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A bond-order analysis of the mechanism for hydrated proton mobility
in liquid water
Hadas Lapid and Noam Agmon
a)
Department of Physical Chemistry and the Fritz Haber Research Center, The Hebrew University,
Jerusalem 91904, Israel
Matt K. Petersen and Gregory A. Voth
b)
Department of Chemistry and the Center for Biophysical Modeling and Simulation, University of Utah,
Salt Lake City, Utah 84112-0850
共Received 8 July 2004; accepted 20 September 2004; published online 14 December 2004兲
Bond-order analysis is introduced to facilitate the study of cooperative many-molecule effects on
proton mobility in liquid water, as simulated using the multistate empirical valence-bond
methodology. We calculate the temperature dependence for proton mobility and the total effective
bond orders in the first two solvation shells surrounding the H
5
O
2
⫹
proton-transferring complex. We
find that proton-hopping between adjacent water molecules proceeds via this intermediate, but
couples to hydrogen-bond dynamics in larger water clusters than previously anticipated. A two-color
classification of these hydrogen bonds leads to an extended mechanism for proton
mobility. © 2005 American Institute of Physics. 关DOI: 10.1063/1.1814973兴
I. INTRODUCTION
Proton mobility in liquid water has attracted much atten-
tion in the last century,
1
with efforts intensifying during the
last decade. This is demonstrated, for example, by the num-
ber of papers on this subject in one special issue.
2–5
The
efforts to elucidate the mechanism of proton mobility in wa-
ter are motivated by the role protons play in acid-base reac-
tions in aqueous solutions, in environmental chemistry, and
in bioenergetics, where energy is transiently stored as trans-
membranal proton gradients.
6
A mechanism for proton mobility was suggested to com-
prises the following ingredients.
1,7,8
共i兲 Cyclic isomerization between the two forms of pro-
tonated water: 共a兲 The more stable H
3
O
⫹
is transiently con-
verted into H
5
O
2
⫹
and back;
1
or else 共b兲 one H
5
O
2
⫹
converts
directly into another.
7
共ii兲 This interconversion is coupled to hydrogen-bond
共HB兲 dynamics in the second solvation shell of the H
3
O
⫹
.
Since the coordination number of H
3
O
⫹
is 3 whereas
that of liquid water is close to 4, it was conjectured that the
transfer event is preceded by HB cleavage to the acceptor
water molecule, and followed by HB formation to the donor
molecule.
1
While this mechanism has found its way into
physical chemistry textbooks,
9
the issue is still under active
investigation through molecular dynamics 共MD兲 simulations.
MD simulations of protonated water are complicated by
the need to find a good representation for the interaction
potential. Two major approaches have been invoked: calcu-
lation of the potential ‘‘on the fly,’’ at every time step, using
density functional theory,
7,8
and use of multistate empirical
valence bond 共EVB兲 potentials.
10–20
The second method is less costly computationally, hence,
allows one to run sufficiently long trajectories for gathering
the required statistics. The initially implemented two-state
EVB 共Refs. 11 and 12兲 has been extended into multistate
共MS兲 EVB, with parameters calibrated to reproduce ab initio
data on small protonated water clusters and for the proton
solvated in bulk water.
15–20
The MS-EVB potential allows
for proton delocalization among several water molecules in a
deterministic fashion. There are important differences be-
tween the MS-EVB approaches of Refs. 13–15 and 16–20
so the reader is referred to these papers for more detail. Ad-
ditional methodologies are rapidly accumulating,
21–23
but
they are of a more phenomenological nature.
Early simulations actually indicated that proton migra-
tion involves a concerted double proton transfer 共PT兲 event,
which converts one H
5
O
2
⫹
moiety into another directly, with-
out a special role for the H
3
O
⫹
cation.
2,7,14,23
However, more
recent Car-Parrinello 共CP兲 path-integral simulations
8
indi-
cated the dominance of single PT events. These convert a
H
3
O
⫹
into a H
5
O
2
⫹
moiety and back, as first suggested in
Ref. 1.
The question of single vs double PT events (H
3
O
⫹
to
H
5
O
2
⫹
vs H
5
O
2
⫹
to H
5
O
2
⫹
) depends on the relative stability of
the two cations. Whereas in Ref. 7 they are nearly isoener-
getic, in the MS-EVB algorithm used below
17,18,20
H
3
O
⫹
is
more stable than H
5
O
2
⫹
by about 1 kcal/mol. One expects
that in trajectories applying this potential, H
5
O
2
⫹
features as a
transient intermediate structure between two more stable
H
3
O
⫹
structures.
This difference in stability, in turn, may depend on dif-
ferences in HB strengths for the different 共protonated兲 water
potentials, with stronger HBs favoring the H
3
O
⫹
ion. For
example, in a recent analysis of the CP methodology for
liquid water,
24
it has been shown that use of a larger fictitious
electronic mass
in the CP Lagrangian artificially produces
a兲
Author to whom correspondence should be addressed. Electronic mail:
agmon@fh.huji.ac.il
b兲
Electronic mail: voth@chem.utah.edu
THE JOURNAL OF CHEMICAL PHYSICS 122, 014506 共2005兲
122, 014506-10021-9606/2005/122(1)/014506/11/$22.50 © 2005 American Institute of Physics
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good self-diffusion coefficients D
water
and radial distribution
functions g(r). When a smaller
is used to better ensure
that the trajectory remains near the Born-Oppenheimer sur-
face, g(r) from the CPMD simulations of water, using two
popular generalized gradient approximations for the density
functional, become too structured compared to experiment
and D
water
decreases significantly relative to the correct
value. These calculations indicate that the HBs produced by
using larger
values are evidently accidentally weaker 关and
this reduces the structure in g(r) and enhances D
water
]. A
large
value 共1100 a.u.兲 was used in the CP simulation of
Ref. 7.
