Y Fan
et al
4
2.2. The 1D nature of the original ABC algorithm
Now let’s revisit the transition network discussed above with the
original ABC method. The ABC algorithm evolves the system
mainly towards the pathway with the lowest barrier, while
neglecting other, higherbarrier pathways. Therefore, for the
same system, ABC will provide an evolution chain that connects
the transition paths in 1D instead of a network (seen in figure
2
).
Since for this 1D chain, each state is only connected with
one state before it and one state after on the nodal network of
states, the residence time for state
p
is given by:
uni03C4
=
+

+
k
k
1
p
p p
p p
,
1
,
1
(6)
And similarly, the evolution time from
i
to
f
can be calcu
lated by solving the following linear equations:
Figure 2.
Illustration of the system evolution from one minimum energy state to another on the PES in the form of a nodal network,
with the initial state ‘
i
’ and the final state ‘
f
’. The collection of states and connectivities are the same as in figure
1
. An evolution chain is
provided by the ABC algorithm as the red path on the left, corresponding to the PES of a 1D chain of transitions illustrated on the right.
Figure 3.
(
a
) (Adapted with permission from [
32
]). A preconstructed 2D PES with a rough landscape, for representing the system
evolution from
i
to
f
with multiple competing processes. (
b
) A variation of the evolution time with temperature sampled by fullcatalog
kMC (with two different
r
c
), ABC, and ABCE.
uni239B
uni239D
uni239C
uni239C
uni239C
uni239C
uni239C
uni239C
uni239C
uni239C
uni239C
uni239C
uni239E
uni23A0
uni239F
uni239F
uni239F
uni239F
uni239F
uni239F
uni239F
uni239F
uni239F
uni239F
uni239B
uni239D
uni239C
uni239C
uni239C
uni239C
uni239C
uni239C
uni239C
uni239C
uni239C
uni239E
uni23A0
uni239F
uni239F
uni239F
uni239F
uni239F
uni239F
uni239F
uni239F
uni239F
uni239B
uni239D
uni239C
uni239C
uni239C
uni239C
uni239C
uni239C
uni239C
uni239C
uni239C
uni239C
uni239E
uni23A0
uni239F
uni239F
uni239F
uni239F
uni239F
uni239F
uni239F
uni239F
uni239F
uni239F
uni03C4
uni03C4
uni03C4

uni22EF
+

+
uni22EF
+

+
uni22EF
uni22EE
uni22F1
uni22EE
uni22EF
+

uni2192
uni2192
uni22EE
uni2192
=

uni22EE



k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
t
t
t
1
1
0
0
0
1
0
0
0
1
0
0
0
0
1
•
.
i
i
i
n n
n n
n f
i
f
f
n
f
i
n
1,
1,
1,2
1,2
1,
1,2
2,1
2,1
2,3
2,3
2,1
2,3
,
1
,
1
,
1
1
(7)
The dimension for the above matrix is the length of the 1D
chain, which is significantly smaller than the total number of
nodes in the system. In addition, the matrix has the banded
tridiagonal structure, which is much simpler than the matrix
in the last section for a full catalog of events and transitions.
kMC can also be employed based on this reduced matrix
to calculate the evolution time
uni2192
t
i
f
. Because of the reduced
dimension and simpler underlying mathematical structure,
the ABCbased kMC method saves substantial computational
load compared to the fullcatalog kMC calculations.