Water-mediated proton transfer: A mechanistic investigation on the example of the hydration of sulfur oxides

Article abstract of DOI:10.1021/jp0038862

We outline general mechanistic features of "water-mediated proton transfer" in the example of the isomerization reaction in hydrogen-bonded sulfur oxide-water supermolecules containing up to three water molecules. The nucleophilic attack of a water oxygen on the sulfur atom occurs concertedly with the (multiple) protontransfer event(s). The protons are transferred according to the well-known hydrogen-bond compression mechanism. However, contrary to "pure" multiple proton-transfer reactions, the protons are transferred asynchronously. These mechanistic features force the reaction to be classical rather than quantum-tunneling-dominated down to rather low temperatures. In the quantum-dominated temperature region, tunneling takes place only if all protons tunnel through the barrier. Straight line corner cutting (large curvature tunneling) does not dominate at any temperature, as the reduced mass corresponding to reaction coordinate motion does not drop to values low enough in the reaction barrier region. The asymmetric nature of the potential energy surface even allows different mechanisms involving transient H₃O⁺ rotation termed "molecular swing" and a H₂SO₃ isomerization to be favorable compared to water-mediated triple proton transfer in the case of three participating water molecules.

Full text of DOI:10.1021/jp0038862

J. Phys. Chem. A 2001, 105, 5137-5145
5137
Water-Mediated Proton Transfer: A Mechanistic Investigation on the Example of the
Hydration of Sulfur Oxides
Thomas Loerting and Klaus R. Liedl*
Institute of General, Inorganic and Theoretical Chemistry, UniVersity of Innsbruck, Innrain 52a,
A-6020 Innsbruck, Austria
ReceiVed: October 20, 2000; In Final Form: February 21, 2001
We outline general mechanistic features of “water-mediated proton transfer” in the example of the isomerization
reaction in hydrogen-bonded sulfur oxide-water supermolecules containing up to three water molecules.
The nucleophilic attack of a water oxygen on the sulfur atom occurs concertedly with the (multiple) proton-
transfer event(s). The protons are transferred according to the well-known hydrogen-bond compression
mechanism. However, contrary to “pure” multiple proton-transfer reactions, the protons are transferred
asynchronously. These mechanistic features force the reaction to be classical rather than quantum-tunneling-
dominated down to rather low temperatures. In the quantum-dominated temperature region, tunneling takes
place only if all protons tunnel through the barrier. Straight line corner cutting (large curvature tunneling)
does not dominate at any temperature, as the reduced mass corresponding to reaction coordinate motion does
not drop to values low enough in the reaction barrier region. The asymmetric nature of the potential energy
surface even allows different mechanisms involving transient H3O+ rotation termed “molecular swing” and
a H2SO3 isomerization to be favorable compared to water-mediated triple proton transfer in the case of three
participating water molecules.
1. Introduction
as the Cope68 and Beckmann69 rearrangements), hydration of
double bonds (e.g., of carbon dioxide,70,71 sulfur oxides,72-76
carbonyl compounds,77,78 ketene imines, or carbodiimides79,80),
hydrolyses (e.g., of carboxylic acid esters81 or methyl chloride82),
to nucleophilic substitutions (e.g., the decomposition of chlorine
nitrate83-88).