It remains to consider the more difficult question,
whether and how the PT events couple to HB dynamics. The
mechanisms discussed in the literature
1,7,8
suggested that the
rate determining step is cleavage of the HB donated from a
second-shell water molecule to the first-shell proton accep-
tor. A variant of this scenario was indeed observed in simu-
lations of proton mobility in ice.
25
Instead of complete cleav-
age and formation of HBs, impeded by the rigidity of the ice
structure, these simulations showed an interplay between
weakening 共increased length兲 followed by the strengthening
共decreased length兲 of the two red HBs in Fig. 1共b兲.
In protonated liquid water simulations, efforts were
made to follow the coordination number of the acceptor and
donor water molecules,
2
or else the angle between the donor,
acceptor, and the HB donated to it.
19,20
While some evidence
for the suggested role of this HB was detected, the average
change in the coordination number was much smaller than
unity. These results indicate that the rate-limiting step for PT
does not reside solely in the specified HB, as previously
suggested.
1,7–9
The assumption that may break down here is that the
first-shell water ligands behave like bulk water, possessing a
coordination number of 3.9. Indeed, femtosecond pump-
probe near-IR measurements suggest that the first-shell water
molecules around cations or anions exhibit slower reorienta-
tional times than bulk water,
26
indicating stronger HBs. Simi-
larly, the three first-shell neighbors of a H
3
O
⫹
ion must form
extra-strong HBs to it.
27
This helps in delocalizing 20%–
30% of the positive protonic charge on the three first-shell
ligands.
20
Consequently, it becomes electrostatically unfavor-
able to donate a HB to these oxygen atoms.
28
The missing
HB leads to the reduction in coordination number, from 3.9
in bulk water to about 3.6 for the first-shell ligands.
3,20
Therefore, cleavage of a HB donated to the acceptor
water molecule cannot be rate limiting, because it simply
does not exist for 40% of the time. Rather, evidence for such
behavior has been detected one water layer further away.
20
Since there are four first-shell neighbors to a H
5
O
2
⫹
, and
these are further engaged in up to 12 additional HBs, this
means that a full description of proton mobility in liquid
water involves the participation of larger water clusters than
previously anticipated, at least as large as depicted in Fig.
1共c兲. This agrees with earlier observations of Ohmine and
collaborators
29,30
that dynamic processes in water are driven
by large-scale collective motions.
The problem is to follow many HBs simultaneously and
average their effect in an appropriate manner. This is ad-
dressed by the ‘‘bond order analysis’’ 共BOA兲 proposed in the
present work. Utilizing it we are able to obtain concrete in-
sight into the microscopic mechanism of proton mobility, at
least within the MS-EVB2 model.
The present work is structured as follows: After briefly
reviewing the simulation methodology 共Sec. II兲, we present
simulation results on the temperature dependence of the pro-
ton diffusion coefficient 共Sec. III兲. From it we conclude that
within a limited range of temperatures 共say 280–310 K兲,
there is little change in the mechanism of proton mobility. To
reduce thermal noise, we choose to work at 280 K. The prin-
ciples of BOA are then outlined in Sec. IV. Its main results
are described in Sec. V. These are based predominantly on
the notion of the total effective bond order 共TEBO兲, which
invokes a two-color classification of HBs. We conclude 共Sec.
VI兲 by suggesting an extended version for the mechanism of
proton mobility in water, which involves the collective reor-
ganization of both types of HBs in the first- and second-
solvation layers of the transferring H
5
O
2
⫹
complex.
FIG. 1. 共Color兲 The proton-transferring complex H
5
O
2
⫹
, and its first two
solvation shells. In the first shell 关panel 共b兲兴, six HBs are tracked 共see Fig. 3
for their color codes兲. In the second shell 关panel 共c兲兴, 12 HBs are tracked.
The two unfavorable HBs from the first shell 共red兲 are not followed onto the
second shell.
014506-2 Lapid
et al.
J. Chem. Phys. 122, 014506 (2005)
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II. SIMULATION METHODOLOGY
Classical MD simulations of a single proton in a cubic
box of 125 water molecules were run using the Schmitt and
Voth MS-EVB program,
4,16,17
version 2.
20
The box linear di-
mension was 15.6 Å 共corresponding to a density of 1.0 gr/cc
at 300 K兲, and periodic boundary conditions were imposed
on its walls. Using time steps of 0.5 fs, a trajectory was
equilibrated for at least 150 ps 共300000 time steps兲 at the
desired temperature 共NVT ensemble兲. Five different tem-
peratures were used for the calculation of the proton diffu-
sion coefficient D
H
⫹ , whereas the mechanistic study was
performed at a single temperature, T⫽ 280 K. After setting
the temperature, the thermostat was turned off and the trajec-
tory was continued at constant energy 共NVE ensemble兲.
In order to quantify D
H
⫹ , the center-of-excess-charge
共CEC兲 coordinate was utilized in the MS-EVB methodology
as outlined in Ref. 20. The coordinates of the CEC were
tracked from time step to time step. If, at any given time
step, the coordinates varied from the previous step by the
approximate length of a periodic image they were accord-
ingly adjusted. This constructed a trajectory where the CEC
effectively diffused through an infinite space of periodic im-
ages relative to an origin in the original MD cell.
D
H
⫹ was calculated at the five temperatures using simi-
lar procedures. For example, at 275 K, 30 starting configu-
rations were collected from a 300 ps NVT trajectory every
10 ps. These configurations were then used to start 500 ps
NVE runs. The mean squared displacement was calculated
for each NVE trajectory and a least squares fit was obtained
between 10 and 50 ps. This interval was chosen so the mea-
surement would be above the nonlinear regime and below
the noisiest portion of the line. The slope was used to esti-
mate D
H
⫹ . This value was then averaged over the 30 runs.
