class3 FF: QM/MM (CHARMM, AMBER)

● Polarizable FF: PIPF, DRF90, AMOEBA, CHARMM, AMBER, OPLS, GROMOS, ORIENT, NEMO

● Reactive FF: ReaxFF, EVB, RWFF

● Coarse-Graining FF: VAMM, MARTINI, SIRAH

FF applications

● AMBER: designed for proteins and nucleic acids

● OPLS: optimized parameters for liquid simulations

● GAFF: generalized amber force field

● COMPASS: condensed-phase optimized FF

● PCFF: organic polymers, metals, zeolites

● ReaxFF: chemical reactions