Molecular dynamics (MD) is a widely
used tool in condensed matter physics, as well as other disciplines
ranging from chemistry to high-energy physics. In MD, one integrates
the equations of motion - Newton's second law for classical particles
- directly, invoking no approximations. To do so requires an interaction
potential energy or forces between atoms. I will discuss both integration
schemes and potential types. MD potentially bridges two length scales
- macroscopic and atomistic - and also links experimental results with
theories. This will be emphasized through a discussion of modern research
in solid-state physics, with each research application highlighting
a different type of interaction potential. I will discuss fluid flow
briefly and highlight some other applications of Lennard-Jones potentials,
surface growth using a Stillinger-Weber potential, and defects in silicon
using tight-binding potentials. I give some references regarding density-functional
theory calculations of these defects.
A different avenue of modern MD research
is in the method itself rather than its application. Much research is
towards developing so-called acceleration methods. By taking advantage
of the physics of condensed matter systems, acceleration methods have
been proposed which extend the time scales accessible by MD by orders-of-magnitude
in many cases. In this article, I will focus on A. F. Voter's methods,
giving their motivation, algorithm, and some derivations.
Molecular dynamics (MD) is a widely
used tool in condensed matter physics, as well as other disciplines
ranging from chemistry (Kityk et al., 1999) to high-energy physics (Bleicher
et al., 1999). This article focuses on applications to solid-state physics;
however, the basic concepts presented should be valuable to any newcomer
to MD, regardless of their field.
One of MD's appeals is that the system
in consideration is simulated directly - without assuming the nature
of transitions or the types of the structures. Moreover, due to increasing
computer power, MD provides atomistic detail for system sizes that are
exponentially increasing to mesoscopic scales. Already, with classical
potentials, simulations are performed with hundreds of millions of atoms
for hundreds of picoseconds (Kadau et al, 2002) or with a more accurate
tight-binding potential simulating about 64 atoms for 0.250 microseconds
(Richie et al., 2002) - nearly a timescale measurable on a stopwatch!
Note that MD bridges two scales - macroscopic
and atomistic - and thus links experimental results, say transport measurements,
with theories governing the interactions of the simulated matter's constituent
atoms. Indeed, conventional "pencil-and-paper" theory generally
cannot evaluate the properties that are of direct consequence in experiments.
In this light, MD forms another bridge, one between experiments and
conventional theory. By iterating the process of performing simulations
and experiments, theories of forces between atoms are refined.
With MD now appropriately evangelized,
the exposition of the fundamentals begins. Traditional (non-accelerated)
MD is conceptually simple. Using some potential interaction between
particles, Newton's second law updates the particles' velocities and
positions; repeating this, the equations of motion are integrated and
the system's trajectories are obtained. In order to be relevant at finite
temperature, pressure, etc., the system must be modified to ensure it
stays at a constant energy, temperature or in whichever thermodynamic
ensemble is desired. Methods of simulating the correct ensemble are
not given here but can be found in (Rapaport, 1997).
Before giving specific examples of research
into new MD methods (acceleration methods), I present the more technical
aspects of MD. Expositions of integration algorithms follow a background
in potentials. Accelerating MD motivates the parallel-replica, hyperdynamics,
and temperature-accelerated dynamics acceleration methods, as well as
on-the-fly kinetic Monte Carlo. Finally, illustrative research at the
forefront of MD simulation in solid-state physics is discussed; each
piece of research highlights a different class of potentials.
To accurately propagate the system, we must have an accurate interaction potential. This potential gives the potential energy as a function of the atomic positions and velocities. Details on existing potentials are given, as well as some general ways of deducing potentials.
POSITION INTEGRATION ALGORITHMS
Some details of potentials
and their calculation are now familiar. With the potential and initial
conditions Newton's second law can be integrated for all the particles.
The verlet and predictor-corrector (PC) methods are common for performing
The integration algorithm
usually does not need to be extremely accurate. Because an extremely
slight position displacement at any time (or an equivalent round-off
error) can cause huge differences in the atom's trajectory at all later
times after a certain time, only quantities which are insensitive to
exact trajectories "matter." This is not particular to MD,
but is characteristic of the natural process itself.
The verlet method gives positions at a short time t after the time corresponding to the supplied positions. Each is derived in a straightforward manner from the Taylor expansion, in time, of the atomic coordinates. The formula for the verlet propagator is (Rapaport, 1997) .
