Chemistry Reference
In-Depth Information
and
m
i
the mass of the
i
th particle. The gradient
r
r
V
is the column vector of
all partial derivatives with respect to particle positions. It is easily verified that
the total energy
¼
r
T
Mr
2
E
þ
V
ð
r
Þ
½
2
is constant along solutions of Eq. [1]:
dE
=
dt
¼
0. The system is simulated from
Þ¼
r
0
, often chosen randomly in
accordance with some appropriate statistical ensemble.
Computer simulation of the system modeled by Eq. [1] requires some
sort of time discretization scheme. The method proposed by Verlet propagates
positions by
r
ð
initial positions and velocities
r
ð
t
0
Þ¼
r
0
,
t
0
r
n
þ1
i
¼
r
n
1
i
2
r
i
þ
h
2
M
1
F
i
þ
½
3
and velocities using
v
i
¼ð
r
n
þ1
r
n
1
i
Þ=
2
h
½
4
i
Here the superscripts denote the indices of time steps, each of which is of size
h
,so
r
i
r
i
ð
t
0
þ
nh
Þ
½
5
F
i
¼r
r
i
V
ð
r
n
and
is a Cartesian vector that gives the sum of forces
acting on particle
i
due to interaction with other particles, evaluated at the
point
r
n
.
Verlet
2
noted in his ground-breaking work the remarkable energy preser-
vation properties of the integrator, reporting ''small irregularities in the total
energy
Þ
but the error is of no consequence.'' The discretization method in
Eqs. [2]-[4], commonly referred to as the
Verlet integrator
, is accurate to sec-
ond order in time, requires only one force evaluation per step, and is
time
reversible
, which is part of the reason for its excellent stability in terms of
near conservation of energy. In fact, it is now known that a more general sym-
metry preservation—the
symplectic
property
6
—of the Verlet method, viewed
as an appropriate mapping of positions and momenta, confers its excellent
long-term energy stability.
7,8
For a thorough review of symplectic numerical
methods, see the monograph of Sanz-Serna and Calvo.
9
The Verlet method
is now regarded as the gold standard for time-stepping schemes in molecular
dynamics. In conformity with modern practice, and to anticipate the algorith-
mic development of multiple time-step methods in the forthcoming sections of
...