For the mechanistic study, 13 NVE trajectories, of length
30 ps each, were run after separate NVT equilibrations. The
atomic coordinates were saved every 25 fs. These were
remapped onto the central unit cell with the H
3
O
⫹
at the
origin. Any water molecule that got broken across the peri-
odic boundaries was reconnected. This gave a corresponding
trajectory file which was suitable for visualization and
analysis.
Using a visualization program 共gOpenMol version 2.2,
by Leif Laaksonen兲, PT events between adjacent water mol-
ecules were identified. To avoid possible correlations be-
tween events, only the first few events were considered from
each 30 ps trajectory 共which was followed by an equilibra-
tion period兲. Aborted or incomplete transfers were not in-
cluded in this study. We have thus collected an arbitrary set
of 25 clear-cut PT events for analysis.
For each PT event, a trajectory segment was rerun start-
ing at 1–2 ps before and ending 1–2 ps after the event. The
coordinates were then saved at 5 fs intervals, to provide a
more detailed picture of the dynamics, and remapped onto
the central cell as above. These refined trajectories form the
data base for our mechanistic study.
Each of the refined trajectories is characterized by one
H
5
O
2
⫹
moiety 共composed of the donor and acceptor water
molecules兲. Its first- and second-shell neighboring water
molecules were identified from the atomic Cartesian coordi-
nates via a minimal distance criterion. This procedure was
repeated every time step, so that if a solvent and bulk water
molecules got interchanged the new solvent molecule was
followed. As depicted schematically in Fig. 1, the coordi-
nates of a total of 20 water molecules are tracked every time
step. From them, we calculate the 20 HB distances shown in
the figure as dashed lines. The HBs are separated into donor
and acceptor types, as indicated by the blue and red colors in
Figs. 1共b兲 and 1共c兲.
III. THE TEMPERATURE DEPENDENCE
OF PROTON MOBILITY
Previous MS-EVB simulations have calculated the protic
CEC mean-square displacement, and hence D
H
⫹ ,at
T⫽300 K 共e.g., Fig. 4 in Ref. 2 and Fig. 8 in Ref. 20兲. Here
we extend these studies to obtain the temperature depen-
dence of D
H
⫹ for the MS-EVB2 model over a range of tem-
peratures, 260–320 K.
Figure 2 shows our results for five temperature values
共squares兲. Due to the classical nuclear dynamics imple-
mented here, the absolute value of D
H
⫹ is too small in com-
parison with experiment. It should be noted that quantization
of the MS-EVB model, using the path integral centroid mo-
lecular dynamics method, reproduces this quantum effect
very well, which is mainly due to simple mode quantization
of the hydrogen bond.
17–19
The ratio of classical to quantum
proton hopping rates was shown in earlier calculations to be
0.56 共Table III in Ref. 17兲. Additionally, there may be some
small contribution to the diffusion rate from correlated pro-
ton hops over more than two water molecules that are not
completely captured by the MS-EVB2 model. Here we find
that when we scale the experimental data
31
by a factor of
0.43, they coincide with our calculated diffusion constants
共see dashed curve兲. Interestingly, the temperature depen-
dence is curved, and this curvature appears to be reproduced
by our calculation, even though the parameters of the MS-
EVB model
20
were adjusted at 300 K.
FIG. 2. Temperature dependence of the proton diffusion coefficient in water
共in units of Å
2
/ps) as calculated from classical MS-EVB2 trajectories, with
their corresponding error bars indicated. The dashed line represents a fit to
the experimental data 共Ref. 31兲, multiplied by 0.43. The slope of the straight
dotted line gives the activation energy of 2.7⫾0.1 kcal/mol.
014506-3 Hydrated proton mobility in water J. Chem. Phys. 122, 014506 (2005)
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A straight line through the calculated points around room
temperature give an Arrhenius activation energy of 2.7 kcal/
mol, slightly larger than the experimental value of about 2.5
kcal/mol.
27
With the inclusion of quantum effects, the calcu-
lated value is expected to decrease by up to 0.4 kcal/mol 共see
Fig. 10 in Ref. 17兲.
It is interesting that a simplified EVB implementation
gets a similar activation energy.
22
This could indicate, as ar-
gued below, that the activation energy for proton mobility
reflects the strength of the HB between bulk water
molecules.
1
As such, the underlying water potential may be
the most crucial element in determining an accurate value for
the activation energy. It should be noted, however, that other
important quantities, such as the actual value of D
H
⫹ 共i.e., its
pre-exponential factor兲 as well as the binding and spectro-
scopic properties of the excess proton, are likely to be more
sensitive to the overall physical accuracy of the model.
IV. BOND-ORDER ANALYSIS
We introduce bond-order 共BO兲 analysis in order to quan-
tify the HB environment around each oxygen atom. The BO
provides a ‘‘gray scale’’ description of HBs, which replaces
their conventional all-or-none definition in terms of cutoff
distances and angles. In addition, it allows us to sum the
contribution from several HBs within a solvation shell, gen-
erating a small number of parameters which we use to de-
scribe the PT process.
Following Pauling,
32
the BO n is related exponentially
to the bond length r,
n⫽exp
关
⫺
共
r⫺r
eq
兲
/a
兴
, 共1兲
where r
eq
is ‘‘the’’ equilibrium bond length, which we take
as the value for OH in gas-phase water, 0.956 Å. The param-
eter a, according to Ref. 33, is 0.35 Å 共the exact value is
immaterial for the qualitative analysis described herein兲. The
Pauling BO varies smoothly between covalent and hydrogen
bonds, with stronger bonds having larger BOs. Typical val-
ues are given in Table I. It has been observed that in adjacent
covalent/hydrogen bond pairs, O–H¯ O, the total BO is
conserved.
