Essentials of Computational Chemistry

80 3 SIMULATIONS OF MOLECULAR ENSEMBLES
had by modeling the larger system stochastically. That is, the explicit nature of the larger
system is ignored, and its influence is made manifest by a continuum that interacts with the
smaller system, typically with that influence including a degree of randomness.
In Langevin dynamics, the equation of motion for each particle is
a(t) =−ζ p(t) + 1
m [Fintra(t) + Fcontinuum(t)] (3.29)
where the continuum is characterized by a microscopic friction coefficient, ζ , and a force,
F, having one or more components (e.g., electrostatic and random collisional). Intramolecular
forces are evaluated in the usual way from a force field. Propagation of position and
momentum vectors proceeds in the usual fashion.
In Brownian dynamics, the momentum degrees of freedom are removed by arguing that
for a system that does not change shape much over very long timescales (e.g., a molecule,
even a fairly large one) the momentum of each particle can be approximated as zero relative
to the rotating center of mass reference frame. Setting the l.h.s. of Eq. (3.29) to zero and
integrating, we obtain the Brownian equation of motion
r(t) = r(t0) + 1
ζ
t
t0
[Fintra(τ) + Fcontinuum(τ)]dτ (3.30)
where we now propagate only the position vector.
Langevin and Brownian dynamics are very efficient because a potentially very large
surrounding medium is represented by a simple continuum. Since the computational time
required for an individual time step is thus reduced compared to a full deterministic MD
simulation, much longer timescales can be accessed. This makes stochastic MD methods
quite attractive for studying system properties with relaxation times longer than those that
can be accessed with deterministic MD simulations. Of course, if those properties involve
the surrounding medium in some explicit way (e.g., a radial distribution function involving
solvent molecules, vide infra), then the stochastic MD approach is not an option.
3.4 Monte Carlo
3.4.1 Manipulation of Phase-space Integrals
If we consider the various MD methods presented above, the Langevin and Brownian
dynamics schemes introduce an increasing degree of stochastic behavior. One may imagine
carrying this stochastic approach to its logical extreme, in which event there are no equations
of motion to integrate, but rather phase points for a system are selected entirely at random.
As noted above, properties of the system can then be determined from Eq. (3.5), but the integration
converges very slowly because most randomly chosen points will be in chemically
meaningless regions of phase space.