JOURNAL OF CHEMICAL PHYSICS
VOLUME 113, NUMBER 13
1 OCTOBER 2000
Structure and vibrational spectra of H¿„H2O…8: Is the excess proton
in a symmetrical hydrogen bond?
Cristian V. Ciobanu
Department of Physics, Ohio State University, 174 West 18th Avenue, Columbus, Ohio 43210
Lars Ojamäe
Physical Chemistry, Arrhenius Laboratory, Stockholm University, S-106 91 Stockholm, Sweden
Isaiah Shavitt and Sherwin J. Singera)
Department of Chemistry, Ohio State University, 100 West 18th Avenue, Columbus, Ohio 43210
共Received 15 May 2000; accepted 23 June 2000兲
The energetics, structure, and vibrational spectra of a wide variety of H⫹共H2O兲8 structures are
calculated using density functional theory and second-order Møller–Plesset ab initio methods. In
these isomers of H⫹共H2O兲8 the local environment of the excess proton sometimes resembles a
⫹
symmetric H5O⫹
2 structure and sometimes H3O , but many structures are intermediate between
these two limits. We introduce a quantitative measure of the degree to which the excess proton
⫹
resembles H5O⫹
2 or H3O . Other bond lengths and, perhaps most useful, the position of certain
vibrational bands track this measure of the symmetry in the local structure surrounding the excess
proton. The general trend is for the most compact structures to have the lowest energy. However,
adding zero-point energy counteracts this trend, making prediction of the most stable isomer
impossible at this time. At elevated temperatures corresponding to recent experiments and
atmospheric conditions 共150–200 K兲, calculated Gibbs free energies clearly favor the least compact
structures, in agreement with recent thermal simulations 关Singer, McDonald, and Ojamäe, J. Chem.
Phys. 112, 710 共2000兲兴. © 2000 American Institute of Physics. 关S0021-9606共00兲30735-8兴
I. INTRODUCTION
contains the excess proton bound symmetrically between two
waters as in H5O⫹
2 . In this work we consider a wider range of
H⫹共H2O兲8 isomers and find no instances in which the minimum of the potential energy surface has the proton located
precisely halfway between two water molecules. Even where
imposed symmetry constraints force the proton into a symmetrical bond, we find the symmetric structure not to be a
true local minimum. Nevertheless, we find some cases where
the excess proton is rather close to a symmetrical arrangement.
The H⫹共H2O兲8 cluster was also one of those previously
chosen by some of us for thermal simulations17 using the
OSS2 empirical potential18 for H⫹共H2O兲n . We generated
many stable isomers of H⫹共H2O兲8 and, in this work, had
hopes of establishing which were likely to be observed in
low temperature beam experiments. We now know that
many of these isomers lie so close in energy that even high
level ab initio calculations do not have sufficient accuracy to
authoritatively identify the global minimum.
The study of protonated water clusters is further motivated by their abundance in the upper atmosphere19 and noctilucent clouds,20–22 and their presence in interstellar
clouds.23 Bondybey and co-workers24,25 have studied the reactivity of HCl with protonated water clusters, a model for
the early steps of ozone depletion reactions. Achatz et al.26
reported measurements of the rates of reaction of acetone
and acetaldehyde with protonated water clusters. The protonated water clusters of interest in current laboratory experiments or in the atmosphere are not particularly cold.
Schindler et al.24 estimated the temperature of their clusters
Compared to their neutral counterparts, relatively little is
known about protonated water clusters, H⫹共H2O兲n . After debate spanning many years 共reviewed by Ratcliffe and
Irish1,2兲, experiment3 and theory4–6 agree that protonated water dimer, H5O⫹
2 , is a symmetric structure in which the
shared proton lies midway between two oxygen atoms.
Ab initio calculations indicate that the excess proton in
H⫹共H2O兲3 and H⫹共H2O兲4 is closely associated with one of
the water molecules in a hydronium unit,5,7,8 although in
H⫹共H2O兲4 there is an isomer only 3.5 kcal mol above the
5,7
global minimum containing a symmetric, H5O⫹
2 -like bond.
⫹
⫹
Similar H3O - and H5O2 -like isomers have been noted for
H⫹共H2O兲6. 7,9 Wei and Salahub7 find the energy difference
between the two forms of H⫹共H2O兲6 to be much smaller than
for H⫹共H2O兲4, while Jiang et al.10 find the symmetric form
to be markedly lower in energy. Beyond these small clusters,
little is known conclusively, either experimentally or theoretically, about the structure of protonated water clusters. In
contrast, the structure of neutral water clusters (H2O) n is
well established through n⫽10.11–15
We have chosen to investigate H⫹共H2O兲8 in detail with
an empirical potential and ab initio techniques for several
reasons. From recent studies involving both vibrational spectroscopy and ab initio calculations, Chang et al.16 and Jiang
et al.10 suggest that, at 170 K, a fraction of their cluster
population of H⫹共H2O兲6 and, tentatively, also H⫹共H2O兲8
a兲
Electronic mail: singer@mps.ohio-state.edu
0021-9606/2000/113(13)/5321/10/$17.00
5321
© 2000 American Institute of Physics
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5322
J. Chem. Phys., Vol. 113, No. 13, 1 October 2000
Ciobanu et al.
FIG. 2. Double-ring structures of H⫹共H2O兲8 observed in basin-hopping
Monte Carlo 共Ref. 27兲 simulations using the OSS2 model 共Ref. 18兲. Structures a, b, and d are actually obtained from DFT/B3LYP structural optimizations, but at this level of depiction the OSS2 and DFT/B3LYP structures
are indistinguishable.
