Chemistry Reference
In-Depth Information
motion can be derived by computing a minimum (more exactly any extre-
mum). This method is called the least action principle. In the optimization
of an action procedure an initial guess for the trajectory is generated by con-
necting two boundary states, and the least action formalism is used to compute
a finite-temperature trajectory.
The first formulation of this methodology was based on a discretized ver-
sion of the classical action:
ð
t
t
0
Þ ¼
Ldt
0
S
½
X
ð
½
28
0
generating the Onsager-Machlup object function:
99-103
2
X
i
X
i
þ
X
i
1
2
X
i
t M
X
i
þ1
dV
dX
i
N
1
2
i
S
SDET
ðf
X
i
g
Þ
þ
¼
t
e
½
29
i
¼
1
t
2
In these equations,
X
is the coordinate vector for the system,
M
is the diagonal
mass matrix,
V
is the potential energy,
X
0
and
X
N
are the fixed boundary con-
formations in the trajectory, and
i
is an error variable. This algorithm, called
stochastic difference equation in time (SDET), has been used to compute
approximate
trajectories by using a large time step
e
t
for long-time events.
These paths are obtained by sampling trajectory space using molecular
dynamics or Monte Carlo techniques according to a Gaussian distribution
of errors (the term between parentheses corresponds to a finite-difference ver-
sion of Newton's equation of motion, i.e., Eq. [2]). Using similar time formal-
isms, Passerone and Parrinello,
104
Passerone et al.,
105
and Bai and Elber
106
have computed
exact
trajectories for relatively short but rare processes.
A variant of the SDET algorithm will be described below in more detail.
In this more recent formulation called SDEL (for stochastic difference equa-
tion in length) the trajectory is parameterized as a function of its arc length
and a unique path is obtained connecting the two boundary conforma-
tions.
97,98
In this sense, the SDEL algorithm is similar to DPS and string meth-
ods because trajectories are computed in configuration space instead of the
space parameterized by time as in normal MD, TPS, and SDET algorithms.
Boundary Value Formulation in Length
The SDEL algorithm allows the computation of atomically detailed tra-
jectories connecting two known conformations of the molecule over long time
scales. In contrast to normal and MTS molecular dynamics algorithms,
step sizes can be increased easily by two or three orders of magnitude without
