Extending the Time Scale in Atomically Detailed Simulations - Reviews in Computational Chemistry - page 392

¼ P ð q

2

l i ; i þ1
(Eq. [32]), which are computed as norms in Cartesian space. Therefore, it is
important to remove overall translations and rotations from the structures
along the trajectory, which can be done by imposing linear constraints using
the Eckart conditions: 112
X

The function S = q Y i Þ depends on the distances

0 X
j

y ij ð y ij y ij Þ¼

y ij ¼ 0 8 i ½ 35

¼ 1 ;...;

j L ¼ 1 ;...; L

where the vector y kl is the mass-weighted Cartesian coordinate of atom l in
structure k , and the vectors y ij represent the coordinates of a reference struc-
ture. Equation [35] provides 6 N linear constraints that are denoted by

s

im

with i ,6.

The gradients of the constraints

¼ 1 ; ... , N and m ¼ 1 ; ...

r s

im and unit vectors in their directions

0
im

g are coordinate independent. Therefore, they only need

be computed once at the beginning of the calculation. These unit vectors are
not necessarily orthogonal for a single structure

¼ðr s =jr s jÞ

im im

but can be
orthogonalized using a Gram-Schmidt procedure. 113 We denote this set of
orthogonalized vectors by

ð g

0
il

g

0
ik

6¼ d lk Þ

N
i

f g im g

1 .

¼

N
i 1 be the set of variable coordinates of the current trajectory

that satisfies the constraints. Let

Let f Y i g

¼

N
i

f d Y i g

1 be a proposed displacement of these
coordinates during the optimization process to generate a new trajectory
f Y i þ d Y i g

¼

N
i

1 . The components of the displacement that satisfy the con-
straints are given by

¼

X

d Y i ¼ d Y i

6 ð d Y i

g im Þ g im 8 i ½ 36

m ¼ 1 ;...;

N
i

A new trajectory with coordinates f Y i þ d Y i g

1 then satisfies the Eckart con- ¼

straints.
The SDEL algorithm has been efficiently parallelized using message pas-
sing interface (MPI) libraries. In the parallelization scheme each node of a clus-
ter of computers calculates the potential energy and derivatives for a particular
path segment. 114 Internode communication is not heavy and the computing
scales favorably with cluster size.
Several advantages of SDEL exist when compared to other methods:

1. The trajectories can be computed at room temperature or any other
temperature of interest and no bias potential is needed. This differs from
methods that rely on high temperature to accelerate the dynamics 115 or
those that modify the potential energy function to drive the trajectory to a
desired outcome. 115-117
2. Both the boundary conditions and the length parameterization enable one
to study very slow processes. This is demonstrated later.