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Molecular Dynamics
Basic Idea
• Solve Newton’s equations of motion
mj
d 2rj
dt
2
 Fj ,
V
Fj  
,
rj
j  1,2, , N
• Choose a force field (specified by a potential V)
appropriate for the given system under study
• Decide a statistical ensemble to use, choice of
boundary conditions; collect statistics of
observables
Commonly Use Force Fields
• Lennard-Jones Potential
– For noble gas and generic fluids
• Tersoff, Brenner, Stillinger-Weber, 3-, 4body potentials
– For C, Si, Ge, …
• AMBER, CHARMM, GROMOS, MM4, etc
– For biomolecules
• GULP, DFT codes, etc
Example of potential used in
biomolecular modeling
2
2
ki
k i
V (r1,r2 , ,rN )    li  li ,0   
i  i ,0 

bonds 2
angles 2
Vn
 
1  cos(n   ) 
torsions 2

   4 ij
i j 

 
 ij
 rij


  ij  
qi q j 
     

 rij   4 0rij 
12
6
Ensembles
• Micro-canonical Ensemble
– Energy is fixed
• Canonical ensemble
– Need to use “thermostat” to fix temperature
• Langevin dynamics
• Nosé-Hoover
• Generalized Langevin
Langevin Dynamics
d 2rj
2

Fj
dt
m
ξ(t )  0,
ξ(t )ξ(t ')
T
  v j  ξ,
j  1,2, , N
2 kBT

I (t  t ')
m
How to correctly implement the white noise
on computer?
Nosé-Hoover Dynamics
rj  v j 
pj
,
m
p j  Fj   p j ,

1K
1
2
  2   1 , K   mv j
  K0 
j 2
Generalized Langevin
t
uF
r

 (t  t ')u(t ')dt ' 

 (t )  0,
 (t ) (t ')  i (t  t ')
†
Σ is known as self-energy
Observables, Statistics
• Equilibrium temperature (in micro-canonical
ensemble) by equipartition theorem.
1
1
kBT 
m j v 2j , ,   x, y, z
2
2
• Pressure of a fluid (for pair potential)
1
PV  NkBT 
d
 (r  r )  F
i j
i
j
ij
Where d is dimension, Fij is the force acting on particle i
from particle j.
Transport Coefficients
• The diffusion constant can be computed
through velocity correlation function


0
v(t )  v(0) dt  lim
t 
r(t )  r(0)
2t
2
 3D
Transport Coefficients
• Thermal conductivity can be computed
through energy-current correlation using
Green-Kubo formula; or nonequilibrium
simulation by directly computing the
energy current
Textbooks on MD
• M P Allen & D J Tildesley, “Computer
Simulation of Liquids,” (Oxford, 1987)
• D Frenkel & B Smit, “Understanding
Molecular Simulation,” 2nd ed (Academic
Press, 2002)
• A R Leach, “Molecular Modeling,
principles and applications” (Addison
Wesley Longman, 1996)
Tutorial Problem Set 12
• Prove the pressure formula (required a
great deal of knowledge of statistical
mechanics).