View
3
Download
0
Category
Preview:
Citation preview
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
Recommended