Chemistry Reference
In-Depth Information
Similarly
V
LJ
is the sum of Lennard-Jones contributions for all pairs of atoms,
and
V
C
is the sum of the Coulombic potential over all charge-charge interac-
tion pairs, although other mathematical functions are frequently used to
account for steric and electrostatic interactions:
X
X
V
LJ
ij
V
LJ
V
C
V
ij
¼
¼
allpairs
allpairs
The functional forms of these terms vary widely. Representative examples can
be found in work by a number of authors and have been reviewed previously
in this topic series.
12-20
A simple, detailed model is presented in the MD Test
Set project.
21
The molecular dynamics (MD) potential energy surface of even small
organic molecules is highly nonlinear, with many local minima. Minimization
of the potential energy is a common task, but the nonpolynomial prolifera-
tion of local minima usually frustrates attempts to determine the lowest
energy states for modeled systems.
22,23
Also, the finite-time dynamics of a
nonlinear multiple-minima system can become trapped in one potential
energy well, which in turn impedes complete conformational sampling. The
terms in the potential must account for a wide range of spatial scales (from
bonds of length 1
˚
10
10
m, to Coulombic lengths that extend throughout
the modeled system) as well as time scales (the fastest bonds have a period of
10 fs
¼
10
14
s, while large-scale conformational interconversions may occur
on the scale of seconds). Time-stepping algorithms like the Verlet method [4]
require a sufficiently short time step (0.5-1.0 fs) to resolve the fastest bonded
motion, meaning that a computed trajectory that spans a time interval of one
nanosecond
¼
10
9
s
requires one million dynamics steps. The great majority
of the computational work in MD simulations is expended in computing the
forces of interaction—for
N
particles, the computational effort is
O
ð
Þ
N
2
.
For simulations in which the long-range force comes only from the rapidly
decaying Lennard-Jones potential, this
N
2
bottleneck can be remedied by
imposing
distance cutoffs
where the potential is assumed to be zero for all
atomic separations greater than a predefined cutoff distance
r
c
.
The electrostatic 1
ð
Þ
r
2
forces, in contrast, are non-negligible at moderate
separations, thus making Coulomb distance cutoffs unphysical and undesir-
able on small- to medium-sized scales. Thus, a molecule's potential energy sur-
face has several characteristics that can impact significantly on the
performance of numerical methods: multiple minima, wide range of time
and space scales, and long-range interactions between many particles. For tra-
ditional molecular dynamics methods that use Eq [6], the most important lim-
itation involves the numerical stability of the integrator. While the
computational resources required for a given numerical simulation could
be lessened by increasing the length of each time step, stability of the time-
stepping algorithm is typically limited by the high-frequency vibrational
=