The verlet propagator is
commonly used for its simplicity and tendency to conserve energy.
The PC methods are more accurate
than verlet, but they are not used as frequently as the simpler verlet-class
propagators. The primary advantage of the PC method over the verlet-like
algorithms is in the ability to change t on the fly, which may be useful
in systems where one set of particles inherently move faster than others.
PC is also useful when constraints (on, say, bond length or angles)
are placed on the system (Rapaport, 1997).
The PC method predicts positions based upon Adams-Bashforth extrapolation, which is exact if they follow monic polynomials. After prediction, correction is made via a different set of formulae. For the (relatively complex) equations see (Rapaport, 1997).
BETTER (FASTER!) MOLECULAR DYNAMICS
Although MD is an increasingly
mature field, there are still continuous advances in methods. Voter
et al. have developed several methods for increasing MD's performance
by many orders of magnitude. Each method requires some assumptions -
usually forms of transition state theory (TST) (Voter, 2002 or Lombardo,
1991) - on the nature of transitions. However, the assumptions are minimal
and their validity can be checked.
There are three common families
of acceleration techniques, namely parallel-replica (PR or par-rep),
temperature-accelerated dynamics (TAD), and hyperdynamics. The families
can be utilized simultaneously for multiplicative performance boosts.
Voter gives an excellent, accessible review concentrating on these acceleration
methods (Voter, 2002).
The limitation that keeps
one from simulating long time scales is the fact that MD is a multi-scale
problem (for most solid-state systems). Specifically, one must use a
small enough integration time step to reproduce the dynamics of the
fast vibrational modes. Since these vibrations occur on the order of
1013 - 1014 times a second, the time step for accurate integration must
be on the order of femtoseconds. A typical time step falls in the range
of 1-5 fs.
On the other hand, transitions
occur infrequently; time scales between interesting transitions range
from picoseconds (quickly diffusing surface atoms) to seconds (dislocation
motion under shearing (Haasen, 1996)). One can conceive of watching
interesting behavior for minutes or hours (for example, in crystal growth),
however the longest MD simulations can now run for only microseconds.
It must be kept in mind that the acceleration methods discussed below
only apply to infrequent event systems, systems in which the time for
transitions between 'sites' or 'structures' is much longer than the
This review does not cover
all classes of research using MD - references are representative rather
than exhaustive. However, it should have given the reader an idea of
some typical applications - fluid dynamics, surface growth, and defect
dynamics. Through examples, connections between experiment, theory,
and MD are emphasized. One now hopefully has a feeling for the variety
and advantages of some potential types.
Also one should now be aware
of some of the growing number of molecular dynamics acceleration methods
- parallel-replica, temperature-accelerated dynamics, hyperdynamics,
and on-the-fly Monte Carlo. Perhaps the most important information presented
is the necessary background to understand what goes into an MD simulation.
With this and the references, a motivated individual could probably
code a simplistic MD simulator in a matter of a week (though given the
increasing number of sophisticated, fast MD programs, writing one from
scratch is probably not advisable except as a teaching tool).
ABOUT THE AUTHOR
Kaden Hazzard is a third-year undergraduate at The Ohio State University planning on graduating in the spring of 2004. He plans on pursuing a doctoral degree in condensed matter physics. He intends to pursue a research career at a university or at a national lab. He has been involved in computational condensed matter physics with Professor John Wilkins's research group since the summer of 2000. He has been researching a variety of topics, mainly defect evolution in silicon; surface growth in silicon; and structure recognition, characterization, and data mining of defects in solids. He has implemented several routines in the group's multi-scale materials simulator (OHMMS), and a new Monte Carlo method for calculating free energies. He also spent the summer of 2002 at Los Alamos National Labs doing low-temperature experimental work.
Everyone in Professor John Wilkin's research group who I have had the fortune to work with deserves thanks for their continued guidance, discussions, and ideas. In particular, I would like to thank Professor Wilkins for his continuous suggestions for improving this manuscript and my writing in general.
text regarding MD in general, along with many techniques especially
for simulations of liquids is
For those interested in finding out more about density-functional theory, a preprint for a good review article accessible to undergraduates who have taken some quantum mechanics, I suggest
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