1
CE 530 Molecular Simulation
Lecture 9
Monte Carlo Simulation
David A. Kofke
Department of Chemical Engineering
SUNY Buffalo
kofke@eng.buffalo.edu
2
Review
 We want to apply Monte Carlo simulation to evaluate the
configuration integrals arising in statistical mechanics
U 
1
N!
 dr
N
U (r N ) e
 U ( r N )
p(rN)
ZN
 Importance-sampling Monte Carlo is the only viable approach
• unweighted sum of U with configurations generated according to
 U
/ ZN
distribution e
 Markov processes can be used to generate configurations
according to the desired distribution p(rN).
• Given a desired limiting distribution, we construct single-step
transition probabilities that yield this distribution for large samples
• Construction of transition probabilities is aided by the use of
detailed balance: p ip ij  p jp ji
• The Metropolis recipe is the most commonly used method in
molecular simulation for constructing the transition probabilities
3
Monte Carlo Simulation
State k
 MC techniques applied to molecular simulation
 Almost always involves a Markov process
• move to a new configuration from an existing one
according to a well-defined transition probability
 Simulation procedure
• generate a new “trial” configuration by making a
perturbation to the present configuration
• accept the new configuration based on the ratio of
the probabilities for the new and old
configurations, according to the Metropolis
algorithm
• if the trial is rejected, the present configuration is
taken as the next one in the Markov chain
• repeat this many times, accumulating sums for
averages
e  U
e
new
  U old
State k+1
4
Trial Moves
 A great variety of trial moves can be made
 Basic selection of trial moves is dictated by choice of ensemble
• almost all MC is performed at constant T
no need to ensure trial holds energy fixed
• must ensure relevant elements of ensemble are sampled
all ensembles have molecule displacement, rotation; atom displacement
isobaric ensembles have trials that change the volume
grand-canonical ensembles have trials that insert/delete a molecule
 Significant increase in efficiency of algorithm can be achieved
by the introduction of clever trial moves
• reptation, crankshaft moves for polymers
• multi-molecule movements of associating molecules
• many more
5
General Form of Algorithm
Monte Carlo Move
Entire Simulation
New configuration
Initialization
Select type of trial move
Reset block sums
“cycle” or
“sweep”
each type of move has fixed
probability of being selected
New configuration
“block”
moves per cycle
Move each atom
once (on average)
Add to block sum
cycles per block
Compute block average
blocks per simulation
Compute final results
100’s or 1000’s
of cycles
Independent
“measurement”
Perform selected trial
move
Decide to accept trial
configuration, or keep
original
6
Monte Carlo in the Molecular Simulation API
User's Perspective on the Molecular Simulation API
Simulation
Space
Controller
IntegratorMC
Meter
Phase
Species
Boundary
Configuration
Potential
Display
Device
MCMove
 IntegratorMC (examine code)
• performs selection of type of move to conduct at each trial
 MCMove (examine code)
• methods to implement the Monte Carlo trial and make acceptance
decision
• different subclasses perform different types of moves
• usually several of these associated with IntegratorMC
7
Displacement Trial Move
1. Specification
 Gives new configuration of same volume and number of molecules
 Basic trial:
•
8
Displacement Trial Move
1. Specification
 Gives new configuration of same volume and number of molecules
 Basic trial:
• displace a randomly selected atom to a point chosen with uniform
probability inside a cubic volume of edge 2d centered on the current
position of the atom
Select an atom
at random
9
Displacement Trial Move
1. Specification
 Gives new configuration of same volume and number of molecules
 Basic trial:
• displace a randomly selected atom to a point chosen with uniform
probability inside a cubic volume of edge 2d centered on the current
position of the atom
2d
Consider a
region about it
10
Displacement Trial Move
1. Specification
 Gives new configuration of same volume and number of molecules
 Basic trial:
• displace a randomly selected atom to a point chosen with uniform
probability inside a cubic volume of edge 2d centered on the current
position of the atom
Consider a
region about it
11
Displacement Trial Move
1. Specification
 Gives new configuration of same volume and number of molecules
 Basic trial:
• displace a randomly selected atom to a point chosen with uniform
probability inside a cubic volume of edge 2d centered on the current
position of the atom
Move atom to
point chosen
uniformly in
region
12
Displacement Trial Move
1. Specification
 Gives new configuration of same volume and number of molecules
 Basic trial:
• displace a randomly selected atom to a point chosen with uniform
probability inside a cubic volume of edge 2d centered on the current
position of the atom
Consider
acceptance of new
configuration
?
13
Displacement Trial Move
1. Specification
 Gives new configuration of same volume and number of molecules
 Basic trial:
• displace a randomly selected atom to a point chosen with uniform
probability inside a cubic volume of edge 2d centered on the current
position of the atom
 Limiting probability distribution
• canonical ensemble
p (r N )dr N 
1   U (r N ) N
e
dr
ZN
Examine underlying
transition
probabilities to
formulate
acceptance criterion
?
• for this trial move, probability ratios are the same in other common
ensembles, so the algorithm described here pertains to them as well
14
Displacement Trial Move
2. Analysis of Transition Probabilities
 Detailed specification of trial move and transition probabilities
Event
[reverse event]
Probability
[reverse probability]
Select molecule k
[select molecule k]
Move to rnew
[move back to rold]
Accept move
[accept move]
Forward-step
transition
probability
1 1
  min(1,  )
N v
1/N
[1/N]
v = (2d)d
1/v
[1/v]
min(1,)
[min(1,1/)]
Reverse-step
transition
probability
1 1
  min(1, 1 )
N v
 is formulated to satisfy
detailed balance
15
Displacement Trial Move
3. Analysis of Detailed Balance
Forward-step
transition
probability
Reverse-step
transition
probability
1 1
  min(1,  )
N v
1 1
  min(1, 1 )
N v
Detailed balance
pi
pij
=
pj
pji
Limiting p (r N )dr N  1 e  U (r N ) dr N
ZN
distribution
16
Displacement Trial Move
3. Analysis of Detailed Balance
Forward-step
transition
probability
Reverse-step
transition
probability
1 1
  min(1,  )
N v
1 1
  min(1, 1 )
N v
Detailed balance
pi
e  U dr N
ZN
old
pij
=
pj
pji
 U
dr N
1 1
 e
 N  v  min(1,  )  
ZN
new
1 1
1 )