FIG. 1. Linear, branched, and single-ring structures of H⫹共H2O兲8 observed
in basin-hopping Monte Carlo 共Ref. 27兲 simulations using the OSS2 model
共Ref. 18兲. Structures a, c, e, and f are actually obtained from DFT/B3LYP
structural optimizations, but at this level of depiction the OSS2 and DFT/
B3LYP structures are indistinguishable.
to be near 140 K, while Jiang et al.10 reported their clusters
at 170 K. The temperature of polar stratospheric cloud particles has been measured19 to be about 190 K. Therefore, our
previously reported thermal simulations17 are relevant to interpretation of these experiments. The present ab initio investigation characterizes the local potential minima which
are visited by clusters near 200 K, and anticipates much
needed experimental investigations of cold H⫹共H2O兲n
clusters.
Our method for generating starting structures using the
OSS2 empirical potential18 is described in Sec. II. Ab initio
methods and results are given in Sec. III. In Sec. IV we focus
on a family of cluster isomers, the double five-membered
rings, to illustrate the issues that arise as we search for local
minima and attempt to characterize the local structure surrounding the excess proton. Having generated data on a relatively large number of H⫹共H2O兲8 isomers, in Sec. V we extract correlations between the local structure of the excess
proton, whether it is H3O⫹- or H5O⫹
2 -like, and the vibrational
spectrum. We indeed find a vibrational signature that distinguishes between the H3O⫹- and H5O⫹
2 -like environments.
Finally, we discuss the relation of our work to previous theoretical and experimental studies.
without side chains 共Fig. 1d兲, single rings with side chains
共Figs. 1c, 1e, 1f兲, double rings with and without side chains
共Fig. 2兲, triple rings 共Figs. 3a–3c兲, and quadruple ring 共Fig.
3d兲 structures. Within each topological class located in the
basin-hopping simulations, there were generally many representatives that differed in the directionality of the hydrogen
bonds and in the two orientations that two-coordinate waters
can assume and still maintain tetrahedral bonding directions
when they simultaneously accept and donate one H bond.
共The latter will be illustrated in Sec. IV.兲 All of the local
minima of the OSS2 potential energy surface obtained from
the basin-hopping simulations remained local minima at den-
II. ISOMERS OF H¿„H2O…8 FROM THE OSS2 MODEL
Our principle source of input structures for ab initio
study of H⫹共H2O兲8 was a basin-hopping Monte Carlo
simulation17 using the OSS2 potential18 for H⫹共H2O兲n , n
⫽8,16. The basin-hopping procedure27 was carried out at
400 K to help overcome the potential barriers separating local minima of H⫹共H2O兲8. It was continued until 1678 different minima were collected. Representative local minima include linear 共Fig. 1a兲, tree-like 共Fig. 1b兲, a single ring
FIG. 3. Multiple-ring structures of H⫹共H2O兲8 observed in basin-hopping
Monte Carlo 共Ref. 27兲 simulations using the OSS2 model 共Ref. 18兲 共a–d兲,
and two cubic structures 共e, f兲 which are local minima of the OSS2 potential
but are not observed in the basin-hopping simulations. Structures a, b, e, and
f are actually obtained from DFT/B3LYP structural optimizations, but at this
level of depiction the OSS2 and DFT/B3LYP structures are indistinguishable.
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J. Chem. Phys., Vol. 113, No. 13, 1 October 2000
sity functional theory using the Lee–Yang–Parr correlation28
and Becke’s three-parameter exchange29–32 functionals
共DFT/B3LYP兲 and second-order Møller–Plesset33–36 共MP2兲
ab initio levels of theory. In fact, the structures are so similar
that OSS2 and ab initio geometries are indistinguishable in
visualizations like Figs. 1–3.
The basin-hopping procedure allowed us to cast a wide
net for locally stable structures but, like any numerical
method, it does not guarantee a complete enumeration of
local minima on the potential energy surface. In fact, cubic
H⫹共H2O兲8 clusters 共Fig. 3, e, f兲 were never located in the
basin-hopping search, although they are indeed local minima
of the OSS2 potential.37 From an earlier study37 of (H2O) n
and H⫹共H2O兲n using graph theoretical methods to enumerate
H-bond topologies for a given structure, we know that there
are precisely 11 possible H-bond arrangements for cubic
H⫹共H2O兲8 with a hydronium ion at one vertex, and four possible H-bond topologies with the proton symmetrically
placed along the bond between two waters. These results
give the possible topologies, but not all may exist as local
minima of a potential energy surface. Using the OSS2
model, we were able to find six 共out of a possible 11兲 actual
minima with a hydronium at a vertex and two 共out of a
possible four兲 actual minima with a symmetrical, H5O⫹
2 -like
bond. The lowest energy representative, according to the
OSS2 model, from each of these two groups was selected for
study with ab initio methods in this work. The cubic structure with an H5O⫹
2 -like bond proved to be the one example
of a local minimum of the OSS2 surface that did not optimize to a corresponding local minimum with ab initio methods. Instead the excess proton migrated to one side to form
one of the eleven possible H3O⫹-containing water cubes. Incidentally, systematic graph theoretical methods can enumerate all the H-bonding topologies possible for each of the
structural types found in our basin-hopping search, or actually all the topologies possible for H⫹共H2O兲8, but given the
magnitude and complexity of analyzing all possible structures, these methods were not employed in this work.
