Chemistry Reference
In-Depth Information
Fig. 2.4 Schematic
illustration of asymmetric
double minimum potentials.
a In condensed matter
intermolecular interactions
tune a symmetric potential.
b In liquid the potential is
continuously perturbed
through surrounding solvents.
c An alternative scheme of H
atom transfer in liquid. The
strong interaction causes the
significantly reduced barrier
where the H atom is
adiabatically transferred
The frequency of this oscillation m
t
, the so-called tunneling frequency, is given by
m
t
ΒΌ
j
H
12
j
p
This periodic oscillation is called the coherent tunneling.
Hydrogen (proton) tunneling has been derived from the theory of the one-
dimensional symmetric double oscillator [
1
]. The separation between the two
states are characterized by DE = hm
t
.(h is Planck's constant). The transfer time s
is related to the tunneling frequency by s = 1/2m
t
-1
. The tunneling splitting
strongly depends on the barrier height and width. The energy level splitting due to
coherent tunneling has been observed in many model systems such as small water
clusters [
2
], NH
3
[
3
], tropolone [
4
,
5
], and malonaldehyde [
6
-
8
] in the gas phase,
using rotation-vibration spectroscopy. The tunneling splitting, namely the tun-
neling frequency, varies from 10
12
Hz to a few Hz depending on the system.
2.1.3 Tunneling in an Asymmetric Double Minimum Potential
Surrounding environments or external perturbations alter the symmetry of the
potential landscape of H transfer. In condensed matter, intermolecular interactions
commonly vary the symmetric potential, resulting in an asymmetric potential as
depicted in Fig.
2.4
a. In this situation, the wave packet of hydrogen is forced to be
localized in one potential well. However, tunneling is still possible at the excited
state. This situation is depicted in Fig.
2.4
a. A vibrational excitation generates the
wave packet in the excited state, which is further transferred to another well via
tunneling. This process is no longer coherent.
In liquid, the situation becomes much more complicated. The effective symmetry
of the potential is continuously altered by surrounding solvents. This situation is