33–36
Here we generalize the definition of BO to give the total
effective BO 共TEBO兲 m around an oxygen center. Since we
are interested in protonated water clusters, such as shown in
Fig. 1, we view the HBs as emanating from the protonated
center. As we ‘‘walk’’ out from this center along the HB
network, HBs that stabilize it are directed from hydrogen to
oxygen. Consider, for example, the hydrogen atom H
*
in
Fig. 3, which is hydrogen bonded to oxygen O
*
in the water
molecule H
2
O
*
. We wish to characterize the effective coor-
dination number of this H
2
O
*
due to all other HBs 共i.e.,
excluding H
*
¯ O
*
H
2
).
Typically there are up to three such bonds, two of which
are donated by the hydrogens of H
2
O
*
共their BOs are de-
noted n
1
and n
2
), whereas the third 共denoted n
3
) is accepted
by O
*
. From the perspective of transferring the proton H
*
to
the nearest oxygen atom O
*
, n
1
and n
2
represent favorable
interactions, which stabilize the transferring proton. In con-
trast, n
3
is an unfavorable, destabilizing interaction. The cu-
mulative effect of these three HBs is thus depicted by the
weighted sum
m⫽n
1
⫹ n
2
⫺ n
3
, 共2兲
in which n
3
receives a negative weight. Note that if H
*
is
positively charged, a bond of type n
3
is less likely to exist.
However, n
3
plays an increasing role as one moves further
away from the positively charged center. Thus, the magni-
tude of m describes how receptive the HB environment
around an oxygen atom is toward accepting a proton. We use
m to characterize the first and second shells around a trans-
ferring proton as follows.
Suppose we ‘‘sit’’ on the transferring proton as depicted
in Fig. 1. This proton is flanked by two water molecules, to
its left, (H
2
O)
l
, and to its right, (H
2
O)
r
. The BOs of the
proton to the two corresponding oxygen atoms are denoted
by n
l
and n
r
, respectively 关Fig. 1共a兲兴. These constitute the
PT coordinates. The first solvation shell of the central H
5
O
2
⫹
is subsequently characterized by the total effective BOs, m
1l
TABLE I. Typical bond orders in water and protonated water. Covalent
bonds are denoted by full lines whereas HBs are dotted.
Bond BO
Gas-phase O–H 1
Liquid-phase O–H 0.9
H
2
O–H
⫹
0.8
H
2
O¯ H
⫹
¯ OH
2
0.55
H
2
OH
⫹
¯ O 0.3
H
2
O¯ H
2
O 0.15
FIG. 3. 共Color兲 The three HBs that enter into the definition of the TEBO in
Eq. 共2兲. n
1
and n
2
, which are donated from the central water molecule, are
depicted in blue. n
3
, which is accepted at the central molecule, is depicted
in red. When H
*
is positively charged, blue and red correspond to favorable
and unfavorable interactions, respectively.
014506-4 Lapid
et al.
J. Chem. Phys. 122, 014506 (2005)
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and m
1r
, respectively 关Fig. 1共b兲兴. These are defined as in Eq.
共2兲. We note that even this first-shell cluster, depicted in Fig.
1共b兲, is larger than previously considered
1,8
because we con-
sider explicitly the dynamics of the two HBs donated by
(H
2
O)
r
and not only of the one accepted by it.
Moving on to the second solvation shell, let us consider
the oxygen atoms l
⬘
and l
⬙
to which O
l
donates HBs. They
are characterized by total effective BOs m
l
⬘
and m
l
⬙
, respec-
tively. Similarly on the acceptor 共right兲 side, see Fig. 1共c兲.
Thus the average m values for the second shell on the left
and right sides are defined by
m
2l
⬅
共
m
l
⬘
⫹ m
l
⬙
兲
/2,
共3兲
m
2r
⬅
共
m
r
⬘
⫹ m
r
⬙
兲
/2.
The 20 HBs generated by our tracking routines have conse-
quently been concatenated into six parameters. These are n
l
,
m
1l
, and m
2l
on the left side of the transferring proton, and
analogously n
r
, m
1r
, and m
2r
on its right. We shall monitor
these parameters during PT events in the MS-EVB simula-
tions.
It is also helpful to consider the differences between the
left and right TEBOs,
⌬m
i
⫽
␣
i
共
m
il
⫺ m
ir
兲
, 共4兲
where
␣
i
is a scaling factor which puts the different layers on
the same scale. One may expect to observe PT when ⌬m
i
⫽ 0, which we verify below.
V. ANALYSIS OF PROTON-TRANSFER EVENTS
Analysis of proton-hopping events was performed at a
single temperature 280 K. While the mechanism of proton
mobility is not expected to change much from 300 K, re-
duced HB fluctuations may make it easier to detect. As Fig.
2 indicates, the MS-EVB model should be applicable over a
whole temperature range around room temperature including
280 K.
We focus first on the PT event within the protonated
water dimer H
5
O
2
⫹
with somewhat more detail than previ-
ously presented. Consequently, we utilize the TEBO vari-
ables to demonstrate how PT within this complex correlates
with the dynamics in the first two solvation shells. All of our
25 PT events are collected as supplementary material,
37
from
which only four are utilized in demonstrating the results be-
low. Because our m
i
values are already averages of 3i HBs
each, the examples presented in the sequel are indeed char-
acteristic.
A. Proton transfer within the central H
5
O
2
¿
moiety
We have monitored the five O–H distances within the
central H
5
O
2
⫹
complex for each PT event. These distances
are depicted in Fig. 4. r
1
and r
2
are the PT coordinates
within this complex, whereas the other four are the covalent
OH bonds of the two participating water molecules.
Figure 5 shows one of the observed PT events 共trajectory
7a, denoted T7a兲. It is clearly indicative of an initial H
3
O
⫹
cation which is converted into another via an intermediate
H
5
O
2
⫹
. Initially, r
1
in panel 共a兲, and r
3
and r
4
in panel 共b兲
assume an average value of 1.055 Å, which is typical to an
FIG. 4. 共Color兲 Five O–H distances within the H
5
O
2
⫹
complex. Color codes
correspond to Fig. 5 below.
