Short Course on Molecular
Dynamics Simulation
Lecture 2:
Potential Energy Functions
Professor A. Martini
Purdue University
High Level Course Outline
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
MD Basics
Potential Energy Functions
Integration Algorithms
Temperature Control
Boundary Conditions
Neighbor Lists
Initialization and Equilibrium
Extracting Static Properties
Extracting Dynamic Properties
Non-Equilibrium MD
Molecular Dynamics Simulation
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Last Updated 8/2009
MD Basics
‰
Process summary
Calculate
Acceleration of
Each Atom
Interaction
Model
Initial
Positions
Calculate Total
Force on N
Atoms
Calculate
Velocity of
Each Atom
Move All Atoms
to New
Positions
Molecular Dynamics Simulation
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Last Updated 8/2009
Interaction Models
ma = F
dv
a=
dt
dr
v=
dt
dU
F=
dr
i
Molecular Dynamics Simulation
Interaction force is the
spatial derivative of the
potential energy
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Last Updated 8/2009
Interaction Models
Potential
Energy
Interatomic
Pair
Coulomb
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EAM
Intramolecular
Reaction
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Bond
Angle
Torsion
Out of
Plane
Last Updated 8/2009
Pair Potentials
‰
Pair potentials – van der Waals interactions
– Attraction
• Act at long distances
• Dispersive forces
• Due to instantaneous dipoles that arise during the
fluctuations in the electron cloud
– Repulsion
• Act at short distances
• Exchange forces or overlap forces
• Overlap of electron clouds such that nuclei are less well
shielded by electrons
Molecular Dynamics Simulation
A. Martini
Last Updated 8/2009
Pair Potentials
r
U(r)
Repulsive
r
0
Attractive
Molecular Dynamics Simulation
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Last Updated 8/2009
Pair Potentials
‰
Pair potentials
– General Form
⎛ ⎛ σ ⎞n ⎛ σ ⎞m ⎞
U (r ) = kε ⎜ ⎜ ⎟ − ⎜ ⎟ ⎟
⎜⎝ r ⎠ ⎝ r ⎠ ⎟
⎝
⎠
Repulsive
n ⎛n⎞
k=
⎜ ⎟
n−m⎝m⎠
m ( n−m)
Attractive
– Lennard-Jones (n=12, m=6)
⎛ ⎛ σ ⎞12 ⎛ σ ⎞ 6 ⎞
U LJ (r ) = 4ε ⎜ ⎜ ⎟ − ⎜ ⎟ ⎟
⎜⎝ r ⎠
⎝ r ⎠ ⎟⎠
⎝
13
7
⎛
ε ⎜ ⎛ σ ⎞ ⎛ σ ⎞ ⎞⎟
dU (r )
FLJ (r ) = −
2⎜ ⎟ − ⎜ ⎟
= 24
⎜
σ ⎝ ⎝ r ⎠ ⎝ r ⎠ ⎟⎠
dr
Molecular Dynamics Simulation
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Last Updated 8/2009
Pair Potentials
r
F(r)
~σ
U(r)
r
0
ε
σ
Molecular Dynamics Simulation
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Last Updated 8/2009
Pair Potentials
‰
Truncation
– The Lennard-Jones force (and similar models)
decays rapidly with distance
– Significant computation time can be saved by
neglecting pair interactions beyond a cut-off
⎧ ⎛⎛ σ
⎪4ε ⎜ ⎜
⎪ ⎜⎝ ⎝ r
U LJ ,t (r ) = ⎨
⎪
⎪
⎩
12
6
⎞
⎛ σ ⎞ ⎞⎟ r ≤ rc
⎟ −⎜ ⎟ ⎟
⎠
⎝r⎠ ⎠
0
r > rc
– Commonly, rc = 2.5 σ
Molecular Dynamics Simulation
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Last Updated 8/2009
Pair Potentials
‰
Truncated and shifted
– The potential energy vanishes at the cut off radius
– No discontinuity Æ no impulsive force
⎧U LJ (r ) − U LJ (rc ) r ≤ rc
⎪
U LJ ,t − s (r ) = ⎨
⎪⎩
r > rc
0
– Shift must be taken into account in all postprocessing calculations
Molecular Dynamics Simulation
A. Martini
Last Updated 8/2009
Pair Potentials
‰
‰
Typical values of LJ constants
Atom
ε/kB (K)
σ (pm)
He
10.22
258
Ne
35.7
279
Ar
124
342
The united atom model
εH , σH
H
εCH4 =137 K
H
CH4
C
εC , σC
Molecular Dynamics Simulation
H
σCH4 =382 pm
H
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Last Updated 8/2009
Pair Potentials
‰
Combination rules
– Relate LJ parameters for an unlike pair (i-j) to the
parameters of two like pairs (i-i and j-j)
1.
2.
3.