In the literature water has been shown to be important as a
catalyst mediating proton transfer. In many proteins hydrogen
atoms are transferred more or less linearly for distances of about
10-50 Å between different amino acid residues along so-called
“proton wires”. Lately, light has been shed on the way protons
are drawn through proteins on the example of ferredoxin I by
a combined fast-scan voltammetry, high-resolution crystal-
lography, and molecular dynamics study.1 Similar trans-
membrane proton pumps or proton relays are ubiquitous in
biochemistry. Bacteriorhodopsin,2-6 alcohol dehydrogenases,7-10
ATP synthase,11-13 cytochrome c oxidase,14-19 rhodobacter
sphaeroides,20,21 ribonucleotide reductase,22,23 and carbonic
anhydrase24-31 are just a few examples of proteins having
hydrogen-bond networks serving the function of long-distance
proton transport. The function of the water molecules is to both
accept and donate a proton. A similar transfer mechanism along
proton wires (commonly referred to as Grøtthus mechanism) is
thought to be the reason for the extraordinarily high conductivity
in liquid water.32-41
Experimental and theoretical studies nowadays focus on a
general, atomistic understanding of the involved proton-transfer
processes. In this theoretical study we analyze in detail the
reaction mechanism and the influence of tunneling on the
example of the hydration of sulfur dioxide and sulfur trioxide
in gas-phase supermolecules constituted of up to three water
molecules. Recently, we have shown that in our atmosphere
sulfur trioxide is hydrated rather than the much more abundant
sulfur dioxide.76 Furthermore, the best agreement in terms of
reaction rate constants with reaction chamber flow studies is
obtained in SO3/H2O complexes of 1:2 and 1:3 stoichiometry.75
Now, we want to show how the proton-transfer mechanism and
the multidimensional tunneling effect are altered when inves-
tigating such an “impure” proton-transfer reaction, i.e., proton-
transfer reaction accompanied by a nucleophilic attack, rather
than a “pure” proton-transfer reaction, which we investigated
previously by the same method for cyclic water clusters89,90 and
the formic acid dimer.91
Another possibility for the protons to be transferred is in a
cyclic rather than linear manner. Such mechanisms referred to
as “water-mediated proton transfer”, “bifunctional water ca-
talysis”, or “water-assisted hydration” were demonstrated to be
of relevance in many organic and anorganic reactions. In
principle, water can be replaced by any hydroxylic solvent, e.g.,
alcohols, so that “solvent-assisted” or “solvent-mediated” proton-
transfer reactions can also be assigned to this type of reaction.
Again, a wealth of examples is known for this category. These
examples range from tautomerization and proton shifts (e.g.,
in DNA/RNA-base analogues like formamidine,42-45 7-aza-
indole,46-50 7-hydroxyquinoline51-53 and others,54-57 in DNA/
RNA-bases,58-50 in peptides,63,64 or in formic acid65-67 as well
2. Methods
2.1. Stationary Points. Geometry optimization of the equi-
librium structures and the transition states was performed both
by hybrid density functional theory (B3LYP/6-31+G(d))92 and
by second-order perturbation theory (MP2/aug-cc-pVDZ),93 as
implemented in Gaussian98.94 The nature of these stationary
points was verified by calculating vibrational frequencies. The
Hessian matrix contains exclusively positive eigenvalues for
10.1021/jp0038862 CCC: $20.00 © 2001 American Chemical Society
Published on Web 05/04/2001

5138 J. Phys. Chem. A, Vol. 105, No. 21, 2001
Loerting and Liedl
Figure 1. Stationary structures involved in the unimolecular conversion
from sulfur dioxide (SO2) to sulfurous acid (H2SO3) in the presence of
one (top), two (middle), and three water molecules (bottom) as found
at the B3LYP/6-31+G(d) level of theory.
minima and exactly one single negative eigenvalue for transition
states. Predictions of reaction dynamics critically depend on the
reaction barrier, i.e., the difference in electronic energy between
transition state and minima, and the tunneling correction factor.
We employed single-point energy calculations at the CCSD-
(T)/aug-cc-pVDZ//MP2/aug-cc-pVDZ95 level of theory to vali-
date the accuracy of the barrier height, where computationally
feasible.
Figure 2. Stationary structures involved in the unimolecular conversion
from sulfur trioxide (SO3) to sulfuric acid (H2SO4) in the presence of
one (top), two (second from top), and three (second from bottom and
bottom) water molecules as found at the B3LYP/6-31+G(d) level of
theory. The mechanism in the third row corresponds to water-mediated
double proton transfer assisted by a rather rigid third water molecule.
The mechanism in the bottom row corresponds to a single proton
transfer, rotation of a H3O+ like transient subspecies, and a second
single proton transfer.
2.2. Ab Initio Reaction Path. Starting from the transition
state the reaction path was generated as the steepest descent
path in mass-scaled coordinates employing a scaling mass of 1
amu throughout. This path, called minimum energy path (MEP)
or intrinsic reaction coordinate (IRC), was generated by using
the Page-McIver algorithm96 by employing a step size of 0.05
bohr (1 bohr corresponds to 0.53 Å). The distance of a point
on the potential energy surface to the transition state is denoted
s and given a “+” sign if on the product side and a “-” sign if
on the educt side. Vibrational frequencies and partition func-
tions were calculated every third point on the hypersurface. On
both branches of the reaction coordinate the path was stopped
when stable minima structures were reached, i.e., when the
gradient vanished. This required altogether about 400 points
and was done using hybrid density functional theory (B3LYP/
6-31+G(d)), which was designed to incorporate electron cor-
relation at a cost comparable to Hartree-Fock calculations,
which do not incorporate electron correlation effects. The use
of such a procedure was found to be successful for “pure”