FIG. 5. 共Color兲 Proton-transfer dynamics within the H
5
O
2
⫹
complex in one
sample trajectory 共PT event T7a兲. The five distances shown correspond to
those depicted in Fig. 4. Dotted horizontal arrows mark typical bond lengths
within the H
5
O
2
⫹
complex, whereas the vertical dashed lines delimit its
existence epoch.
014506-5 Hydrated proton mobility in water J. Chem. Phys. 122, 014506 (2005)
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OH bond within H
3
O
⫹
. At the same time, r
2
is a HB formed
between H
3
O
⫹
and water, having a typical value of 1.6 Å.
The two O–H bond lengths of the acceptor water molecule
are then around 0.98 Å, as seen in panel 共c兲. Due to water-
water interactions in the condensed phase, this value is
slightly larger than the gas-phase bond length of 0.96 Å.
At about 1 ps, a rather tight H
5
O
2
⫹
complex is formed,
with both r
1
and r
2
fluctuating around 1.2 Å 关dotted arrow in
panel 共a兲兴. Interestingly, the four covalent bonds also assume
an intermediate value, around 1.015 Å 关dotted arrows in pan-
els 共b兲 and 共c兲兴. Thus all the O–H bonds give testimony to
the formation of the complex. The complex attempts to sepa-
rate several times, but succeeds in doing so only at about 3
ps. It is thus rather long lived 共ca.2ps兲. This lifetime varies
from event to event, see Sec. V B below.
In the second half of the transfer event, the complex
dissociates to form the product H
3
O
⫹
. Then r
2
becomes a
covalent bond whereas r
1
is converted into a HB. At the
same time, r
3
and r
4
further reduce to the characteristic wa-
ter value of 0.98 Å, whereas r
4
and r
5
increase to around
1.05 Å. The PT act is then completed. As previously
suggested,
1
it can indeed be regarded as isomerization from a
donor H
3
O
⫹
, via an intermediate H
5
O
2
⫹
, to the acceptor
H
3
O
⫹
.
We mention that in the 25 PT events monitored in this
study there was no indication of a concerted double-proton
transfer event that converts a donor H
5
O
2
⫹
directly into an
acceptor H
5
O
2
⫹
, as suggested in some MD work.
2,7,14,23
As
discussed in the Introduction, this may depend on the poten-
tial used in the simulations. For MS-EVB2 potential utilized
here,
20
particularly when the trajectories are classical, H
3
O
⫹
is more stable 共hence longer living兲 than H
5
O
2
⫹
, and this
makes a double PT event less probable. Nevertheless, we do
observe that after PT the OH bonds to the product H
3
O
⫹
remain excited and may participate in further transfer at-
tempts 关see, for example, the jump in r
5
in Fig. 5共c兲,at4.2
ps兴. At least for the relatively low temperature considered
here 共280 K兲, such secondary PT attempts occur only well
after the main PT event has terminated.
Finally, it is interesting to consider the amplitude of the
vibrations in the various bonds as time proceeds. The general
anticipation is that the longer bonds are weaker and hence
fluctuate more readily. Panel 共a兲 shows very large fluctua-
tions in the ‘‘soft’’ HBs, which become considerably more
restricted as a HB is converted into a ‘‘rigid’’ covalent bond.
A similar trend is seen in the covalent bonds. For example,
the OH bonds in the donor H
3
O
⫹
molecule, panel 共b兲, fluc-
tuate wildly. Their fluctuations become more tamed at long
times, after the PT event. Possibly, this may also be due to
the fact that the pyramidal H
3
O
⫹
disturbs the tetrahedral
water structure around it. This disturbance is mostly allevi-
ated once it is converted into a first-shell water molecule.
B. Lifetime distribution of the protonated dimer
We can use the (n
l
,n
r
) data collected in the supplemen-
tary material
37
共upper panels兲 to compute a lifetime distribu-
tion for the H
5
O
2
⫹
complex. This complex was assumed to
exist between the first and last time that n
l
⫽ n
r
. The 25
lifetimes 共
兲 thus determined average to
具
典
⫽ 375 fs. They
were binned into five equal intervals between 0–1 ps, as
shown in Fig. 6. Two trajectories had
⬎ 1 ps and three had
⫽ 0. In the latter case H
5
O
2
⫹
is better described as a transi-
tion state of a direct reaction.
Although the statistics generated by a small number of
events is not particularly good, the lifetime distribution p(
)
does seem to obey an exponential law,
p
共
兲
⫽ A exp
共
⫺
/
具
典
兲
, 共5兲
and the best fit gives 具
典⫽ 367 fs, very close to its numerical
value. Exponential lifetime distribution is what one expects
from first order kinetics A B, where A and B are the two
forms of protonated water. The average lifetime of the
H
5
O
2
⫹
, about 370 fs, is around one order of magnitude
shorter than that of the H
3
O
⫹
cation, which is in accordance
with their energy difference 关ca. 1 kcal/mol 共Ref. 20兲兴. This
ratio may diminish if the nuclear coordinates are propagated
by quantal rather than classical MD.
C. Correlated dynamics within the solvation shells
The main mechanistic result of the present study is the
correlation between the PT dynamics observed in Fig. 5, and
the HB dynamics in the first- and second-solvation shells of
the protonated dimer. These are best depicted by the TEBO
parameters m
1
共for first shell兲 and m
2
共for second shell兲 de-
fined above. In order to eliminate the fast hydrogen vibra-
tions, these curves were smoothed using a three-point mov-
ing average filter, through which the data were run three
consecutive times. What is left is the ‘‘backbone’’ oxygen
fluctuations, so that the crossing of corresponding ‘‘left’’ and
‘‘right’’ curves is a better indication of an attempted transfer
event. Results are presented as supplementary material for all
the 25 trajectories,
37
three of which are discussed below.
