Chemistry Reference
In-Depth Information
Computing exact trajectories using Eq. [30] is more expensive than in
normal MD because the evaluation and optimization of the action entails
knowing the entire trajectory. However, if the aim is to obtain an approximate
trajectory with a large step size between successive structures, optimization of
Eq. [30] becomes a more feasible task and a more stable trajectory than in a
straightforward MD algorithm can be generated. An approximate trajectory is
computed from Eq. [30] numerically when a large step size is used, i.e., when
l
dl
. Specifically, after replacing
dl
!
l
a discrete version of the action in
Eq. [30] is obtained:
X
p
2
S
ð
E
V
ð
Y
i
ÞÞ
l
i
;
i
þ1
½
31
i
¼
0
;...;
N
where the action is now a function of the coordinates of the
N
intermediate
structures in the path,
N
i
f
Y
i
g
(with the coordinates of
the structures
¼
1
Y
0
l
i
;
i
þ1
is the mass-weighted distance
separating consecutive structures in the trajectory
Y
u
and
Y
N
þ1
Y
f
held fixed), and
. The
trajectory that makes
S
stationary is determined by optimization. The opti-
mized trajectory thus represents a sequence of structures connecting the initial
and final state of the system. Explicitly, after optimization the following
expression is obtained:
108
ð
l
i
;
i
þ1
¼j
Y
i
Y
i
þ1
jÞ
2
Y
i
q
Y
i
q
S
1
l
2
ð
Y
i
Þ
½r
V
ðr
V
e
i
Þ
e
i
¼
0
½
32
2
½
E
V
with
2
Y
i
2
Y
i
Y
i
þ
1
Y
i
1
Y
i
þ1
Y
i
1
j
Y
i
þ1
e
i
l
2
¼
and
½
33
l
2
Y
i
1
j
where
l
can be made a constant in the calculation and therefore independent of the
index
i.
The first term of Eq. [32] (equivalent to the acceleration term in
Newton's equation of motion, Eq. [1]) depends on the step size. At larger
step sizes, this ''acceleration'' contribution becomes smaller. In the limit
when this inertial term can be neglected, Eq. [32] becomes
e
i
is a unit vector tangential to the path at slice
i
, and the length step
d
1
S
d
Y
i
ð
Y
i
Þ
½r
V
ð
Y
i
Þðr
V
ð
Y
i
Þ
e
i
Þ
e
i
¼
0
2
½
E
V
!r
V
ð
Y
i
Þ½r
V
ð
Y
i
Þ
e
i
e
i
¼
0
8
i
½
34
Equation [34] generates a path in which the force is minimized in all directions
except the tangential direction of the path. This is one of the definitions of the
minimum energy path (MEP).
110
Equation [34] suggests that SDEL provides a
physically meaningful trajectory even at low resolution (large step sizes).