III. MP2 AND DFTÕB3LYP STUDIES
We study the protonated water clusters at two different
levels of theory, DFT/B3LYP28–32 and MP2. The geometry
optimizations were performed with the ab initio packages,
38
NWCHEM version 3.2 for MP2 共with the largest gradient at
convergence set to 0.0008 E h /a 0 兲 and GAUSSIAN9839 for
DFT/B3LYP 共largest gradient 0.000 45 E h /a 0 兲. Geometry
optimization was rather slow for the MP2 calculations, the
number of steps taken being in the range of 100–200. The
DFT calculations were less demanding and the geometry optimization was faster, so we could afford smaller tolerances
for the energy gradient. Erratic geometrical optimization
steps were observed for GAUSSIAN98 with symmetryconstrained DFT/B3LYP optimization unless we used a finer
integration grid than the default choice. The starting structures for the geometry optimizations were local minima of
the OSS2 energy surface.
All geometry optimizations in this study, both B3LYP
and MP2, were carried out with the aug-cc-pVDZ* basis
Structure of H⫹(H2O)8
5323
previously used in calculations on water clusters by Tsai and
Jordan.40,41 This basis is a modified form of the aug-ccpVDZ basis set of Dunning and co-workers.42,43 The modification consists of the deletion of the diffuse p functions on
the hydrogen atoms, leaving a diffuse s function on each
hydrogen and diffuse s, p, d functions on oxygen. Tsai and
Jordan justified the deletion of the diffuse p functions on the
hydrogen atoms by the observation that it had no significant
effect on the binding energy of the water dimer. In conventional notation the primitive and contracted components of
the basis can be described as (10s5 p2d)→ 关 4s3p2d 兴 on
oxygen and (5s1 p)→ 关 3s1 p 兴 on hydrogen. In an attempt for
more economical calculations, we also tested the smaller ccpvDZ set 共without diffuse functions兲 in the DFT calculations.
We found that the geometries and the energy ordering of the
structures were greatly affected by using the smaller basis,
and our basis of choice remains aug-cc-pVDZ*.
Only Cartesian Gaussians, with six components for each
d function, were available in the computer program38 used to
perform all MP2 calculations. This choice resulted in a total
basis size of 302 contracted functions for H⫹共H2O兲8. The
B3LYP calculations, performed with a different package,39
used spherical Gaussians, with five d components, resulting
in a basis size of 286 contracted functions.
Diffuse basis functions, at least on nonhydrogen atoms,
were found to be important in calculations on hydrogenbonded systems by Del Bene and Shavitt.44,45 In particular,
including diffuse functions was found to reduce the effects of
basis-set superposition errors 共BSSEs兲. Correcting for BSSEs
in calculations on the H⫹共H2O兲8 cluster, particularly during
geometry optimization, would have been rather difficult, and
therefore it was considered important to reduce such errors at
the outset. Previous work on hydrogen-bonded systems45
共see also a detailed analysis by Xantheas46兲 showed that
much of the BSSE can be eliminated with the use of the
aug⬘-cc-pVTZ basis, which is a modified form of the aug-ccpVTZ basis of Dunning and co-workers.42,43 The modification in this case consists of omitting all diffuse functions on
the hydrogen atoms. This basis was found to give binding
energies close to those obtained with bigger basis sets.45
Single-point MP2 energy calculations were carried out with
the aug⬘-cc-pVTZ basis for many of the H⫹共H2O兲8 structures
at the optimized geometries obtained with the aug-cc-pVDZ*
basis in order to obtain more reliable relative energies. The
aug⬘-cc-pVTZ basis can be described in the conventional
notation as (11s6 p3d2 f )→ 关 5s4 p3d2 f 兴 on oxygen and
(5s2 p1d)→ 关 3s2 p1d 兴 on hydrogen, and using Cartesian
Gaussians, with six d components and ten f components,
adds up to 695 contracted functions for H⫹共H2O兲8.
Table I shows the MP2 energies of a number of clusters
calculated using the two basis sets, measured with respect to
the cubelike structure 共f兲 of Fig. 3. The relative energies
differ very little in the two cases 共less than 10%兲. We also
find that the dissociation energies 共the last two columns of
Table I兲 are in even better agreement, their values differing
by less than 1% when we change the basis set. We conclude
that our calculations are not affected by unacceptable basis
set superposition errors.
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5324
Ciobanu et al.
J. Chem. Phys., Vol. 113, No. 13, 1 October 2000
TABLE I. MP2 energies (E h ) relative to the energy of the protonated cubelike structure 共3e兲 and dissociation energies 共E h , last two columns兲 calculated using the double-zeta and triple-zeta basis sets. See the text for a
discussion of these basis sets.
