1. Fundamentals of Molecular Dynamics Simulations
Fundamentals of MD Simulations
Fundamentals of Molecular Dynamics Simulations
Presented by: Mohamad Raad, Ph.D. Student Dec.10 , Tutors: Dr. H. Behnejad
Faculty of Science, University of Tehran

3. Fundamentals of MD Simulations

7. Why MD simulations? Link physics, chemistry and biology
Model phenomena that cannot be observed experimentally
Understand protein folding…
Access to thermodynamics quantities (free energies, binding energies,…)

8. Introduction
Molecular dynamics is a form of simulations in which atoms and molecules are allowed to interact for a period of time by approximations of known physics, giving a view of the motion of the particles.
Main justification of the MD method is that statistical ensemble averages are equal to time averages of the system.
MD simulation generates a sequence of points in phase space connected in time, result in trajectory of all particles.
Use current positions, velocity and acceleration of atoms and find the position and velocity in next time step.
We combine these infinitesimal steps to find the trajectory of the particle.

9. Steps in Performing an MD Simulation
Selection of interaction model
Selection of boundary conditions
Selection of initial conditions (positions,
velocities )
Selection of ensemble (NVE, NVT, NPT )
Selection of target temperature,
density/pressure . . .
Selection of integrator, thermostat, barostat . . .
Perform simulation until equilibration is
reached (property dependent)
Perform production simulation to collect
thermodynamic averages, positions, velocities
Analyze the results via post-processing
Initialize: Initial particle
positions, velocities ...
Calculate forces from
particle positions
Loop
over
time-
steps
Solve Newton’s eqns.
of motion (integrate)
Perform analysis /
Write data to disk

10. Basic MD Simulation : The Idea
Follow the exactly same procedure as real experiments
Prepare sample
prepare N particles
solve equation of motions
Connect sample to measuring instruments (e.g. thermometer, viscometer,…)
after equilibration time, actual measurement begins
Measure the property of interest for a certain time interval
average properties
Example : measurement of temperature

12. How does MD work Initialize Extract data Approach to equilibrium
The algorithm of a standard MD simulation for studying systems in equilibrium is:
Initialize
Approach to equilibrium
Extract data

13. Initialize
Parameters in use: the number of particles, the form of interaction, the boundary condition, the temperature etc.
Positions, e.g. the sites of a Bravais-fcc lattice
Momenta, e.g. the Maxwell distribution

14. Approach to equilibrium
The Verlet algorithm (original form):
h: the time step, typically ~ seconds

15. Extract data
The expectation value of a static physical quantity is determined as a time average according to
The correlation function (in an isotropic system) is defined by
n(r): the average number of pairs within [r, r+Δr]

26. MD as a tool for minimization
Energy
Molecular dynamics
uses thermal energy
to explore the energy
surface
State A
State B
position
Energy minimization
stops at local minima

32. Fundamentals of MD Simulations
neighbor lists
force calculation is the most time-consuming part in a MD simulation
→ save CPU time by using neighbor lists
Verlet list:
cell list:
update when a particle has
made a displacement
> (rskin-rcut)/2
most efficient is a combination
of Verlet and cell list 34

33. Equation of Motion Classical Newton’s equation of motion
Three formulation
Newtonian
Lagrangian
Hamiltonian
Hamiltonian preferred for many-body systems
solution of 2N differential equations
Solution methods : Finite Difference Method

34. Newton‘s equations of motion
Fundamentals of MD Simulations
Newton‘s equations of motion
classical system of N particles at positions
Upot : potential function, describes interactions
between the particles
simplest case: pairwise additive interaction between point particles
solution of equations of motion yield trajectories of the particles, i.e.
positions and velocities of all the particles as a function of time 36

35. The Lagrange and Hamilton Forms
Fundamentals of MD Simulations
The Lagrange and Hamilton Forms
These equations also apply to the center of mass motion of a
molecule with fi representing the total force on molecule i.
The generalized momentum pk conjugate to qk is defined as
An alternative formulation uses Hamilton’s equations
For Cartesian coordinates Hamilton’s equations become

36. What the integration algorithm does
Fundamentals of MD Simulations
What the integration algorithm does
Advance the system by a small time step Dt during which forces are considered constant
Recalculate forces and velocities
Repeat the process
If Dt is small enough, solution is a reasonable approximation

