1. Overview of Molecular Dynamics Simulation Theory
Dr. Mohamed Shajahan Gulam Razul

2. Introduction In the 1950’s two landmark pioneering publications:
The first by Metropolis et al.
N. Metropolis, A.W. Rosenbluth, M. N. Rosenbluth, A.H. Teller and E. Teller, J. Chem. Phys., 21, 1087 (1953).
The second by Alder and Wainwright
B. J. Alder and T. E. Wainwright, J. Chem. Phys., 27, 1208 (1957).
Indelibly laid the foundation for the Monte Carlo method and molecular
dynamics, respectively. The molecular simulation world was forever
changed
Computer simulations have provided a bridge between the abstract
theoretical construct of statistical mechanics to the macroscopic level of
experimental measurements of thermodynamics

3. So we have coordinates in
Theory
Consider a system of N atoms
Positions
Momenta
So we have coordinates in
6N Dimensional space
or phase space

4. Theory so a point in phase space can be given by
and an instantaneous property, Aint Aint()
therefore an experimentally observable “macroscopic”property
Aobs ,over a long time, tobs

5. Theory
The Ergodic Hypothesis states that the time average in the previous
slide can be given by an ensemble average.
The ensemble average for a closed system can be regarded as a
collection of points, , where in phase space

6. Theory Where Aobs via the ensemble average is given by
Partition function
Where Aobs via the ensemble average is given by
Now lets replace rens by a weight function, wens()

7. Theory Therefore in the the canonical ensemble (constant N,V,T)
rens is proportional to
with the partition function expressed in quasi-classical form
given by

8. Theory Many other connections are also made in different
ensembles (the basis of equilibrium statistical mechanics)
Connection of statistical mechanics and thermodynamics
The connection to a thermodynamic function is made through
the Helmholtz free energy

9. Theory Many other connections are also made in different
ensembles (the basis of equilibrium statistical mechanics)
Connection of statistical mechanics and thermodynamics
The connection to a thermodynamic function is made through
the Helmholtz free energy

10. Theory No let’s turn our attention to the link between statistical
mechanics and molecular dynamics
Liouville’s theorem: the rate of change of phase space density is
given by
An alternative formulation can be derived from a phase point
traveling in space, the rate of change of phase space density, ,
in this case is now 0,

11. Theory Force exerted on a particle i that includes all particles
in the system calculated by
pairwise interactions

12. Theory and 6N first order differential equations as opposed to
Newton’s 3N second order equations
The Newtonian equation can be written in Hamiltonian form,
and

13. Computation
There are two principal techniques utilized to solve the equations of motion: the predictor-corrector and the Taylor series methods.
All these methods allow one to obtain numerical solutions to differential equations.
Which means time had to be discretized, thus the origin
of the MD timestep

14. Computation
As an example, a general predictor-corrector algorithm is outlined below:
at a time t, use the current positions and current velocities and their time derivatives to predict new positions and velocities at a new time (t + t);
at the new positions, forces are evaluated;
utilizing these forces, correct the positions, velocities
and their derivatives;
d) finally, variables of interest are calculated and time
averages are performed; the process now iterates by
returning to step a).

15. Computation
As an example, a general predictor-corrector algorithm is outlined below:
at a time t, use the current positions and current velocities and their time derivatives to predict new positions and velocities at a new time (t + t);
at the new positions, forces are evaluated;
utilizing these forces, correct the positions, velocities
and their derivatives;
d) finally, variables of interest are calculated and time
averages are performed; the process now iterates by
returning to step a).

16. Computation
As an example, a general predictor-corrector algorithm is outlined below:
at a time t, use the current positions and current velocities and their time derivatives to predict new positions and velocities at a new time (t + t);
at the new positions, forces are evaluated;
utilizing these forces, correct the positions, velocities
and their derivatives;
d) finally, variables of interest are calculated and time
averages are performed; the process now iterates by
returning to step a).

17. Computation
As an example, a general predictor-corrector algorithm is outlined below:
at a time t, use the current positions and current velocities and their time derivatives to predict new positions and velocities at a new time (t + t);
at the new positions, forces are evaluated;
utilizing these forces, correct the positions, velocities
and their derivatives;
d) finally, variables of interest are calculated and time
averages are performed; the process now iterates by
returning to step a).

18. Simple Verlet algorithm
Computation
Simple Verlet algorithm

19. Computation
PBC

20. Computation
PBC

21. Computation
PBC

22. PBC helps model bulk systems, restricting the PBC to 2D or none at all
Computation
PBC
PBC helps model bulk systems, restricting the PBC to 2D or none at all
is also possible!

23. Computation Therefore the simulation “box” shape is important
PBC
Therefore the simulation “box” shape is important
other shapes are supported as well
example-dodecahedron

24. The minimum image convention
Computation
PBC
The minimum image convention

25. Computation
PBC

26. Computation PBC Usually half the box Length for short
range interactions
PBC

27. Computation
Short range
Long range

28. Computation Simple Short range Complicated Long range
Ewald method Particle mesh ewald

29. Computation Simple Short range Complicated Long range
Ewald method Particle mesh ewald

30. Simulation

31. Simulation
How?

32. Simulation
How?

33. Simulation
How?

34. Simulation

35. Key Properties
Temperature
Pressure