⌬E 0
⌬E 0
E diss
E diss
Structure aug-cc-pVDZ* aug⬘-cc-pVTZ aug-cc-pVDZ* aug⬘-cc-pVTZ
3b
2b
1e
1f
2d
6b
1c
1a
0.003 163 99
0.004 795 85
0.005 051 23
0.005 982 16
0.006 017 02
0.006 309 11
0.007 285 19
0.020 679 43
0.002 836 70
0.004 604 22
0.004 869 25
0.005 722 65
0.005 651 89
0.006 210 61
0.007 034 34
0.019 682 15
0.207 602 25
0.205 970 40
0.205 715 02
0.204 784 09
0.204 749 22
0.204 457 12
0.203 485 82
0.190 086 81
0.207 932 77
0.206 165 26
0.205 900 23
0.205 046 82
0.205 117 58
0.204 558 86
0.203 735 14
0.191 087 33
A. Energetics
We performed geometry optimization on a diverse array
of H⫹共H2O兲8 structures, ranging from the floppy linear isomer 共1a兲 to rigid cubes like 共3e, 3f兲 共Figs. 1–3兲. The data are
presented in Tables II–V. The general trends are captured in
Fig. 4. The structures in Fig. 4 are arranged from left to right
roughly from most floppy to most compact. Actually, they
are ordered according to increasing Gibbs free energy at 190
K, which also places them almost perfectly in order of increasing zero-point energy. The general trend is for the most
compact of the H⫹共H2O兲8 structures to have the lowest energy, as depicted by the E 0 values in Fig. 4, but not as
distinctly favored as their neutral counterparts.11–15 As a result, when zero-point energy is added the result shows no
clear trend and all the structures fall within about 2000 cm⫺1
TABLE III. Dissociation energy and structural data for H⫹共H2O兲8 calculated at the DFT/B3LYP level. Structures are ordered going from H5O⫹
2 to
H3O⫹ character. The designations in the first column refer to the figures in
which the structures are shown. Structures followed by a point group symbol were constrained to have that symmetry during optimization. The two
energy columns give the dissociation free energy at 190 K 共including ZPE兲
and the dissociation energy 共no ZPE included兲 to 7H2O⫹H3O⫹. The first
three entries have at least one imaginary frequency and optimize to a structure with a slightly assymetrical H5O⫹
2 unit upon release of symmetry constraints. The parameter f is defined in Eq. 共1兲.
⌬G diss 共kcal/mol兲
⌬E diss(E h )
R OO⬘ 共Å兲
f
1f (C s )
⫺46.3361
0.204 859 51
2.4055
0.0000
⫹ ⫺
6兵⫺
兩 ⫹ 其 (C 2 )
⫺49.0051
0.205 331 19
2.4106
0.0000
⫺ ⫺
6兵⫺
兩 ⫺ 其 (C 2 )
⫺49.5099
0.205 242 25
2.4111
0.0000
⫹ ⫺
6兵⫺
兩 ⫹其
⫺48.4701
0.205 329 28
2.4106
0.0365
1a
1f
6b
3f
3b
3a
3e
1e
1c
⫺43.0456
⫺46.1049
⫺50.0313
⫺54.6953
⫺52.3350
⫺51.4745
⫺55.1680
⫺47.0298
⫺46.8874
0.194 807 84
0.206 278 16
0.205 560 05
0.206 577 26
0.208 158 42
0.201 887 71
0.206 217 22
0.208 011 72
0.205 306 26
2.4154
2.4210
2.4484
2.4866
2.4720
2.5138
2.5236
2.5195
2.5229
0.0619
0.0705
0.1130
0.1413
0.1437
0.1688
0.1737
0.1796
0.1841
Structure
of each other. These energies are too close to each other to
make a clear prediction of the global minimum energy isomer of H⫹共H2O兲8.
According to the estimated Gibbs free energy at either
170 or 190 K 共Table II兲, the more open structures are clearly
favored at elevated temperature. The temperatures 170 and
TABLE II. B3LYP 共aug-cc-pVDZ兲* energies of various H⫹共H2O兲8 geometries, reported in ascending order of
total free energy at 190 K 共second column兲. Columns 3–6 represent free energy at 170 K, electronic energy,
zero-point energy 共ZPE兲, and the sum of electronic and ZPE, respectively. Free energies are given in kcal/mol.
The designations in the first column refer to the figures in which the structures are shown. Structures followed
by a point group symbol were constrained to have that symmetry during optimization. Such a constraint would
result in a perfectly symmetric H5O⫹
2 unit, but in each case we found such constrained structures to be saddle
⫹ ⫺
⫹ ⫺
points. The fact that structure 6 兵 ⫺
兩 ⫹ 其 (C 2 ) lies lower in energy than 6 兵 ⫺
兩 ⫹ 其 is apparently an artifact of
numerical grid effects within GAUSSIAN98 共Ref. 39兲. When the grid parameter is increased from default (75
⫻302) to high (100⫻24⫻48) the energy of the symmetry-constrained structure is higher than the unconstrained structure, ⫺612.024 010 697 vs ⫺612.024 014 162 E h .
G 共190 K兲
G 共170 K兲
E 0 (E h )
ZPE (E h )
E 0 ⫹ZPE
1a
1f
1f (C s )
1c
1e
92.5526
95.6119
95.8431
96.3944
96.5368
95.5808
98.4450
98.5283
99.2398
99.3657
⫺612.013 517 349
⫺612.024 987 669
⫺612.023 569 021
⫺612.024 015 774
⫺612.026 721 227
0.202 838
0.205 105
0.204 073
0.207 138
0.207 093
⫺611.810 679 349
⫺611.819 882 669
⫺611.819 496 021
⫺611.816 877 774
⫺611.819 628 227
⫹ ⫺
6兵⫺
兩 ⫹其
97.9771
100.5954
⫺612.024 038 787
0.206 887
⫺611.817 151 787
⫹ ⫺
6兵⫺
兩 ⫹ 其 (C 2 )
98.5121
101.0247
⫺612.024 040 702
0.206 557
⫺611.817 483 702
⫺ ⫺
6兵⫺
兩 ⫺其
99.0169
101.4261
⫺612.023 951 762
0.206 232
⫺611.817 719 762
99.5383
100.9815
101.8420
104.2023
104.6750
102.0656
103.4190
104.2019
106.4027
106.8418
⫺612.024 269 555
⫺612.020 597 225
⫺612.026 867 931
⫺612.025 286 769
⫺612.024 926 731
0.208 724
0.210 198
0.210 810
0.213 057
0.213 410
⫺611.815 545 555
⫺611.810 399 225
⫺611.816 057 931
⫺611.812 229 769
⫺611.811 516 731
Structure
6b
3a
3b
3f
3e
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Structure of H⫹(H2O)8
J. Chem. Phys., Vol. 113, No. 13, 1 October 2000
5325
TABLE IV. Dissociation energies to 7H2O⫹H3O⫹ of a family of double
five-ring H⫹共H2O兲8 clusters calculated using the OSS2 potential model. The
structures are shown in Fig. 6.
