Chemistry Reference
In-Depth Information
Figure 7
(a) Schematic energy diagram as a function of two coordinates
q
and
q
0
with
reactant region
A
and product region
B.
The chain of beads represents a discretized path
in the TPS method. (b) Schematic free energy along reaction coordinate
q.
(Adapted
from Dellago et al.
69
)
states can be elucidated, and rate constants can be determined using the time
derivative of the time correlation function:
64,69
h
h
B
ð
t
Þi
A
;
H
B
ð
T
Þ
dC
ð
t
Þ
Þ¼
h
h
A
ð
x
0
Þ
h
B
ð
x
t
Þi
k
TPS
AB
k
TPS
AB
t
0
Þ
ð
t
Þ¼
C
ð
t
ð
t
Þ¼
C
ð
h
ð
x
0
Þi
h
ð
t
0
Þi
A
;
H
B
ð
T
Þ
dt
h
A
h
B
½
23
in which
h
A
ð
x
Þ
are indicator functions that state if the system is in
the phase space regions
A
or
B
:
and
h
B
ð
x
Þ
h
A
ð
x
Þ¼
1if
x
2
A
else
h
A
ð
x
Þ¼
0
½
24
h
B
ð
x
Þ¼
1if
x
2
B
else
h
B
ð
x
Þ¼
0
Here
x
is a phase space vector,
H
B
ð
is
an average over the ensembles of paths that start in
A
and go to
B
at least
once during a fixed length
T.
An order parameter is introduced to describe
T
Þ¼
max
0<
t
<
T
h
B
ð
x
t
Þ
,and
hi
A
;
H
B
ð
T
Þ