Corresponding to the three panels in Fig. 1, we show in
the three panels of Fig. 7 the smoothed TEBO parameters for
FIG. 6. Lifetime distribution for the H
5
O
2
⫹
complex 共histogram兲, with the
exponential fit 共line兲 of Eq. 共5兲.
014506-6 Lapid
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J. Chem. Phys. 122, 014506 (2005)
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the PT coordinates and the first-two solvation layers, for the
trajectory denoted T9b2. Panel 共a兲 shows time evolution of
the PT coordinates. This is the same as Fig. 5共a兲 in terms of
BOs 共but smoothed and for a different PT event兲. The for-
mation and cleavage of the protonated dimer is identified as
the first and last crossing of the two curves. The correspond-
ing times are indicated by the dashed vertical lines. During
the epoch between these lines, there are periods of a tightly
bound dimer 共the lines depicting n
l
and n
r
lie close together兲
and other periods of wide fluctuations which may almost be
considered as involving an intervening backward PT step.
This constitutes a ‘‘fluxional complex,’’
8
which samples
many substates during its lifetime.
The new feature revealed by Fig. 7 is the correlated mo-
tion of the different hydration layers. The figure shows that
an analogous behavior to that in panel 共a兲 is observed for the
m
i
values of layers 1 and 2. Hence one concludes that PT
occurs when all water layers respond in concert. As Ohmine
and collaborators have observed in simulated water
dynamics,
29,30
processes in water are driven by large-scale
collective motions. In this paper a cooperative behavior is
demonstrated for proton mobility in water.
A more careful inspection reveals that the curves for the
first layer in panel 共b兲 approach and separate more slowly
than the PT coordinates enter or exit their first and last cross-
ings in panel 共a兲, respectively. In comparison, the second
layer in panel 共c兲 shows an even more sluggish response, the
two lines appearing to ‘‘stick together’’ beyond the epoch
defining the protonated dimer complex. In a sense, the sur-
rounding solvent is ‘‘preparing itself’’ to the PT reaction first,
much as Marcus has envisioned the occurrence of a charge-
transfer reaction. As Onsager once commented, reactive
events in water take place from the outside in, as an ‘‘in-
verted snow ball’’ 共second shell first, inner core last兲.
Additional insight on how the different water-layers par-
ticipate in the PT event may be gleaned by overlaying the
bottom two panels in Fig. 7. Maintaining the same color
code, Fig. 8 shows that before the complex is formed the two
green curves roughly coincide, m
2l
⬇m
1r
共encircled兲. This
corresponds to a symmetric solvent environment around the
donor H
3
O
⫹
. After the complex disintegrates, the two blue
curves roughly coincide, m
1l
⬇m
2r
共encircled兲. This indi-
cates the formation of a symmetric solvent environment
around the acceptor H
3
O
⫹
moiety. The following relation
donor: m
2l
⬇m
1r
,
dimer: m
1l
⬇m
1r
, m
2l
⬇m
2r
, 共6兲
acceptor: m
1l
⬇m
2r
,
summarizes our observation. It is best appreciated with ref-
erence to Fig. 1.
The isomerization times namely, the time to actually
convert H
3
O
⫹
into H
5
O
2
⫹
or vice versa can be estimated
qualitatively from Fig. 8. It is seen that the donor environ-
ment loses its threefold symmetry about 50–100 fs before
the protonated H
5
O
2
⫹
is formed 共circle兲. Similarly, the accep-
tor environment gains its threefold symmetry at about 50–
100 fs after the H
5
O
2
⫹
dissociates 共circle兲. These times are
appreciably faster than the 370 fs dimer lifetime or the few
FIG. 7. 共Color兲 Proton-transfer dynamics correlates with the HB dynamics
within the first two solvation layers surrounding the H
5
O
2
⫹
complex 共see
Fig. 1 for definitions and color codes兲. Panel 共a兲 depicts PT event T9b2,
which is analogous to PT event T7a in Fig. 5共a兲, only in BO coordinates.
The first and last crossings of n
l
and n
r
delimit the existence of the complex
共vertical dashed lines兲. The zero of time is set at the middle of this interval.
The two BO parameters 关panel 共a兲兴 and four TEBO parameters 关panels 共b兲
and 共c兲兴 have been smoothed to eliminate fast hydrogen atom vibrations.
FIG. 8. 共Color兲 Overlay of the bottom two panels in Fig. 7. Bold and dashed
lines correspond to first and second layers, respectively. See text for further
discussion.
014506-7 Hydrated proton mobility in water J. Chem. Phys. 122, 014506 (2005)
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picoseconds which elapse between proton hops. Thus the
rate limiting process is the concerted reorganization of the
HB environment as revealed by the TEBO analysis.
To see that the above conclusions are not trajectory spe-
cific, we present the TEBO parameters along two other tra-
jectories. Figure 9 共event T10c兲 presents an extreme case of
a single crossing event 共at t⫽ 0), corresponding to a ‘‘direct’’
H
3
O
⫹
to H
3
O
⫹
transition 共without a long-lived H
5
O
2
⫹
inter-
mediate兲. Figure 10 共event T13a兲 is one of the ‘‘worst’’ cases
in our collection in terms of demonstrating the correlation
between the solvation layers. It is similar to Fig. 9 in exhib-
iting a direct transition, but the curves depicting m
2l
and m
2r
fail to separate after this transition. A closer inspection of the
upper panel reveals that this is probably connected with re-
peated 共nonreactive兲 re-encounters, when n
l
and n
r
in the
upper panel nearly touch.
The two cases are compared in Fig. 11, which depicts the
difference ⌬m, between the two curves in each panel, see
Eq. 共4兲. More precisely, we show ⌬m
0
⬅n
l
⫺ n
r
共inner core,
black兲, ⌬m
1
⬅1.5(m
1l
⫺ m
1r
) 共first layer, red兲, and ⌬m
2
⬅3(m
2l
⫺ m
2r
) 共second layer, green兲. The scaling factors
共1.5 and 3兲 were applied to put the data from all layers on a
similar scale. PT is represented by a crossing of the ⌬m
⫽ 0 line.
