4.2 A Phenomenological Model for the Broadband Dielectric Response
135
of water.
7 In other words, the finite lifetime of H 2 O molecules depends on the rate of
the generation and recombination of excess protons (H 3 O
+ ) and proton holes (OH
− ).
The de Broglie wavelength is inversely proportional to particle momentum [14],
thus, tunneling becomes noticeable in small masses and at low temperatures. However, if the process is diffusion controlled (as in Frenkel’s concept of diffusion), the
conditions change, and tunneling during hydrogenation/deuteration is accompanied
by a semi-classical kinetic isotope effect with a negligible difference between the
transfer probability of hydrogen and deuterium. Hama et al. [14] suggest that for
this reason, the tunneling, which may account for the unexplained fast dynamics
in water and aqueous systems, has been unrecognized in a variety of experimental
studies. This, in particular, may apply to the relaxation part of the water spectrum
below 10 THz, which demonstrates no significant isotopic shift after deuteration (see
Sect. 2.7).
Bakker et al. [17] showed by pump–probe spectroscopy that the vibrational potential of the O–H stretch vibrations of water has extreme anharmonicity that arises from
the O–H· · · O interaction, and they observed that the energy required for this delocalization of the proton between the oxygen atoms of two neighboring water molecules
is unexpectedly low. They found that the dissociation energy of the O–H bond of the
water molecule in liquid water is at least 20% lower than that in the water vapor, which
explicitly assumes the quantum-mechanical nature of the proton transfer. It was later
confirmed [18] that Frenkel’s thermally activated motion of water molecules leads
to pronounced structural fluctuations, underlying processes ranging from sub-100 fs
librational motions to picosecond intermolecular transformations. This timescale is
in line with the results by Eigen [8], who also studied the dynamics at the same time
interval.
Using ultrafast IR spectroscopy, Carpenter et al. [19] showed that the transport
of an excess proton is followed by the reconstruction of its hydration shell,
8 which
goes through the orientational relaxation of molecular dipoles within τ rot ≈ 2.5 ps.
The authors attributed the slow decay of the hydrated proton anisotropy to the proton
intermolecular transfer kinetics that result in the loss of the memory of the prior
configuration within the τ rot . In other words, the hydration shell of the H 3 O
+ ion
relaxes to the equilibrium conditions after the perturbation caused by the intermolecular proton transfer in a picosecond time interval. The time decay τ rot corresponds
7 Interestingly, proton-transfer reactions have been found dramatically less common in simulations
based on an empirical force field [16]. The problem is that extreme fluctuations of the O–H stretching
vibrations are stabilized by the rearrangement of the electron density. This rearrangement is usually
unaccounted for by the empirical potentials, which are defined by the X-ray diffraction of molecules
of water vapor, where proton transfer is extremely rare. As a result, autoprotolysis is usually excluded
from standard simulations of molecular dynamics (see Sect. 1.6.). However, this does not mean that
the process is impossible in reality.
8 The hydration shell is the layer of water molecules around the ionic species with perturbed structure
and dynamics. The hydration shell provides an additional polarization degree of freedom in water
and aqueous solutions.
135
of water.
7 In other words, the finite lifetime of H 2 O molecules depends on the rate of
the generation and recombination of excess protons (H 3 O
+ ) and proton holes (OH
− ).
The de Broglie wavelength is inversely proportional to particle momentum [14],
thus, tunneling becomes noticeable in small masses and at low temperatures. However, if the process is diffusion controlled (as in Frenkel’s concept of diffusion), the
conditions change, and tunneling during hydrogenation/deuteration is accompanied
by a semi-classical kinetic isotope effect with a negligible difference between the
transfer probability of hydrogen and deuterium. Hama et al. [14] suggest that for
this reason, the tunneling, which may account for the unexplained fast dynamics
in water and aqueous systems, has been unrecognized in a variety of experimental
studies. This, in particular, may apply to the relaxation part of the water spectrum
below 10 THz, which demonstrates no significant isotopic shift after deuteration (see
Sect. 2.7).
Bakker et al. [17] showed by pump–probe spectroscopy that the vibrational potential of the O–H stretch vibrations of water has extreme anharmonicity that arises from
the O–H· · · O interaction, and they observed that the energy required for this delocalization of the proton between the oxygen atoms of two neighboring water molecules
is unexpectedly low. They found that the dissociation energy of the O–H bond of the
water molecule in liquid water is at least 20% lower than that in the water vapor, which
explicitly assumes the quantum-mechanical nature of the proton transfer. It was later
confirmed [18] that Frenkel’s thermally activated motion of water molecules leads
to pronounced structural fluctuations, underlying processes ranging from sub-100 fs
librational motions to picosecond intermolecular transformations. This timescale is
in line with the results by Eigen [8], who also studied the dynamics at the same time
interval.
Using ultrafast IR spectroscopy, Carpenter et al. [19] showed that the transport
of an excess proton is followed by the reconstruction of its hydration shell,
8 which
goes through the orientational relaxation of molecular dipoles within τ rot ≈ 2.5 ps.
The authors attributed the slow decay of the hydrated proton anisotropy to the proton
intermolecular transfer kinetics that result in the loss of the memory of the prior
configuration within the τ rot . In other words, the hydration shell of the H 3 O
+ ion
relaxes to the equilibrium conditions after the perturbation caused by the intermolecular proton transfer in a picosecond time interval. The time decay τ rot corresponds
7 Interestingly, proton-transfer reactions have been found dramatically less common in simulations
based on an empirical force field [16]. The problem is that extreme fluctuations of the O–H stretching
vibrations are stabilized by the rearrangement of the electron density. This rearrangement is usually
unaccounted for by the empirical potentials, which are defined by the X-ray diffraction of molecules
of water vapor, where proton transfer is extremely rare. As a result, autoprotolysis is usually excluded
from standard simulations of molecular dynamics (see Sect. 1.6.). However, this does not mean that
the process is impossible in reality.
8 The hydration shell is the layer of water molecules around the ionic species with perturbed structure
and dynamics. The hydration shell provides an additional polarization degree of freedom in water
and aqueous solutions.