1
CE 530 Molecular Simulation
Lecture 23
Symmetric MD Integrators
David A. Kofke
Department of Chemical Engineering
SUNY Buffalo
kofke@eng.buffalo.edu
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Molecular Dynamics Integration
 Equations of motion in cartesian coordinates
dr j
dt
dp j
dt

pj
r  (rx , ry )
m
p  ( px , p y )
 Fj
2-dimensional space (for example)
N
F j   Fij
pairwise additive forces
i 1
i j
 Previously, we examined basic MD integrators
F
• Verlet family
Verlet; Leap-frog; Velocity Verlet
• Popular because of their simplicity and effectiveness
 Today we will consider
• Symmetry features that make the Verlet methods work so well
• Multiple-timestep extensions of the Verlet algorithms
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Integration Algorithms
 Features of a good integrator
•
•
•
•
minimal need to compute forces (a very expensive calculation)
good stability for large time steps
good accuracy
conserves energy and momentum
noise less important than drift
 The true (continuum) equations of motions display certain
symmetries
• time-reversible
• area-preserving (symplectic)
 Good integrators can be constructed by paying attention to
these features
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Symmetry
 An object displays symmetry if some transformation leaves
it (or something about it) unaltered
Original
shape
90º rotation
Horizontal
reflection
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Time Symmetry
 Path traced by a mechanical
system is unchanged upon
reversal of momenta or time
• e.g., motion in a constant
gravitational field
x(t )  x0  v0t  12 gt 2
v(t )  v0  gt
 Another view
dr j
• Substitution
dt
dp j
p  p
t  t
dt
leaves equations
of motion unchanged

pj
m
 Fj
An Irreversible Integrator
 Forward Euler
• well known to be quite bad
r (t  d t )  r (t )  p(t )d t  12 F(t )d t 2
unit mass
p(t  d t )  p(t )  F(t )d t
 Examine time reversibility
• Assume we have progressed forward in time an increment dt;
positions and momenta are now r f (t ), p f (t )
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An Irreversible Integrator
 Forward Euler
• well known to be quite bad
r (t  d t )  r (t )  p(t )d t  12 F(t )d t 2
unit mass
p(t  d t )  p(t )  F(t )d t
 Examine time reversibility
• Assume we have progressed forward in time an increment dt;
positions and momenta are now r f (t ), p f (t )
• Reverse time, and step back to original condition
rr (to  d t  d t )  r f (to  d t )  p f (to  d t )(d t )  12 F(to  d t )(d t ) 2
p r (to  d t  d t )  p f (to  d t )  F(to  d t )(d t )
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An Irreversible Integrator
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 Forward Euler
• well known to be quite bad
r (t  d t )  r (t )  p(t )d t  12 F(t )d t 2
unit mass
p(t  d t )  p(t )  F(t )d t
 Examine time reversibility
• Assume we have progressed forward in time an increment dt;
positions and momenta are now r f (t ), p f (t )
• Reverse time, and step back to original condition
rr (to  d t  d t )  r f (to  d t )  p f (to  d t )(d t )  12 F(to  d t )(d t ) 2
p r (to  d t  d t )  p f (to  d t )  F(to  d t )(d t )
• Insert from above for r f (to  d t ), p f (to  d t )
rr (to )  r f (to )  p(to )d t  12 F(to )d t 2   p(to )  F(to )d t  (d t )  12 F(to  d t )( d t ) 2


p r (to )  p f (to )  F(to )d t   F(to  d t )(d t )
An Irreversible Integrator
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 Forward Euler
• well known to be quite bad
r (t  d t )  r (t )  p(t )d t  12 F(t )d t 2
unit mass
p(t  d t )  p(t )  F(t )d t
 Examine time reversibility
• Assume we have progressed forward in time an increment dt;
positions and momenta are now r f (t ), p f (t )
• Reverse time, and step back to original condition
rr (to  d t  d t )  r f (to  d t )  p f (to  d t )(d t )  12 F(to  d t )(d t ) 2
p r (to  d t  d t )  p f (to  d t )  F(to  d t )(d t )
• Insert from above for r f (to  d t ), p f (to  d t ); Cancel
rr (to )  r f (to )  p(to )d t  12 F(to )d t 2   p(to ) 12F(to )d t  (d t )  12 F(to  d t )( d t ) 2