In both cases, the first layer follows the inner core very
closely: even the backbone fluctuations in ⌬m
1
and ⌬m
0
are
nearly identical. Thus the correlation between these two is so
strong that their response is essentially in concert. The cor-
relation with the second layer is only in the average trend,
having ⌬m
2
⬎ 0 before the PT, decreasing to ⌬m
2
⬍ 0 after-
ward. This seems to hold on the average even in the worst
case of T13a. When ⌬m
2
lingers around 0, we obtain re-
peated transfer events or a period with a tight H
5
O
2
⫹
com-
plex.
Figure 12 shows the average of ⌬m
i
over our 25 trajec-
tory data bases. There is some arbitrariness in averaging over
different trajectories. For example, the result depends on the
choice of t⫽ 0 for each trajectory. Here we maintain the
assignment of t⫽ 0 at the middle of each interval defining
the H
5
O
2
⫹
complex. The resulting curves in this figure dem-
onstrate that the collective behavior of the core and first two
layers holds also on the average. Again the PT event is seen
to correlate more strongly with the dynamics in the first layer
than with the second 共the curves for i⫽ 0 and 1 nearly over-
lap兲. On the other hand, the averaging procedure has com-
pletely obliterated the H
5
O
2
⫹
intermediate. Since in each tra-
jectory it lives for a different time duration
, the flat step at
⌬m
i
⫽ 0 has been replaced by a gradual slope. The resulting
apparently slow (⬇1 ps) conversion of reactant to product
H
3
O
⫹
conceals the much faster (⬇100 fs) interconversion
between H
3
O
⫹
and H
5
O
2
⫹
共which is driven by the slower
reorganization of the environment兲. The figure thus demon-
FIG. 9. 共Color兲 Same as Fig. 7 for PT event T10c, which shows a very short
H
5
O
2
⫹
lifetime.
FIG. 10. 共Color兲 Same as Fig. 7 for PT event T13a, which shows some of
the worst correlations with second-shell dynamics.
014506-8 Lapid
et al.
J. Chem. Phys. 122, 014506 (2005)
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strates both the utility and drawback of applying averaging
procedures in mechanistic studies.
D. Contribution from individual hydrogen bonds
The above analysis focused on the average BO contribu-
tion to the first two solvation shells surrounding the H
5
O
2
⫹
intermediate. It is instructive to bisect this into typical con-
tributions from the different types of HBs in the two hydra-
tion layers. Doing so, we lose the self-averaging property of
the m
i
so that the results discussed below show much larger
variability from one trajectory to another.
The behavior of the first layer, Fig. 1共b兲, resembles to
some extent the behavior observed for PT in ice.
25
The
‘‘good’’ HBs (n
1
and n
2
, see Fig. 3兲 seldom break. As the
proton migrates from left to right, they expand on the left
and shrink on the right.
1
This leads to the observed decrease
of m
1l
共predominantly before PT兲 and the increase in m
1r
afterwards. In ice, the two ‘‘bad’’ HBs 共of type n
3
) show the
opposite trend: The one on the left lengthens whereas that on
the right contracts.
25
In liquid water, due to the delocalization
of the positive charge, these two bonds are not frequently
observed.
20
When they do exist, we often find that the one on
the right cleaves before the transfer and the one on the left
forms afterwards, as suggested in Ref. 1.
In the second layer, one may break down m
2
into the
contribution from n
1
⫹ n
2
and n
3
. Figure 13 shows the result
for event T10c on the donor side for each of the two water
molecules involved (l
⬘
and l
⬙
in Fig. 1兲. It can be seen that
before the PT event 共vertical dashed line兲, one or more of the
‘‘good’’ HBs cleave, reducing n
1
⫹ n
2
abruptly 共blue line兲.
After the PT event, the ‘‘negative’’ HB, n
3
becomes active
共red line兲. A mirror image of this scenario often holds on the
acceptor side.
FIG. 11. 共Color兲 TEBO parameter difference for the transferring proton and
the two solvation layers, for the trajectories shown in Figs. 9 共top兲 and 10
共bottom兲. These represent favorable and unfavorable cases 共respectively兲 in
terms of the correlation with the second shell.
FIG. 12. 共Color兲 Averaged TEBO difference for the core and two solvation
layers. Same as Fig. 11, only averaged over the 25 trajectories in the supple-
mentary material 共Ref. 37兲. For the second shell 共green兲, a scaling of
␣
2
⫽ 4.5 was used.
FIG. 13. 共Color兲 The contribution from individual HBs to m
2l
for PT event
T10c 关see Fig. 9共c兲兴. The TEBO m
2l
is half the sum of the four lines in this
figure.
014506-9 Hydrated proton mobility in water J. Chem. Phys. 122, 014506 (2005)
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The contribution of n
3
to m
2
is very much in line with
the observations of Day et al.
19,20
By monitoring the HO¯ H
angle of the red HBs in Fig. 1共c兲, they have observed that the
ones on the right break, on the average, concomitant with the
PT event. Thus the HB cleavage event suggested in Ref. 1 as
a rate limiting step for proton mobility does occur, only one
water molecule further away from the protonated center.
What was not previously anticipated is the contribution
of the good HBs, n
1
and n
2
. These, on the average, tend to
cleave on the donor side 共left兲 and reform on the acceptor
side 共right兲. Evidently, their behavior is just the opposite of
that of n
3
. Consequently, the H
3
O
⫹
first-shell coordination
numbers fluctuate over the larger range of 1–4, rather than
just between 3 and 4 as assumed before. Clearly, not all these
individual HB cleavage and formation events occur in each
trajectory. What is required for PT is just enough of them to
tilt the balance from the donor to the acceptor side.
E. The extended picture of proton mobility
in liquid water
From the above discussion, an extended picture of pro-
ton mobility emerges. A schematic summary of HB rear-
rangements coupled to the proton hopping act is given in Fig.
