Dr. Harpreet Singh
Head, PG Department of Bioinformatics,
Hans Raj Mahila Maha Vidyalaya,
Jalandhar, Punjab, India
e-module
Introduction to Molecular Dynamics Simulations
What is Molecular Dynamics (MD) ?
Numerical method for studying many-particle systems such as molecules, clusters, and even macroscopic systems such as gases, liquids and solids
Used extensively in materials science, chemical physics, and biophysics/biochemistry
First reported MD simulation:
Alder + Wainwright (1957): Phase diagram of a hard-sphere gas
Introduction
Calculate how a system of particles evolves in time
Consider a set of atoms with positions /velocities
and the potential energy function of the system
Predict the next positions of particles
over some short time interval
by solving Newtonian mechanics
Introduction
Set initial conditions and
Get new forces
Solve the equations of motion
numerically over a short step
Is ?
Calculate results and finish
Basic MD Algorithm
Constructing neighboring cells
Simulation Cell
Boundary Condition
Initial atom velocities
MD Time step
Temperature Control
Simulation Setup
Open boundary
for a molecule or nanocluster in vacuum
not for a continuous medium
usually using orthogonal cells
Fixed boundary
fixed boundary atoms
completely unphysical
Periodic boundary conditions
obtaining bulk properties
Simulation Cell
An atom moving out of boundary
comes back on the other side
considered in force calculation
Periodic Boundary Conditions
pair potential calculation
atoms move per time step
not necessary to search all atoms
Verlet neighbor list
containing all neighbor atoms within
updating every time steps
where
skin
Constructing Neighbor Cells
Linked cell method
divide MD cell into smaller subcells :
The length of subcell is chosen so that
: the length of MD cell
going through 27 atom pairs
instead
where
26 skin cells
reducing it to
Constructing Neighbor Cells
Constructing neighboring cells
Simulation Cell
Boundary Condition
Initial atom velocities
MD Time step
Temperature Control
Simulation Setup
The probability of finding a particle with speed
Maxwell-Boltzmann distribution
Generate random initial atom velocities
scaling T with equipartition theorem
Initial Velocities
1/20 of the nearest atom distance
In practice fs.
MD is limited to <~100 ns
Too long : energy is not conserved
MD Time Step
Velocity Scaling
Nose-Hoover thermostat
Scale velocities to the target T
Efficient, but limited by energy transfer
Larger system takes longer to equilibrate
Fictitious degree of freedom is added
Produces canonical ensemble (NVT)
Unwanted kinetic effects from T oscillation
Temperature Control
Verlet Method
Predictor-Corrector
Finite difference method
Numerical approximation of the integral over time
Better long-tem energy conservation
Not for forces depending on the velocities
Long-term energy drift (error is linear in time)
Good local energy conservation (minimal fluctuation)
Integration Method
From the initial ,
Obtain the positions and velocities at
Integration Method: Verlet
The force on an atom is determined by
: potential function
: number of atoms in the system
: vector distance between
atoms i and j
Integration Method: Force Calculation
Classical Potential
: Single particle potential
Ex) external electric field, zero if no external force
: Pair potential only depending on
: Three-body potential with an angular dependence
MD Potential
Born-Oppenheimer Approximation
Consider electron motion for fixed nuclei ( )
Assume total wavefunction as
: Nuclei wavefunction
: Electron wavefunction
parametrically depending on
The equation of motion for nuclei is given by
(approximated to classical motion)
Using Classical Potential
A snapshot of the MD Simulation Protocol
Empirical Potential
Semi-empirical Potential
Ab-initio MD
functional form for the potential
fitting the parameters to experimental data
Ex) Lennard-Jones, Morse, Born-Mayer
calculate the electronic wavefunction
for fixed atomic positions from QM
Ex) EAM, Glue Model, Tersoff
direct QM calculation of electronic structure
Ex) Car-Parrinello using plane-wave psuedopotential
MD Protocol Models
References and Links
Thanks