Structure
⌬E diss(E h )
⫹ ⫺
兩 ⫹其
兵⫺
0.195 153
⫺ ⫺
兩 ⫹其
兵⫺
0.194 997
⫹ ⫺
兩 ⫺其
兵⫹
0.194 818
⫺ ⫺
兩 ⫺其
兵⫺
0.194 798
⫹ ⫹
兩 ⫺其
兵⫺
0.194 637
⫺ ⫹
兩 ⫹其
兵⫹
0.194 555
190 K were chosen because they correspond to conditions
reported for recent experiments10 and for polar stratospheric
clouds,19 respectively. The free energies clearly favor the
less compact structures at elevated temperature, and provide
some confirmation of the break-up of ring-containing structures in this temperature range, as was predicted in thermal
simulations using the OSS2 potential.17 However, our reported free energies should only be taken as a rough guide.
The free energies reported in Table II are based on a harmonic model for internal motion within the cluster. Thermal
simulations17 indicate that very large-amplitude motion, including barrier crossings between local minima of the potential energy surface, is prevalent in this temperature range.
Therefore, free energies calculated from a harmonic model
have to be used with caution in selecting the most probable
structure at elevated temperature.
DFT/B3LYP, MP2, and OSS2 are in rough agreement
with respect to the trends in energy values (E 0 ). The major
discrepancy between the various models occurs in the energy
of the most compact structures. MP2 calculations favor the
protonated cubes 共Figs. 3e, 3f兲 as the lowest energy structure,
while DFT/B3LYP and OSS217 both favor the somewhat
more open three-ring structure 3b. We do not report zeropoint energies at the MP2 level since the size of our basis
sets made frequency calculations at this level impractical.
TABLE V. Energies and structural data for H⫹共H2O兲8 calculated at the MP2
⫹
level. Structures are ordered going from H5O⫹
2 to H3O character. The designations in the first column refer to the figures in which the structures are
shown. The two energy columns give the energy and the dissociation energy
共no ZPE included兲 to 7H2O⫹H3O⫹. The parameter f is defined in Eq. 共1兲.
Structure
E 0 (E h )
⌬E diss(E h )
R OO⬘ 共Å兲
f
⫹ ⫺
6兵⫺
兩 ⫹其
⫺610.575 261 5
0.206 000 571
2.422 52
0.069 412
1f
1a
6b
3f
3b
3a
3e
1e
2d
1c
2b
⫺610.574 045 0
⫺610.559 347 7
⫺610.573 718 0
⫺610.579 995 9
⫺610.576 863 1
⫺610.572 379 9
⫺610.580 027 1
⫺610.574 975 9
⫺610 574 975 9
⫺610 572 746 7
⫺610 575 231 3
0.204 784 089
0.190 086 807
0.204 457 123
0.210 734 953
0.207 602 248
0.203 119 028
0.210 766 237
0.205 715 018
0.204 749 216
0.203 485 817
0.205 970 396
2.425 21
2.427 70
2.456 56
2.487 16
2.482 94
2.516 14
2.526 72
2.521 36
2.527 16
2.532 00
2.535 00
0.083 857
0.087 312
0.126 399
0.144 927
0.156 992
0.172 954
0.177 666
0.180 728
0.185 331
0.191 245
0.193 930
FIG. 4. Energy trends among isomers of H⫹共H2O兲8. The designations along
the horizontal axis refer to the figures in which the structures are shown. The
three solid curves depict the energy E 0 of various structures using DFT/
B3LYP 共䊊兲, MP2 共䉭兲, and the OSS2 potential 共䊐兲. The zero point energy
共dot-dash兲, E 0 ⫹ZPE 共long dash兲, and Gibbs free energy at 190 K 共short
dash兲 are all calculated by DFT/B3LYP. All energies are referred to the
corresponding energy of structure la to facilitate comparison. The general
trend is for structures to become more compact going from left to right. E 0
generally decreases in this direction, but, counteracting this trend, zero-point
energy increases and entropy decreases.
B. Is the excess proton in a symmetrical H bond?
There is continued speculation as to whether the excess
proton is located symmetrically or nonsymmetrically between two oxygen atoms for the larger proton hydrate clusters and, by extension, for the excess proton in bulk. In reality there is a continuum of possible structures between the
hydronium and H5O⫹
2 limits. We identified the hydronium or
the H5O⫹
2 unit within our structures by monitoring bond
length. The degree to which the configuration of the excess
proton within our set of H⫹共H2O兲8 clusters resembles H3O⫹
or H5O⫹
2 is quantified by defining a simple measure of symmetry of the excess proton’s environment 共Fig. 5兲. By inspection of bond lengths, all the hydrogen atoms in our
H⫹共H2O兲8 clusters can be associated with a single oxygen,
forming either water or hydronium units, or if a hydrogen is
equally close to two oxygen neighbors, a perfectly symmetric H5O⫹
2 unit. When a hydronium unit can be identified, let
O⬘ designate the closest oxygen atom to the oxygen of the
hydronium, so that O–H¯O⬘ is the shortest hydrogen bond
formed with the hydronium, and let r OH and r OH⬘ be the
oxygen–hydrogen bond distances within that hydrogen bond,
as shown in Fig. 5. Note that by definition r OH⭐r O⬘ H⬘ , with
the equality applicable in the symmetrical limit. Then a dimensionless measure of the symmetry of the excess proton’s
environment is
f⬅
r O⬘ H⫺r OH
,
r OH⬘ ⫹r OH
共1兲
which smoothly interpolates from values near 0.2 in the
H3O⫹ limit to zero as the excess proton’s environment approaches the symmetrical, H5O⫹
2 -type limit.