ε ij = ε iε j
σ ij = σ iσ j
ε ij = ε iε j
ε ij =
Molecular Dynamics Simulation
2σ σ
3
i
σ ij =
3
j
ε iε j
σ i6 + σ 6j
A. Martini
σi +σ j
2
⎛σ +σ
σ ij = ⎜⎜
2
⎝
6
i
6
j
⎞
⎟
⎟
⎠
1
6
Last Updated 8/2009
Coulombic Interactions
‰
Coulombic interactions
– Included if electrostatics between atoms are
significant
– Atomic charges qi and qj
U coulomb =
1
qi q j
rij
4πε 0 rij
qi
Molecular Dynamics Simulation
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qj
Last Updated 8/2009
Coulombic Interactions
‰
In general, the contribution of the tail of a
truncated potential is
∞
U tail
‰
‰
Nρ
2
=
U
(
r
)
4
π
r
dr
∫
2 rc
Coulombic force decays slower than r-3
Methods for calculating the long-range
contributions
– Ewald summation
– Fast multipole methods
– Particle-mesh-based methods
Molecular Dynamics Simulation
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Last Updated 8/2009
Coulombic Interactions
‰
Ewald sums
– Assume every particle i with charge qi is surrounded
by a diffuse charge distribution of opposite sign
qi
– Electrostatic potential due to i is exclusively due to
the fraction of qi that is not screened by the cloud
– This fraction rapidly decays to zero
Molecular Dynamics Simulation
A. Martini
Last Updated 8/2009
Coulombic Interactions
– Contribution to the electrostatic potential due to a set
of screened charges can be found by direct
summation because it decays rapidly with distance
– But we want to evaluate the contribution from point
charges, not screened charges
– So we correct for the screening clouds by
introducing a smooth, compensating charge density
point charges = (point charges + screening cloud) + compensating cloud
+
=
Molecular Dynamics Simulation
A. Martini
Last Updated 8/2009
Coulombic Interactions
– Compensating charge distribution surrounding i is a
Gaussian of width 2 α
ρ Gauss (r ) = −qi (α π ) exp(− αr 2 )
3
2
– Finally
U Coulomb
1
=
2V
− (α π )
1
2
4π
2
2
k
k
−
ρ
(
)
exp
4α
∑
2
k ≠0 k
(
N
∑q
i =1
2
i
(
1 N qi q j erfc α rij
+ ∑
2 i≠ j
rij
Molecular Dynamics Simulation
)
)
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Last Updated 8/2009
Embedded Atom Model
‰
Metals have ionized atom cores with a “sea” of
delocalized valence electrons
Pair Potential
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Electron Sea
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Last Updated 8/2009
Embedded Atom Model
‰
Model formulation
–
–
–
–
‰
ρi : electron density at atom i
F(ρi) : embedding function
φ(rij) : pair potential between atoms i and j
f(rij) : electron density function at atom i due to atom j
Three look-up tables
Molecular Dynamics Simulation
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Last Updated 8/2009
Reactive Potentials
‰
Some potentials model chemical reactions
– Bond formation
– Bond disassociation
‰
Common examples
– Tersoff
– REBO (reactive empirical bond order)
‰
Models developed for
–
–
–
–
Silicon
Carbon
Oxygen
Hydrogen
Molecular Dynamics Simulation
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Last Updated 8/2009
Intramolecular Interactions
‰
For multi-atomic molecules, models describe
the behavior of covalent bonds
Bond
Molecular Dynamics Simulation
Angle
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Torsion
Last Updated 8/2009
Intramolecular - Bond
‰
Bond stretching models
– Harmonic bond model where k is the “spring
constant”
U bond
r0
k
2
= (r − r0 )
2
– What is r0
• Reference bond length – value
of bond when all other terms in
the force field are zero
• Equilibrium bond length – value
when other terms contribute
Molecular Dynamics Simulation
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Bond
r0 (A)
K
(kcal/mol/A2)
C-C
1.523
317
C=C
1.337
690
Last Updated 8/2009
Intramolecular - Bond
‰
Bond stretching models
– Morse bond potential where D is the potential well
depth, α is a stiffness constant, and r0 is the
equilibrium bond distance
– Better handling of large displacements
[
U bond = D 1 − e
Molecular Dynamics Simulation
]
−α ( r − r0 ) 2
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r0
Last Updated 8/2009
Intramolecular - Angle
‰
Bond angle models
– Harmonic angle model
U angle = k (θ − θ 0 )
2
θ0
– Cosine angle models
U angle = k [1+ cos(θ )]
– For all of these, k is energy constant and θ0 is the
equilibrium angle
Molecular Dynamics Simulation
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Last Updated 8/2009
Intramolecular - Torsion
‰
Torsion models (dihedrals)
– Harmonic torsion angle
U torsion = k [1+ d cos(nφ )]
φ
is the angle between
Plane 1 and Plane 2
Plane 2
defined by
B-C-D
C
A
Plane 1
defined by
A-B-C
Molecular Dynamics Simulation
B
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D
Last Updated 8/2009
Intramolecular – Out of Plane
‰
Out of plane models
– The least common of the intramolecular potentials
– Describes energy associated with the displacement
of atoms out of their equilibrium plane
– Relevant only to parts of molecules where atoms are
known to lie in the same plane
U oop = kh 2
Molecular Dynamics Simulation
h
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Last Updated 8/2009