proton-transfer reactions, which are more difficult to describe
in terms of quantum tunneling.90,91,97
2.3. Quantum Mechanical Tunneling and Corner Cutting.
The tunneling correction factor was calculated in the framework
of canonical variational transition state theory (CVTST). Four
different correction schemes were employed. The Wigner
correction98 is calculated directly from the imaginary frequency
at the transition state without using any information on the
reaction path. Zero-curvature tunneling (ZCT) involves tunneling
along the minimum energy path, small-curvature tunneling
(SCT)99-101 involves adiabatic tunneling at the inner turning
Figure 3. Transition state interconverting two isomers of the dihydrate
of sulfurous acid as found at B3LYP/6-31+G(d) level of theory.
points of the concave side of the minimum energy path,102 and
large-curvature tunneling (LCT) involves vibrationally non-
adiabatic straight line tunneling through the reaction swath103-106
by allowing tunneling into all available vibrationally excited
states.103-105,107,108 The latter three corrections were all calculated
by employing the semiclassical approximation109 as implemented
in the program Polyrate8.2,110,111 written at the University of
Minnesota in the group of Donald Truhlar. SCT and LCT allow
us to take reaction paths different from the MEP, which are
shorter, but more demanding in terms of energy. Such paths
may be favored compared to the MEP, as they involve more
hydrogenic motion and are, therefore, accelerated by quantum
mechanical tunneling.

Hydration of Sulfur Oxides
J. Phys. Chem. A, Vol. 105, No. 21, 2001 5139
TABLE 1: Electronic Energies (kcal/mol) for the Hydration of SOx (x ) 2, 3) Assisted by n ) 1-3 Water Molecules,
Respectively
B3LYP/6-31+G(d)
MP2/aug-cc-pVDZ
CCSD(T)/aug-cc-pVDZ
MP2/aug-cc-pVTZ
Sulfur Dioxide Hydration
n ) 1
n ) 2
n ) 3
SO2/H2O
SO2‚H2O
TS
5.26
0.00
33.34
6.94
16.60
0.00
19.96
5.77
27.86
0.00
13.02
5.52
4.50
0.00
35.44
11.59
14.01
0.00
22.95
9.38
23.54
0.00
5.00
0.00
34.19
6.82
14.74
0.00
22.12
4.47
4.34
0.00
34.40
9.00
H2SO3
SO2/2H2O
SO2‚(H2O)2
TS
H2SO3‚H2O
SO2/3H2O
SO2‚(H2O)3
TS
H2SO3‚(H2O)2
8.31
Sulfur Trioxide Hydration
n ) 1
SO3/H2O
SO3‚H2O
TS
9.96
0.00
27.60
-7.03
22.48
0.00
10.03
-7.71
36.50
0.00
8.72
0.00
28.11
-3.74
19.40
0.00
11.61
-4.60
31.39
0.00
9.09
0.00
28.15
-6.74
19.81
0.00
H2SO4
n ) 2
SO3/2H2O
SO3‚(H2O)2
TS
H2SO4‚H2O
SO3/3H2O
SO3‚(H2O)3
TS
H2SO4‚(H2O)2
SO3/3H2O
SO3‚(H2O)3
TS
11.34
-7.70
n ) 2 + 1
n ) 3
9.35
-6.44
36.43
0.00
31.12
0.00
3.99
H2SO4‚(H2O)2
-3.22
a In each group, the first lines correspond to the separated SOx and nH2O molecules, the second lines correspond to the SOx‚nH2O minima (set
to 0.00 kcal/mol by definition), the third lines correspond to the transition states to the concerted nucleophilic attack/proton-transfer reaction (TS),
and the last lines correspond to the H2SOx+1‚(n - 1)H2O minima. CCSD(T) energies rely on MP2/aug-cc-pVDZ geometries.