min(1,
 
 N v

Limiting p (r N )dr N  1 e  U (r N ) dr N
ZN
distribution
17
Displacement Trial Move
3. Analysis of Detailed Balance
Forward-step
transition
probability
Reverse-step
transition
probability
1 1
  min(1,  )
N v
1 1
  min(1, 1 )
N v
Detailed balance
pi
e  U dr N
ZN
old
pij
=
pj
pji
 U
dr N
1 1
 e
 N  v  min(1,  )  
ZN
new
e  U
old
 e
  e  U
1 1
1 )


min(1,
 
 N v

new
  (U new U old )
Acceptance probability
Displacement Trial Move
4a. Examination of Java Code
public void thisTrial(Phase phase) {
double uOld, uNew;
if(phase.atomCount==0) {return;}
18
//no atoms to move
int i = (int)(rand.nextDouble()*phase.atomCount); //pick a random number from 1 to N
Atom a = phase.firstAtom();
for(int j=i; --j>=0; ) {a = a.nextAtom();}
//get ith atom in list
uOld = phase.potentialEnergy.currentValue(a); //calculate its contribution to the energy
a.displaceWithin(stepSize);
//move it within a local volume
phase.boundary().centralImage(a.coordinate.position()); //apply PBC
uNew = phase.potentialEnergy.currentValue(a); //calculate its new contribution to energy
if(uNew < uOld) {
//accept if energy decreased
nAccept++;
return;
}
if(uNew >= Double.MAX_VALUE || //reject if energy is huge or doesn’t pass test
Math.exp(-(uNew-uOld)/parentIntegrator.temperature) < rand.nextDouble()) {
a.replace();
//...put it back in its original position
return;
}
nAccept++;
//if reached here, move is accepted
}
Have a look at a simple MC simulation applet
19
Displacement Trial Move
4b. Examination of Java Code
 Atom methods
public final void displaceWithin(double d) {
workVector.setRandomCube();
displaceBy(d,workVector);}
public final void displaceBy(double d, Space.Vector u) {
rLast.E(r);
translateBy(d,u);}
public final void translateBy(double d, Space.Vector u) {r.PEa1Tv1(d,u);}
public final void replace() {r.E(rLast);}
 Space.Vector methods
public void setRandomCube() {
x = random.nextDouble() - 0.5;
y = random.nextDouble() - 0.5;
}
public void E(Vector u) {x = u.x; y = u.y;}
public void PEa1Tv1(double a1, Vector u) {
x += a1*u.x;
y += a1*u.y;}
20
Displacement Trial Move
5. Tuning
 Size of step is adjusted to reach a target rate of acceptance of
displacement trials
• typical target is 50%
• for hard potentials target may be lower (rejection is efficient)
Large step leads to
less acceptance but
bigger moves
Small step leads to
less movement but
more acceptance
21
Volume-change Trial Move
1. Specification
 Gives new configuration of different volume and same N and sN
 Basic trial:
•
22
Volume-change Trial Move
1. Specification
 Gives new configuration of different volume and same N and sN
 Basic trial:
• increase or decrease the total system volume by some amount within ±dV,
scaling all molecule centers-of-mass in proportion to the linear scaling of
the volume
+dV
dV
Select a random
value for volume
change
23
Volume-change Trial Move
1. Specification
 Gives new configuration of different volume and same N and sN
 Basic trial:
• increase or decrease the total system volume by some amount within ±dV,
scaling all molecule centers-of-mass in proportion to the linear scaling of
the volume
Perturb the total
system volume
24
Volume-change Trial Move
1. Specification
 Gives new configuration of different volume and same N and sN
 Basic trial:
• increase or decrease the total system volume by some amount within ±dV,
scaling all molecule centers-of-mass in proportion to the linear scaling of
the volume
Scale all positions in
proportion
25
Volume-change Trial Move
1. Specification
 Gives new configuration of different volume and same N and sN
 Basic trial:
• increase or decrease the total system volume by some amount within ±dV,
scaling all molecule centers-of-mass in proportion to the linear scaling of
the volume
Consider acceptance
of new configuration
?
26
Volume-change Trial Move
1. Specification
 Gives new configuration of different volume and same N and sN
 Basic trial:
• increase or decrease the total system volume by some amount within ±dV,
scaling all molecule centers-of-mass in proportion to the linear scaling of
the volume
 Limiting probability distribution
• isothermal-isobaric ensemble
Examine underlying
transition
probabilities to
formulate
acceptance criterion