p r (to )  p f (to )  F(to )d t   F(to  d t )(d t )
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An Irreversible Integrator
 Forward Euler
• well known to be quite bad
r (t  d t )  r (t )  p(t )d t  12 F(t )d t 2
unit mass
p(t  d t )  p(t )  F(t )d t
 Examine time reversibility
• Assume we have progressed forward in time an increment dt;
positions and momenta are now r f (t ), p f (t )
• Reverse time, and step back to original condition
rr (to  d t  d t )  r f (to  d t )  p f (to  d t )(d t )  12 F(to  d t )(d t ) 2
p r (to  d t  d t )  p f (to  d t )  F(to  d t )(d t )
• Insert from above for r f (to  d t ), p f (to  d t ); Simplify
rr (to )  r f (to )  12  F(to  d t )  F(to ) d t 2
p r (to )  p f (to )   F(to  d t )  F(to ) d t
dt
• Equal only in limit of zero time step
Inequality indicates lack of time reversibility
Verlet integrators are time reversible
F(t )
F(t  d t )
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Symplectic Symmetry 1.
 Consider motion of a single particle in 1D
• Described using a 2D phase space (x,p)
• Hamiltonian H ( p, x)  12 p 2  V ( x)
• Equations of motion dx
H
dp
dt
• Phase space

p
p
dt

unit mass
H
 F ( x)
x
p
{p(t),x(t)} trajectory
x
• Forthcoming result are easily generalized to higher dimensional
phase space, but hard to visualize
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Symplectic Symmetry 2.
 Consider motion of a differential element through phase space
• Shape of element is distorted by motion
• Area of the element is preserved
p
p
x
p
p
x
p
x
x
p
x
x
 This is a manifestation of the symplectic symmetry of the equations
of motion
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Classical Harmonic Oscillator
 Exactly solvable example
dx
 p
dt
dp
 x
dt
H ( p, x)  12 p 2  12 x 2
x(0)  x0
p(0)  p0
 Solution
2
1
x   x0 cos t  p0 sin t
p   x0 sin t  p0 cos t
H (t )  12 p02  12 x02  constant
0
-1
-2
-2
-1
0
1
2
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Liouville Formulation 1.
 Operator-based view of mechanics
 Very useful for deriving symplectic, time-reversible
integration schemes
 Consider an arbitrary function of phase-space coordinates
• and thereby a function of time
f  x(t ), p (t ) 
f
f
• time derivative is f  x x  p p
 Define the Liouville operator
iL  x
 So


p
x
p
f  iLf
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Liouville Formulation 2.
 Operator form
f
 iLf
t
 This has the solution
f (t )  eiLt f (0)
• in principle this gives f at any time t
• in practice it is not directly useful
 Let f be the phase-space vector
f  Γ(t )   x(t ), p(t ) 
T
• then the solution gives the trajectory through phase space
 Harmonic oscillator
 x   cos t sin t   xo 
 p     sin t cos t   p 
  
 o 
 cos t sin t 
eiLt  

  sin t cos t 
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Separation of Liouville Operator 1.
 Position and momentum parts


p
x
p
 iLx  iL p
iL  x
 Propagator of either part can be solved analytically by itself
iLx Γ  x
  x   x

x  p   0 
 x(t )   x0  p0t 

 p(t )    p

 

0
  x 0

p  p   p 
 x (t )   x0 
 p (t )    p  p t 

  0
0 
iL p Γ  p
1 t 
eiLx t  

 0 1
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Separation of Liouville Operator 2.
 Liouville components do not commute
• result differs depending on order in which they are applied
 iLx   iL p    iL p   iLx 
 Otherwise we could write
iLxt iL p t
eiLt  e
iL p t
 eiLxt e
This is incorrect
 We could then apply each propagator in sequence to move
the system ahead in time eiLt Γ  eiLxt eiLpt Γ


Advance
momentum
Advance position
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Trotter Expansion
 The following relation does hold

e A B  lim e A / 2 P e B / P e A / 2 P
P 

P
• for example, if P = 2



  e A / 4e B / 2e A / 4e A / 4e B / 2e A / 4 
  e A / 4e B / 2e A / 2e B / 2e A / 4 
e A  B  e A / 4e B / 2e A / 4 e A / 4e B / 2e A / 4
 We cannot work with infinite P
• but for large P
e
A B

 e
 
2
A / 2P B / P A / 2P P O P
e
e

e
…plus a correction
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Formulating an Integrator
 Using the large-P approximation

e A B  e A / 2 P e B / P e A / 2 P

P
 To advance the system over the time interval T
• break Liouville operator into displacement parts iLx + iLp
iL T iL T
• write eiLT  e x p
• apply the Trotter expansion, and interpret T/P as a discretized
time dt
iLT
e