3. Results
3.1. Stationary Structures. The educt minimum, the product
minimum, and the transition state found after full geometry
optimization at the B3LYP/6-31+G(d) level of theory are
depicted in Figure 1 for the SO2 hydration and in Figure 2 for
the SO3 hydration. Additionally, a transition state interconverting
H2SO3‚2H2O structures is shown in Figure 3. A main difference
between the two reactions is that SO3 hydration is exothermic
and SO2 hydration is endothermic (cf. Table 1). As observed
experimentally, “sulfurous acid” is unstable in any hydration
state by at least 5 kcal/mol relative to loosely hydrated sulfur
dioxide species. Zero-point correction tends to further increase
this instability by approximately 1 kcal/mol. Nevertheless, it is
still possible that oligomeric “sulfurous acid” (H2SO3)n species
are approximately as stable as loosely hydrated sulfur dioxide
species. Such a situation occurs for structurally similar carbon
dioxide hydrates and carbonic acid dimers (H2CO3)2.112 Ac-
Figure 4. Electronic energy along the minimum energy path for the
conversion from sulfur dioxide to sulfurous acid, as calculated at the
cording to the Hammond postulate113,114 exothermic reactions
exhibit early transition states resembling educts, whereas endo-
thermic reactions exhibit late transition states resembling prod-
ucts. In this case the postulate proves to be useful. For SO3 hy-
dration the nucleophilic attack is still in its initial stages at the
transition state. The SO3 subunit is not distorted very much from
the planar coordination toward a tetrahedral coordination in the
transition state. On the other hand the nucleophilic attack is
more or less complete in the transition state in the case of SO2
hydration. This can be seen best on the example of three par-
ticipating water molecules in Figures 1 and 2. For the thermo-
neutral isomerization reaction the proton is positioned exactly
in the middle between the two oxygen atoms (cf. Figure 3).
The reaction barrier to SOx hydration (as summarized in Table
1) clearly decreases in going from single proton transfer (n )
B3LYP/6-31+G(d) level of theory.
1) to water-mediated double proton transfer (n ) 2) and double
proton transfer alleviated by a “molecular swing” (n ) 3). The
decrease on going from n ) 1 to n ) 2 is directly explainable
by ring strain in the transition state. Whereas in the case of n
) 1 an unfavorable four-membered transition state is encoun-
tered, a six-membered transition state is involved in the case of
n ) 2, which is more free of ring strain. The amount of ring
strain of cyclobutane compared to cyclohexane is lower by 26
kcal/mol,115 which is clearly more than the calculated barrier
reduction of 12 kcal/mol (SO2) and 17 kcal/mol (SO3). This is
probably due to the sulfur atom, which is not restricted to a
tetrahedral coordination geometry as is the case for carbon atoms

5140 J. Phys. Chem. A, Vol. 105, No. 21, 2001
Loerting and Liedl
Figure 5. Distances (Å) between two selected atoms along the minimum energy path (bohr) for the hydration of sulfur dioxide by one (top left),
two (top right), or three (bottom left) water molecules and for the isomerization of “sulfurous acid” in the presence of two water molecules (bottom
right). The numbering scheme is shown as an inset in the respective plots.
due to the missing d orbitals. The reduction of the reaction
barrier on going from n ) 2 to n ) 3 amounts to 6-7 kcal/
mol. This reduction is supposedly due to a different reaction
mechanism (described in the next section) bringing the system
to an energetically favorable atomic arrangement in the transition
state region.
3.2. “Classical” Reaction Coordinate. All investigated
hydration reactions involve concerted atomic movement, because
no additional transition states besides the one at s ) 0.0 bohr
can be found. From the mechanistic viewpoint this means that
the nucleophilic attack of oxygen on sulfur and all involved
proton transfers happen in a single step without any intermedi-
ates existing for longer than a vibrational period.116 However,
the multiple proton-transfer steps occur not synchronously, but
rather asynchronously. Both for the n ) 2 and n ) 3 reactions
two protons are transferred at different reaction coordinates. This
is quite surprising for the SOx hydration involving three water
molecules, as one could suspect the triple proton transfer along
three water molecules to be favored most from ring strain
considerations. Obviously, the task of the third water molecule
must be a different one.
In Figures 5 and 6 selected distances between pairs of atoms
are depicted along the classical reaction coordinate for the
different hydration reactions of SOx and the isomerization of
the unstable dihydrate of sulfurous acid. Commonly, the protons
involved in OH‚‚‚O hydrogen bonds stay at a rather constant
distance of slightly less than 1 Å from the donor oxygen atom
for a great portion of the reaction coordinate. At the same time
the O- -O distance reduces from the equilibrium distance of 2.7-
2.9 Å to about 2.5 Å. Then there is a point on the reaction
coordinate where the proton suddenly starts the transfer to the
acceptor oxygen atom, which can be seen from the sudden
increase of the proton distance to the acceptor oxygen atom.