p (Vs)
N
)
1  U  (Vs) N )   PV N N
 e
V ds dV

Remember how volumescaling was used in
derivation of virial formula
27
Volume-change Trial Move
2. Analysis of Transition Probabilities
 Detailed specification of trial move and transition probabilities
Event
[reverse event]
Select Vnew
[select Vold]
Accept move
[accept move]
Probability
[reverse probability]
1/(2 dV)
[1/(2 dV)]
Min(1,)
[Min(1,1/)]
Forward-step
transition
probability
Reverse-step
transition
probability
1
 min(1,  )
2d V
1
 min(1, 1 )
2d V
 is formulated to satisfy
detailed balance
28
Volume-change Trial Move
3. Analysis of Detailed Balance
Forward-step
transition
probability
Reverse-step
transition
probability
1
 min(1,  )
2d V
1
 min(1, 1 )
2d V
Detailed balance
pi
Limiting
distribution
pij
=

p (Vs)
N
)
pj
pji
1  U  (Vs) N )   PV N N
 e
V ds dV

29
Volume-change Trial Move
3. Analysis of Detailed Balance
Forward-step
transition
probability
Reverse-step
transition
probability
1
 min(1,  )
2d V
1
 min(1, 1 )
2d V
Detailed balance

pi
   (U old + PV old ) old
V
e

N

)
N
pij
=
pj
pji


   (U new + PV new ) new
V
  1  min(1,  )    e
 
  2d V
N


Limiting
distribution

p (Vs)
N
)
)
N

  1  min(1, 1 ) 
 
  2d V


1  U  (Vs) N )   PV N N
 e
V ds dV

30
Volume-change Trial Move
3. Analysis of Detailed Balance
Forward-step
transition
probability
Reverse-step
transition
probability
1
 min(1,  )
2d V
1
 min(1, 1 )
2d V
Detailed balance
pi

   (U old + PV old ) old
V
e



e
)
N
pij
=
pj
pji


   (U new + PV new ) new
V
  1  min(1,  )    e
 
  2d V



  (U old + PV old )
V )
old N
 e
  (U new + PV new )
  exp   (U + PV ) + N ln(V new / V old ) 
)
N

  1  min(1, 1 ) 
 
  2d V


V )
new N
Acceptance probability
31
Volume-change Trial Move
4. Alternative Formulation
 Step in ln(V) instead of V
• larger steps at larger volumes, smaller steps at smaller volumes
1  U  (Vs) N )   PV N +1 N
Limiting
N
p (Vs)  e
V
ds d ln V
distribution


)
Trial move
d lnV
V new  V old e  )
Acceptance
probability
min(1,)
  exp   (U + PV ) + ( N + 1)ln(V new / V old ) 
 ln V )new   ln V )old + d  ln V )
32
Volume-change Trial Move
5. Examination of Java Code
public void thisTrial(Phase phase) {
double hOld, hNew, vOld, vNew;
vOld = phase.volume();
hOld = phase.potentialEnergy.currentValue() //current value of the enthalpy
+ pressure*vOld*Constants.PV2T; //PV2T is a conversion factor
double vScale = (2.*rand.nextDouble()-1.)*stepSize; //choose step size
vNew = vOld * Math.exp(vScale);
//Step in ln(V)
double rScale = Math.exp(vScale/(double)Simulation.D); //evaluate linear scaling
phase.inflate(rScale);
//scale all center-of-mass coordinates
hNew = phase.potentialEnergy.currentValue() //new value of the enthalpy
+ pressure*vNew*Constants.PV2T;
if(hNew >= Double.MAX_VALUE ||
//decide acceptance
Math.exp(-(hNew-hOld)/parentIntegrator.temperature+(phase.moleculeCount+1)*vScale)
< rand.nextDouble())
{
//reject; put coordinates back to original positions
phase.inflate(1.0/rScale);
}
nAccept++;
//accept
}
Have a look at a simple NPT MC simulation applet
33
Summary
 Monte Carlo simulation is the application of MC integration
to molecular simulation
 Trial moves made in MC simulation depend on governing
ensemble
• many trial moves are possible to sample the same ensemble
 Careful examination of underlying transition matrix and
limiting distribution give acceptance probabilities
• particle displacement
• volume change
 Next up: molecular dynamics