The length R OO⬘ of the shortest hydrogen bond associated with the hydronium unit correlates quite well with f, our
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5326
Ciobanu et al.
J. Chem. Phys., Vol. 113, No. 13, 1 October 2000
FIG. 5. The hydrogen bond to the excess proton contracts as the excess
proton changes from H3O⫹- to H5 O⫹
2 -like in character. The parameter f
measures the asymmetry of this bond, and vanishes in the H5 O⫹
2 -like limit.
R OO ⬘ is the length of the shortest H bond to the hydronium unit. Closed and
open points are from DFT/B3LYP and MP2 calculations, respectively. The
nearly overlapping points at f ⫽0 are from symmetry-constrained optimizations, and correspond to saddle-point structures. The smooth curve is a
quadratic fit to the data R OO⬘ (Å)⫽2.405⫹3.595f 2 . Agmon 共Ref. 47兲 has
used bond-energy–bond-order 共BEBO兲 relations 共Refs. 48–50兲 to correlate
oxygen–hydrogen bond lengths within a hydrogen bond with the overall
hydrogen bond length. The dashed line shows the prediction using Agmon’s
empirical parameters for water clusters and assuming that the hydrogen
bonds depart uniformly from linearity by ⬃11°, as discussed in the text.
measure of H3O⫹/H5O⫹
2 character, as shown in Fig. 5. The
relationship is approximately fit by R OO⬘ (Å)⫽2.405
⫹3.595f 2 , for both MP2 and DFT/B3LYP data. Also apparent from Fig. 5 is that the H⫹共H2O兲8 isomers we have chosen
to study yield a distribution of f values, more or less dense
⫹
between the H5O⫹
2 and H3O limits. We have seen in previously reported thermal simulations17 that above about 180 K
the H⫹共H2O兲8 cluster is found almost exclusively in
hydronium-like configurations. Here we find there is no
shortage of local minima of either H3O⫹-, H5O⫹
2 -like, or
intermediate character. However, as mentioned previously,
we have not discovered a perfectly symmetrical H5O⫹
2 -like
local minimum of H⫹共H2O兲8.
It is possible to provide some interpretation of the trend
shown in Fig. 5. Agmon47 has used bond-energy–bond-order
共BEBO兲 relations48–50 to correlate oxygen–hydrogen bond
lengths within a hydrogen bond with the overall hydrogen
bond length. In our terminology, the BEBO relations become
n⫽exp关 ⫺ 共 r OH⫺r s 兲 /a 兴 ,
n ⬘ ⫽exp关 ⫺ 共 r O⬘ H⫺r s 兲 /a 兴 , n⫹n ⬘ ⫽1,
共2兲
where n and n ⬘ are the bond orders of the colavent and
H-bonded oxygen–hydrogen bonds, whose sum is con-
served. Agmon found that r s ⫽0.956 Å and a⫽0.385 Å fit
data from water clusters.47 Relations 共2兲 are sufficient to predict R OO⬘ as a function of f provided we assume a linear
hydrogen bond (r OH⫹r O⬘ H⫽R OO⬘ ). Compared with our calculations, this gives the trend in R OO⬘ vs f quite well, although the values of R OO⬘ are about 8.5% too large. Scaling
the BEBO prediction of R OO⬘ uniformly by 0.9815, that is,
assuming that the hydrogen bonds uniformly depart from
linearity by ⬃11°, produces the best agreement with the ab
initio data and is shown by the dashed line in Fig. 5. While
the hydrogen bond involving the excess proton does depart
from linearity by this magnitude in some of the isomers, like
3b and 3e, the departure from linearity is usually not this
great. The necessity for the scaling factor more accurately
reflects that the BEBO parameters should be slightly altered
for the clusters studied in this work.
Since bond lengths clearly indicate an H3O⫹ or H5O⫹
2
unit as special within our optimized structures, we speak of
these units as containing the ‘‘excess proton.’’ This language
is flawed because it suggests that the excess charge resides
on one or several hydrogens of the H3O⫹ or H5O⫹
2 unit. In
actuality the excess charge, as estimated by Mulliken
charges,51,52 is dispersed over the entire H3O⫹ or H5O⫹
2 unit
and beyond, as illustrated by the following discussion. In our
density functional calculations, the H3O⫹ typically contains
excess charge close to 0.78e (e⫽ 兩 electron charge兩 ), with the
rest of the excess charge 共from Mulliken population analysis兲
spread over neighboring water molecules. Water molecules
which are far from the excess charge have Mulliken populations of roughly ⫹0.22e and ⫺0.44e on hydrogen and oxygen, respectively. We estimate how the 0.78e excess charge
is distributed among the hydrogen and oxygen atoms of the
hydronium unit by comparing their Mulliken populations
with those of waters far from an excess charge, and find that
the excess charge is rather evenly distributed among the atoms of the hydronium unit. From ⬃0.2e to ⬃0.25e resides
on the oxygen and the remainder is located on the three
hydrogens of the H3O⫹ unit. For example, the Mulliken
charges associated with the H3O⫹ unit of the protonated cube
structure, Fig. 3e, are ⫺0.26e on the oxygen and ⫹0.28e,
⫹0.37e, and ⫹0.37e for the hydrogens. The charge on the
hydrogens is especially variable, and it is not uncommon for
the charge on hydrogens far from the excess proton to exceed
the charge on one of the hydrogens of the H3O⫹ unit. Turning now to the symmetrical limit, one typically finds 0.85e
on the H5O⫹
2 unit with ⬃0.1e on each of the oxygens,
⬃0.20e on the central hydrogen, and the remainder on the
other four hydrogens of the H5O⫹
2 unit.