After the proton transfer has been accomplished, the O- -O
distance again relaxes to the equilibrium distance of 2.7-2.9
Å. Obviously, the proton transfer is triggered by very short
O- -O distances. This phenomenon has been noted both by NMR
Another low-energy transition state (cf. Figure 3) that can
be found in the H2SO3‚2H2O water system corresponds to the
transfer of a hydrogen atom of “sulfurous acid” via double
proton transfer along a two water chain to the oxygen atom
previously double bonded to the sulfur atom. Of course the
educts and products of this isomerization are energetically equal,
as seen in Figure 4. Nevertheless, this mechanism provides a
possibility for oxygen isotope exchange. The marked *O atom
moves from the hydroxy group (H*O)(OdS)(OH) to the double-
bonded position (HO)(*OdS)(OH). The reaction barrier to this
process amounts to about 9 kcal/mol, which is clearly lower
than all hydration barriers. However, using the SO2‚3H2O
complex as the zero of energy, as exemplified in Figure 4, it
becomes clear that the barrier-top for hydration (n ) 3) lies
lower by about 1.5 kcal/mol. When taking into account zero-
point correction for both structures, the barrier-top to isomer-
ization is lower by 0.5 kcal/mol. This similarity in absolute
energies of the transition states indicates that for the backward
reaction there is a competition between isomerization and
decomposition to sulfur dioxide. Indeed, the geometries found
on the minimum energy path are absolutely identical for both
reaction channels between 6 and 4 bohr. At the reaction coor-
dinate of about 4 bohr there is a point at which the system “de-
cides” statistically or from the available distribution of thermal
energy into vibrational excitations, which path will be tracked.

Hydration of Sulfur Oxides
J. Phys. Chem. A, Vol. 105, No. 21, 2001 5141
Figure 6. Distances (Å) between selected two atoms along the minimum energy path (bohr) for the hydration of sulfur trioxide by one (top left),
two (top right), two active and one passive (bottom left), or three water molecules (bottom right). The numbering scheme is shown as an inset in
the respective plots.
methods and from electronic structure calculations for many
different hydrogen bonds in the literature and received the name
“hydrogen-bond compression mechanism”.117-120 It has been
shown in numerous publications that the potential describing
the proton-transfer process changes from a double-well potential
involving a barrier to a single-well potential involving no barrier
on decreasing the distance of the donor and acceptor atoms
subsequently.121-129 When more than one proton has to be
transferred, all O- -O distances vary at the same time; i.e., the
process becomes highly cooperative. The accomplishment of
the first proton transfer yields a H3O+ containing, unstable
transient species. To reach a stable state again, the next proton
transfer is triggered and so forth. According to IUPAC
nomenclature such a behavior is called asynchronous.130 On
inspecting the S- -O distance, which decreases along the whole
hydration coordinate, it becomes clear that the proton transfer
is concerted with the nucleophilic attack.
This hydrogen-bond compression mechanism, which works
best in linear OH‚‚‚O arrangements and at O- -O distances of
2.5 Å, explains why H9 is transferred to O10 rather than to O4
in the case of H2SO3 isomerization. Whereas the O3-O4 atom
pair can only reach a distance of 2.8 Å, the O3-O10 atom pair
can reach a better orientation for hydrogen transfer. The
hydrogen atom takes its way to the potential valley correspond-
ing to the isomerization channel, therefore. The S1- -O7 distance
varies by ≈0.1 Å, corresponding to rehybridization from a
formal single bond (S-O) to a double bond (SdO).
sical” water-mediated proton transfer, the hydronium ion like
subpart of the supermolecule performs a slight rotational
movement in order to bring one proton to a position from which
transfer to the neighbor oxygen atom is energetically favorable.
Although the third water molecule acts as spectator, i.e., remains
in about the same position during the whole reaction, it has the
important function of stabilizing the H3O+ subunit by providing
the oxygen atom O2 (SO2 case), which acts as hydrogen-bond
acceptor. Figuratively spoken, this oxygen atom is the anchor
for the second hydrogen bridge “rope” to which the molecular
water “swing” carrying a proton is fastened. And when the
“swing” reaches the highest point, the proton has gained enough
energy to be able to “jump” to the oxygen atom O10 (SO2 case).