iL pd t / 2 iLxd t iL pd t / 2 P
 e
e
e
• application of eiLpd t / 2eiLxd t eiLpd t / 2 P times advances the system
(approximately) through T
 An integrator formulated this way will be both timereversible and symplectic
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Examination of Integrator 1.
 Consider effect of one time step on positions and momenta
e
iL pd t / 2 iLxd t iL pd t / 2 
e
x
 p
 
e
 First apply exp(iLpdt/2)
x(0)

x(0)  
 p(0)    p(0)  p(0) d t 

 
2
iL pd t / 2 
e
 Then apply exp(iLxdt)
e
 
  x(0)  p d2t d t 


  
d
t
d
t
 p (0)  p (0) 2   p (0)  p (0) 2 
iLxd t 
x(0)
 Finally apply exp(iLpdt/2) again
e
iL pd t / 2
 
 
 x(0)  p d t d t  
x(0)  p d2t d t
2


 p (0)  p (0) d t   p (0)  p (0) d t  p d t

2
2
2  
 


dt 
2 
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Examination of Integrator 2.
 One time step
e
 
x(0)  p d2t d t
x 

 p  
dt  p dt
 
p
(0)

p
(0)
2
2

iL pd t / 2 iLxd t iL pd t / 2 
e
e
 


dt 
2 
 Examine effect on coordinate and momentum

p (0)  p (0)  d2t F (0)  F
x(0)  x(0)  d2t x
 d2t 
 d2t  
 x(0)  x(0)d t  12 F (0) d t 
2
 It’s the Velocity Verlet integrator!
 Higher-order algorithms can be derived systematically by
including higher orders in the Trotter factorization
• Not appealing because introduces derivatives of forces
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A Deep Truth
 Verlet integrator replaces the true Liouville propagator by an
approximate one
iLpd t / 2 iLxd t iL pd t / 2
eiLd t  e
e
e
 We can make these equal by saying the approximate
propagator is obtained as the propagator of an approximate
Liouville operator
eiLd t   e
iLpseudot
 or
iLpd t / 2 iLxd t iLpd t / 2
e
e
e
iLpseudo  iL   / d t
 This corresponds to some unknown Hamiltonian
• and this Hamiltonian is conserved by the Verlet propagator
• the Verlet algorithm will not likely give rise to drift in the true
Hamiltonian, since this “shadow” Hamiltonian is conserved
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Other Decompositions
 Other choices for the decomposition of the Liouville
operator can be worthwhile
 Some choices:
• Separation of short- and long-ranged forces
• Separation of fast and slow time-scale motions
 Approach involves defining a reference system that is solved
more precisely (more frequently)
 Difference between real and reference is updated over a
longer time scale
 RESPA
• (Reversible) REference System Propagator Algorithm
• Uses numerical solution of reference
 NAPA
• Numerical Analytical Propagator Algorithm
• Uses a reference that can be solved analytically
Martyna, Tuckerman, Berne
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RESPA: Force Decomposition 1.
 Decompose force into short (Fs) and long (Fl) range
contributions
F ( x)  S ( x) F ( x)  1  S ( x)  F ( x)
 Fs ( x)  Fl ( x)
• S(x) is a switching function that turns off the force at some
distance
 Liouville operator



 Fs ( x )  Fl ( x )
x
p
p

 iLs  Fl ( x )
p
iL  x
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RESPA: Force Decomposition 2.
 Liouville operator
iL  iLs  Fl ( x)

p
 Trotter factorization of propagator
e
iLt
e
t F ( x ) 
2 l
p
e
iLs t
e
t F ( x ) 
2 l
p
 Decompose term treating short-range forces
e
iLs t

 e

iLs , p d2t
e
iLs , xd t
e
iLs , p d2t


n
t
dt 
n
 Long-range forces are computed n times less frequently than
short-range ones
• Long-range forces vary more slowly
• They are more expensive to calculate, because more pairs
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RESPA: Force Decomposition 3.
 Full RESPA propagator
e
iLt
e
t F ( x ) 
2 l
p
n
 d2t Fs ( x ) p d tx x d2t Fs ( x ) p  2t Fl ( x ) p
e
e
e
 e


 Procedure
• Repeat for n steps
update short-range force, evaluate new momenta
evaluate new positions
• Evaluate long-range forces, update momenta
• Repeat
t
dt 
n
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RESPA: Time-Scale Decomposition
 Many systems display disparate time scales of motion
• Massive particles interacting with light ones
helium in argon
• Stiff and loose potentials
intramolecular and intermolecular forces
 Approach works as before
• Integrate fast motions (degrees of freedom) using short time step
• Integrate slow motions using long time step