14. HBs break and form predominantly within the second
hydration shell 共curly arrows兲, whereas in the first shell they
typically only extend or contract 共straight arrows兲, therefore
showing much stronger correlation with the inner core. The
rate limiting step is likely to be the HB cleavage events
which occur within the second shell. This conclusion is
qualitatively as suggested in Refs. 1 and 7, except that it is
not possible to implicate one single HB as the key player in
the mechanism.
The two types of HBs show opposite behaviors within
the second shell. Prior to PT 关Fig. 14共a兲兴, good HBs cleave
on the donor side and bad HBs cleave on the acceptor side. A
sufficient number of these bonds should remain simulta-
neously broken in order to break the threefold symmetry
around the donor H
3
O
⫹
. The cleavage events themselves are
fast 共say, 50–150 fs兲, so that they occur consecutively rather
than simultaneously. The last of these finally tilts the balance
from reactants to products. This explains why the activation
energy for proton mobility 共Sec. III兲 is so close to the HB
strength in liquid water 共2.6 kcal/mol兲,
38
although it is a col-
lective breaking of HBs and not the cleavage of a single HB
that drives the PT.
Finally, following the PT event, good HBs form on the
acceptor side, whereas the bad ones reform on the donor side
关see Fig. 14共b兲兴. This terminates the fluctuations of the pro-
ton within the H
5
O
2
⫹
complex. It localizes the proton on the
acceptor H
3
O
⫹
moiety and establishes a new threefold sym-
metry around it. The temporal division may be less sharp
than depicted, as some of the HB dynamics occurs also dur-
ing the lifetime of the complex.
VI. CONCLUSIONS
The elucidation of the mechanism of proton mobility in
water is a basic problem of physical chemistry,
9
with far
reaching consequences, for example, in fuel-cell technology
and biology. Previous discussions were based on the combi-
nation of experiment and chemical intuition,
1
on qualitative
visualization of short trajectories,
7,8
or focused on a single
HB.
2,19,20
A method for systematic analysis of the coopera-
tive behavior of many HBs was lacking. The present work
introduced BOA, a method which is both intuitive and easy
to implement.
FIG. 14. 共Color兲 HB dynamics couples to proton mobility in water. Before
PT, panel 共a兲, HBs mainly break 共second shell, curly orange arrows兲 or
expand 共first shell, straight orange arrows兲. Good HBs may break on the
donor side 共left兲 whereas bad ones break on the acceptor side 共right兲.Around
the donor H
3
O
⫹
, two bonds expand, whereas the central O–O bond con-
tracts to form the H
5
O
2
⫹
complex 共not shown兲. As the complex disintegrates
to form the product H
3
O
⫹
, panel 共b兲, this central bond expands, whereas the
two other HBs in the first shell contract 共straight green arrows兲. HBs now
form in the second solvation shell 共curly green arrows兲, predominantly the
good ones on the acceptor side and the bad ones on the donor side. Some
bonds may have reformed during the lifetime of the H
5
O
2
⫹
intermediate
共bottom left兲.
014506-10 Lapid
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J. Chem. Phys. 122, 014506 (2005)
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Utilizing BOA, we could track the cooperative behavior
of the 18 HBs surrounding the transferring H
5
O
2
⫹
complex.
We find that this complex is almost always an intermediate
separating the donor and acceptor H
3
O
⫹
moieties. The tran-
sition between these structures indeed couples to HB dynam-
ics, but within considerably larger water clusters than previ-
ously anticipated.
As PT progresses, the TEBO diminishes on the donor
side and increases on the acceptor side. This appears to occur
in concert within the first and second solvation shells. In the
first shell one observes predominantly stretching and con-
traction of HBs, and the correlation with the PT act is very
strong. In the second shell there are more HB cleavage and
formation events, and the correlation is only on average. One
may nevertheless conjecture that the ‘‘rate limiting step’’ lies
in the second shell. The collective accumulation of several
consecutive HB cleavage events there eventually tilts the
balance from one form of protonated water to the other.
The TEBO is constructed by dividing the HBs into two
types: The blue bonds emanate from the protonated center
and thus stabilize it, whereas the red bonds which are di-
rected toward the protonated center destabilize it. The anal-
ogy with Moses parting the Red Sea
1
is now two colored:
The red sea parts in front of the proton and closes behind its
back, whereas the blue sea parts in his rear and forms up
front.
The above conclusions may depend on several aspects of
our simulation methodology. One concern may be the utili-
zation of one particular MS-EVB potential. Judging from the
excellent value obtained for the activation energy of proton
mobility 共2.7 kcal/mol at room temperature兲, this potential
appears to yield quite realistic results. A second concern is
the neglect of quantum effects on the nuclear dynamics. We
anticipate that these could reduce the activation energy by
about 0.3 kcal/mol but, as previously noted,
7
the influence on
the mechanism itself may not be dramatic. An increase in
temperature 共from the value of 280 K considered herein兲
could also make a difference in the details of the mechanism.
Given the above reservations, the conclusions in this
work are not expected to be the final word on the proton
mobility mechanism. The classical MS-EVB2 calculations
serve to demonstrate the utility of our TEBO approach. Since
it is easily applied to trajectories of any origin, it may be
interesting to apply this method to different temperatures and
simulation methodologies in the future. Finally, the BOA
may be useful in analyzing proton mobility near and through
biological membranes and channels, topics of great interest
for proton-driven bioenergetics.
6
ACKNOWLEDGMENTS
This research was supported by Grant No. 98-00083
from the United States-Israel Binational Science Foundation
共BSF兲, Jerusalem, Israel. G.A.V. acknowledges support from
the United States National Science Foundation 共NSF兲 Grant
No. CHE-0317132.
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014506-11 Hydrated proton mobility in water J. Chem. Phys. 122, 014506 (2005)
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