37. Verlet algorithm

41. A widely-used algorithm: Leap-frog Verlet
Fundamentals of MD Simulations
A widely-used algorithm: Leap-frog Verlet
Using accelerations of the current time step, compute the velocities at half-time step:
v(t+Dt/2) = v(t – Dt/2) + a(t)Dt v
t-Dt/2 t
t+Dt/2
t+Dt
t+3Dt/2
t+2Dt

42. A widely-used algorithm: Leap-frog Verlet
Fundamentals of MD Simulations
A widely-used algorithm: Leap-frog Verlet
Using accelerations of the current time step, compute the velocities at half-time step:
v(t+Dt/2) = v(t – Dt/2) + a(t)Dt
Then determine positions at the next time step:
X(t+Dt) = X(t) + v(t + Dt/2)Dt v X
t-Dt/2 t
t+Dt/2
t+Dt
t+3Dt/2
t+2Dt

43. A widely-used algorithm: Leap-frog Verlet
Fundamentals of MD Simulations
A widely-used algorithm: Leap-frog Verlet
Using accelerations of the current time step, compute the velocities at half-time step:
v(t+Dt/2) = v(t – Dt/2) + a(t)Dt
Then determine positions at the next time step:
X(t+Dt) = X(t) + v(t + Dt/2)Dt v v X
t-Dt/2 t
t+Dt/2
t+Dt
t+3Dt/2
t+2Dt

52. Fundamentals of MD Simulations

54. CONSTANT TEMPERATURE (NVT) METHODS
Andersén method
Langevin thermostat
Dissipative Particle Dynamics thermostat
Nosé-Hoover method
Nosé-Hoover Chain Method

55. CONSTANT PRESSURE AND TEMPERATURE (NPT)
Andersén Pressure Control
Parrinello-Rahman Pressure Control

57. simulations at constant temperature: a simple thermostat
idea: with a frequency ν assign new velocities to randomly selected
particles according to a Maxwell-Boltzmann (MB) distribution
with the desired temperature
simple version: assign periodically new velocities to all the particles
(typically every 150 time steps)
algorithm:
take new velocities from distribution
total momentum should be zero
scale velocities to desired temperature

58. Where is the Physics? Potential energy functions Interaction force
Cut-off distance
Integrators
Energy conservation
Time reversibility
Physical constraints
Constant temperature
Constant pressure

60. How do you run a MD simulation?
For each time step:
Compute the force on each atom:
Solve Newton’s 2nd law of motion for each atom, to get new
coordinates and velocities
Store coordinates
Stop
X: cartesian vector
of the system
M diagonal mass matrix
.. means second order
differentiation with
respect to time
Newton’s equation cannot be solved analytically:
Use stepwise numerical integration

64. MC vs. MD MC Probabilistic simulation technique Limitations
require the knowledge of an equilibrium distribution
rigorous sampling of large number of possible phase-space
gives only configurational properties (not dynamic properties !) MD
Deterministic simulation technique
Fully numerical formalism
numerical solution of N-body system

65. Disadvantages: MD simulations compared with MC simulations (II)
More limited scale
Difficult to be incorporated with quantum mechanics
Advantages:
Dynamical phenomena
Nonequilibrium properties
Structural properties

66. Scales in Simulations Time Scale 10-6 S 10-8 S 10-12 S 10-10 M 10-8 M
continuum
mesoscale
Monte Carlo
Time Scale
10-6 S
molecular
dynamics
domain
10-8 S
quantum
chemistry
exp(- D
E/kT)
10-12 S
F = MA
Hy = Ey
10-10 M
10-8 M
10-6 M
10-4 M
Length Scale

68. M. Allen, D. J. Tildesley, Computer Simulation of Liquids
(Clarendon Press, Oxford, 1987)
D. Rapaport, The Art of Molecular Dynamics
(Cambridge University Press, Cambridge, 1995)
D. Frenkel, B. Smit, Understanding Molecular Simulation: From
Algorithms to Applications
(Academic Press, San Diego, 1996)
K. Binder, G. Ciccotti (eds.), Monte Carlo and Molecular Dynamics
of Condensed Matter Systems
(Societa Italiana di Fisica, Bologna, 1996)

69. Thank you for your attention!
J. M. Thijssen, Computational Physics, Cambridge University Press (1999)
T. Pang, An Introduction to Computational Physics, Cambridge University Press (1997)
and the references therein.
Thank you for your attention!