A mechanism differing in this way from water-mediated triple
proton transfer is possible, as the reaction takes place on a highly
complex potential energy surface showing many valleys and
ridges because of the energy asymmetry of the reaction and the
according “impurity” of the proton transfer. The “rope” function
can also be seen in Figure 6. Whereas the O6- -O5 distance
remains rather constant after the transfer of the first proton, the
O6- -O3 distance is shortened. Atom H10 rotates into the
O6- -O3 line and is transferred, when the O6- -O3 distance
reaches 2.5 Å. A slight elongation of the O6- -H11 bond is
required to make this rotation energetically feasible. Instead of
H11 being transferred to O5 the second proton switch involves
H10 being transferred to O3, therefore.
3.3. Influence of Quantum-Tunneling and Corner-Cutting.
Now the question remains, how long is “classical” reaction
coordinate motion expected to be the dominating motion
compared to other motions significantly enhanced by quantum-
For the n ) 3 cases a reaction mechanism clearly distinct
from the n ) 2 and n ) 2 + 1 cases can be observed, which
we term the “molecular swing mechanism”. Instead of “clas-

5142 J. Phys. Chem. A, Vol. 105, No. 21, 2001
Loerting and Liedl
TABLE 2: Tunneling Correction Factors K at 100, 200, and
300 K to the Rate Constants Obtained from Classical
Transition State Theory for the Reaction SOx‚nH2O H
H2SOx+1‚(n - 1)H2O (x ) 2, 3; n ) 1-3)
temp, K Wigner
ZCT
SCT
LCT
Sulfur Dioxide Hydration
n ) 1
n ) 2
n ) 3
iso
100
200
300
100
200
300
100
200
300
100
200
300
24.9
7.0
3.7
7.9
2.7
1.8
1.1
1.0
1.0
4.1
1.8
1.3
9.9 × 1017 6.4 × 1023 4.1 × 1021
1.3 × 103 1.9 × 105 7.4 × 104
13.3
76.4
27.1
4.4 × 104 6.0 × 107 4.7 × 108
6.3
2.1
3.4
1.4
1.2
6.1
1.4
1.2
39.1
4.2
20.9
1.6
68.2
3.3
25.8
1.5
1.2
1.2
5.8 × 102 3.9 × 104
2.1
1.3
5.1
1.4
Sulfur Trioxide Hydration
n ) 1
100
200
300
100
200
300
100
200
300
100
200
300
24.9
7.0
3.7
3.7
1.7
1.3
3.9
1.7
1.3
1.7
1.2
1.1
2.7 × 1021 4.8 × 1026 6.1 × 1024
6.5 × 103 1.1 × 106 3.1 × 105
19.2
113.4
35.3
n ) 2
3.2 × 103 1.9 × 106 3.8 × 106
3.3
1.6
8.9
2.2
7.2
1.8
n ) 2 + 1
n ) 3
6.1 × 103 2.6 × 106 8.3 × 106
4.5
1.8
1.5
1.1
1.0
17.0
2.9
2.0
1.2
1.1
15.1
2.2
3.4
1.3
1.1
a B3LYP/6-31+G(d) was employed throughout.
