Chemistry Reference
In-Depth Information
Fig. 2.2 Schematic diagram
of one-dimensional smooth
potential barrier
The transmission probability decreases exponentially with an increase of the
mass, m
1/2
, and the barrier width, L. Thus a significant isotope effect between H
and D (deuterium) is expected in the dynamical processes.
The one-dimensional rectangular barrier is an oversimplified model system.
The barrier that smoothly varies, as shown in Fig.
2.2
, is a more realistic question.
Using the WKB approximation, the tunneling probability D(E) of a particle that
has the total energy E in a one-dimensional potential barrier V(x) is expressed by
Z
x
2
p
2m
ð
V
ð
x
Þ
E
Þ
D
ð
E
Þ¼
exp
2
h
dx
ð
2
:
9
Þ
x
1
where x
1
and x
2
is the classical turning points, and (x
1
-x
2
) corresponds to the
barrier length. In the WKB approximation the Planck constant is regarded as a
small number. Therefore this approximation is called semi-classical and valid
when the total energy is significantly lower than the barrier. It is noted that the
WKB approximation is restricted to one-dimensional problems, thus it cannot be
applied to multi-dimensional problems that are much more likely in chemical and
biological processes.
In addition to the mass effect, the zero point energy (ZPE) must be considered
when we treat the kinetics of H and D. Assuming a harmonic potential, the ZPE is
given by
r
k
m
E
0
¼
1
2
hx
;
x
¼
ð
2
:
10
Þ
where x is the vibrational frequency and k is the spring constant. According to
(
2.10
), the ZPE of D atom is smaller by the factor of (1/2)
1/2
than that of H. This
reduction of the ZPE also lowers the transmission probability of D atom through
the increase of the effective barrier.