With the caveat that the location of what we call the
‘‘excess proton’’ does not indicate a localization of the excess charge on a single atom, and for lack of better terminology and measure of excess charge,52 we shall continue to use
this language. The principle advantage of singling out the
hydrogen in the shortest H bond to the hydronium is that it
affords us a definition that smoothly connects the hydronium
and H5O⫹
2 -type limit.
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Structure of H⫹(H2O)8
J. Chem. Phys., Vol. 113, No. 13, 1 October 2000
5327
⫺ ⫺
兩 ⫺ 其 , according to whether the dangling hydrogens point in
兵⫺
FIG. 6. A small sampling of the many local minima of the OSS2 potential
with the topology of two edge-sharing five-membered rings. The top six
structures have the same overall hydrogen bond topology, and only differ in
the orientation of the dangling hydrogen on the the four single-acceptor/
single-donor waters. The notation describes this orientation, with a plus or
minus sign indicating the position of the dangling hydrogen with respect to
the plane of the rings and the horizontal line separating the two rings. The
OSS2 potential has local minima for a wide variety of other edge-sharing
double five-rings, and the two structures at the bottom are of possible interest either because of near C 2 symmetry 共a兲 or because an H5 O⫹
2 -type bond
appears to be forming in a position different from the edge shared by the
two five-rings 共b兲.
IV. DOUBLE FIVE-RING H¿„H2O…8
A family of double five-ring isomers 共Fig. 6兲 illustrates
the subtleties of locating local minima and characterizing the
symmetry of the excess proton’s environment. A double
five-ring structure of this type for H⫹共H2O兲8 was, to our
knowledge, first identified by Cheng.9 This structure was
also examined recently by Jiang et al.,10 and identified as
their prime candidate for the isomer of H⫹共H2O兲8 observed
in their beam experiments. Actually, the structures depicted
by Cheng and by Jiang et al. are not the same, and a wide
variety of local minima based on the simple double five-ring
structure are found, some with a clear hydronium-like or
H5O⫹
2 -like excess proton and most with intermediate character. In some structures the excess charge can be found in the
H bond shared between the two rings, and in others on one of
the two rings.
In optimizations using both the OSS2 potential and the
ab initio calculations, the double five-ring tends to pucker.
We define the ⫹z direction as the direction perpendicular to
the shared H bond which makes the puckered rings concave
upward. We first examine a family of double five-ring structures in which the end waters furthest from the shared H
bond are double acceptors 共the top six structures of Fig. 6兲.
The four water molecules which are neither end waters nor
part of the shared H bond are single acceptors/single donors.
If one follows the simple notion that water molecules accept
H bonds to one of two lone pairs tetrahedrally disposed
around the oxygen, as is borne out in ab initio and experimental data,53 then the dangling hydrogens of the four
single-acceptor/single-donor waters will point either above
or below the five-ring, depending on which lone pair accepts
the H bond. We label the seven possible arrangements within
⫹ ⫹
⫹ ⫹
⫹ ⫺
⫹ ⫺
⫹ ⫹
⫺ ⫺
兩 ⫹其 , 兵 ⫺
兩 ⫹其 , 兵 ⫹
兩 ⫺其 , 兵 ⫺
兩 ⫹其 , 兵 ⫺
兩 ⫺其 , 兵 ⫹
兩 ⫺其 ,
this family as 兵 ⫹
the ⫹z or ⫺z direction. The horizontal line separating the
two rows corresponds to the shared edge of the two fiverings, and the plus signs and minus signs give the orientation
of the dangling hydrogens with the ⫹z direction pointing
from the page to the reader. Also note that structures like
⫹ ⫺
⫺ ⫹
兩 ⫹ 其 and 兵 ⫹
兩 ⫺ 其 are enantiomers.
兵⫺
Six of the possible seven structures listed above were
located as minima of the OSS2 potential energy surface, the
one omission being the structure with all plus signs. The
energies of these structures under the OSS2 potential model
are given in Table IV. The structures are quite close in energy. Associating an energy cost with dangling hydrogens in
close proximity accounts for the trend in energy, just as we
found for other clusters.37 Dangling hydrogens that are adjacent, either across a five-ring or found on the same side of
two five-rings, prefer to be on opposite sides of the rings,
⫹ ⫺
兩 ⫹ 其 is most stable. Dangling hydrogens on
explaining why 兵 ⫺
the same side of the five-rings are best tolerated on the convex side of the structure than the concave side, since dangling hydrogens are closer together on the concave side. This
fact may explain why having all dangling hydrogens on the
⫹ ⫹
兩 ⫹ 其 produces an unstable structure rather
concave side in 兵 ⫹
than a local minimum on the OSS2 surface. Three of the
⫹ ⫺
⫺ ⫺
⫹ ⫺
兩 ⫹其 , 兵 ⫺
兩 ⫺ 其 , and 兵 ⫹
兩 ⫺ 其 , because of their
located structures, 兵 ⫺
near C 2 , C 2 v , and C s symmetry, respectively, are candidates for having the excess proton in a perfectly symmetrical
environment within the shared H bond. If any of the local
minima turned out to have precisely these point group symmetries, then the central H5O⫹
2 -like bond would have been
perfectly symmetrical. However, the optimized OSS2 potential structures broke symmetry in each case, and hence the
H5O⫹
2 unit departs slightly from perfect symmetry. We attempted to optimize the first two such structures under C 2
symmetry using DFT/B3LYP, and in both cases symmetryconstrained optimization led to structures that had imaginary
vibrational frequencies and were not true local minima. We
have included the symmetry-constrained minima as data
points in Fig. 5 to discover trends in H-bond lengths and
vibrational spectra versus f.