tunneling? Obviously, the answer to this question depends on
the temperature. In Table 2 the magnitude of the quantum-
mechanical tunneling correction κ to the classical rate constant
is listed for the hydration reactions of SO2 or SO3 and the
isomerization mechanism (labeled Iso) described above at three
temperatures (100, 200, and 300 K). There is a clear correlation
between the magnitude of the imaginary frequency at the
transition state and the tunneling correction factor. For the lowest
imaginary frequency observed, namely 124i cm-1 for the SO2
hydration involving three water molecules, tunneling is rather
unimportant. In the direct corner-cutting (LCT) mechanism, the
reaction is accelerated by a factor of 25.8 at 100 K. At higher
temperatures tunneling ceases, i.e., κ approaches 1.0. In
comparison the isotope exchange mechanism is preferred by
tunneling. This can be explained by the thinner energy barrier
in Figure 4 and the higher absolute value of the imaginary
frequency, namely 597i cm-1. Similarly, for the SO3 hydrations,
on increasing the number of microsolvating water molecules
the importance of quantum mechanical tunneling diminishes due
to the broadening nature of the reaction barrier, which can be
seen in the decrease of the “imaginary frequency” from 1663i
to 564i cm-1 (582i cm-1) and 288i cm-1 for n ) 1, n ) 2 (n
) 2 + 1), and n ) 3, respectively. Most tunneling enhancement
is observed for the classically slowest reactions involving one
and two water molecules. Especially at 100 K acceleration
factors of 1024 and 108 are found from the calculation for n )
1 and n ) 2, respectively. However, this acceleration is still
not sufficient to outweigh the classical disadvantage in terms
of reaction rates. Assuming uncertainties of 1 kcal/mol in the
reaction barrier leading to uncertainties of a factor of 5 (300
K), 12 (200 K), and 150 (100 K) in the classical rate constant,
it becomes clear that tunneling is a negligible effect for n ) 3
down to 100 K, important below about 250 K for n ) 2, and
important even above room temperature for n ) 1. The LCT
tunneling corrections are not significantly higher than the SCT
Figure 7. Representative relative tunneling energy referring to the
barrier top as the zero of energy for the hydration reactions of SO2
(top) and SO3 (bottom) in water clusters of different sizes.
TABLE 3: Approximate Crossover Temperatures above
Which the Classical Over-Barrier Reaction Dominates and
below Which Quantum-Mechanical Tunneling through the
Barrier Dominates, As Calculated by the Multidimensional
Optimized Tunneling Method at a B3LYP/6-31+G(d)
Hypersurface
SO2
SO3
n ) 1
375 K
225 K
100 K
175 K
350 K
175 K
75 K
n ) 2
n ) 3
iso/n ) 2 + 1
200 K
corrections. This is in strict contrast to the results found in
carboxylic acid dimers,91 cyclic HF,97,131 and water clusters.90
Whereas these systems involve “pure” proton transfer of reduced
mass 1 amu, in the case of sulfur oxide hydration, heavy atom
movement accompanies the reaction. This renders the proton
transfer “impure”, so that the effective reduced mass becomes
1 amu nowhere along the reaction coordinate and even does
not drop below 5 amu in the n ) 1 systems, although a proton
is transferred.
In Table 3 the crossover temperatures, i.e., the border
temperatures between overbarrier and tunneling-dominated
reaction, are summarized. Especially for the larger supermol-
ecules investigated here the crossover temperatures are very low.
Even for the n ) 1 reactions the crossover temperatures are
still about 100 K lower than the crossover temperatures found
for “pure” proton-transfer reactions.91,131
In Figure 7 the representative tunneling energies relative to
the barrier top are depicted. An energy of 0 kcal/mol implies

Hydration of Sulfur Oxides
J. Phys. Chem. A, Vol. 105, No. 21, 2001 5143
an overbarrier mechanism, whereas negative values indicate
tunneling. It can be seen that for n ) 2 and n ) 3 the overbarrier
mechanism prevails down to rather low temperatures. The kinks
seen in this plot are indicative of the sudden change to tunneling
of all transferred protons. For n ) 1 the representative tunneling
energy changes rather continuosly to lower values on decreasing
the temperature, as the whole temperature region shown is in
the tunneling domain.
The difference in tunneling mechanisms compared to “pure”
proton transfer also has consequences for the choice of the
method of calculating the tunneling correction. Whereas for
“pure” proton-transfer reactions LCT is clearly the dominant
tunneling contribution, in the case of “impure” proton transfer
SCT is the dominant tunneling mechanism (even for n ) 1).
Only in some cases does the LCT scheme provide slightly higher
corrections, e.g., for n ) 2 at 100 K. However, the additional
effort for the calculation is not outweighed by such a marginal
increase. The other corrections (ZCT and Wigner) are clearly
insufficient to describe tunneling enhancement of such proton-
transfer reactions properly. As a consequence, we recommend
employing the LCT scheme for “pure” proton-transfer reactions
and to employ the SCT scheme for “impure” proton transfer in
order to reach agreement with experimental data.
Acknowledgment. We are grateful to Markus Loferer for
help in the initial stage of the work. Support by the Austrian
Science Fund (project no. P14357-TPH) is gratefully acknowl-
edged. T.L. is indebted to the Austrian Academy of Sciences
(DOC-program) for financal support.
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