Structure a in Fig. 6 illustrates that there are other double
five-ring structures that are candidates for a perfectly symmetrical H5O⫹
2 -like bond besides the family considered to
this point. However, like all other structures, perfect symmetry, C 2 in this case, is broken in structure a within the OSS2
model. Structure b in Fig. 6 illustrates that an H5O⫹
2 -like
excess proton may be found in various locations, not just the
shared edge of the double five-ring.
V. VIBRATIONAL SPECTRA
Our data set of H⫹共H2O兲8 local minima, spanning a
range of structures, allows us to explore whether there is a
spectroscopic signature of the symmetry of the excess proton’s environment in the vibrational spectrum. The answer is
that there is indeed such a signature in the vibrational modes
below 2800 cm⫺1.
In Fig. 7 the vibrational spectra of 12 isomers of
H⫹共H2O兲8, obtained from DFT/B3LYP calculations in the
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5328
J. Chem. Phys., Vol. 113, No. 13, 1 October 2000
Ciobanu et al.
FIG. 7. Infrared spectra of H⫹共H2O兲8 clusters, arranged
from top to bottom according to decreasing symmetry
of the excess proton’s environment as measured by the
parameter f 关Eq. 共1兲兴. The spectra are obtained from
DFT/B3LYP calculations in the harmonic approximation. The label where the corresponding structure can
be located is given toward the left-hand side of each
spectrum. The mode with imaginary frequency in the
top spectrum is placed on the negative frequency axis.
The intense vibrations below 2800 cm⫺1 move to
higher frequency with increasing f, while the vibrations
above 2800 cm⫺1 are relatively insensitive to the symmetry of the excess proton’s environment. This trend is
quantified in Fig. 8.
harmonic approximation, are arranged in order of increasing
f, that is, in order from most symmetrical to least symmetrical excess proton. The top spectrum of Fig. 7 is that of the
⫹ ⫺
兩 ⫹ 其 in Fig. 6兲 optimized under
double five-ring 共labeled 兵 ⫺
the constraint of C 2 symmetry. The excess proton in this
structure is exactly equidistant between two oxygens. The
intense infrared active modes in the 690–1000 cm⫺1 range
are true proton transfer modes: The proton shuffles back and
forth between the two oxygens, its motion strongly coupled
to bends and wags of the outer hydrogens of the H5O⫹
2 unit.
These modes are least likely to be harmonic in nature, since
the effective potential for the central proton is known to be
rather flat.5,54 The strong IR-active modes above 2800 cm⫺1
involve very little central proton motion. They are principally stretching of the outer OH bonds of the H5O⫹
2 unit.
As mentioned earlier, the double five-ring designated as
⫹ ⫺
兩 ⫹ 其 is not stable in C 2 symmetry. The symmetry breaking
兵⫺
can be viewed as a consequence of the effective potential for
the central excess proton exhibiting a double-well character
with a maximum at the midpoint. The structure, energy, and
distribution of vibrational frequencies in the symmetry con⫹ ⫺
兩 ⫹ 其 共the two top
strained and unconstrained versions of 兵 ⫺
spectra in Fig. 7兲 are quite similar, so we suppose that the
double-well character is not very pronounced. As one proceeds to structures with increasing values of f 共toward spectra at the bottom of Fig. 7兲 either the double-well character
becomes increasingly pronounced, and the excess proton is
gradually localized on one side of the double well, or the
symmetric well shifts to one much deeper toward one side.
The cluster geometry determines which scenario applies.
What was a proton transfer mode for the near-symmetric
cluster turns, with increasing f, to localized vibrations of an
H3O⫹ unit 共with tunnel splittings if the effective double-well
is symmetric兲.
The trend from symmetric to nonsymmetric excess pro⫹
ton, from an H5O⫹
2 - to H3O -like configuration, is clearly
reflected in the IR-active modes below 2800 cm⫺1. As the
motion of the excess proton evolves from proton transfer to
H3O⫹ stretch in character, the locus of intensely IR-active
modes shifts to higher frequency and merges with what
were, at lower f, stretches of the outer OH bonds of an H5O⫹
2
unit. This is to be expected, since in the H3O⫹-like limit
there is no distinction between a central proton and outer
hydrogens. This trend is quantified in Fig. 8. The vibrational
modes were separated into groups above and below 2800
cm⫺1 and the intensity-weighted average frequency
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Structure of H⫹(H2O)8
J. Chem. Phys., Vol. 113, No. 13, 1 October 2000
FIG. 8. The intensity-weighted average vibrational frequency of modes below and above 2800 cm⫺1 共closed and open symbols, respectively兲 are plotted as a function of f, the measure of the symmetry of the excess proton’s
environment. The solid lines are linear least-squares fits to the data. The
average frequency of the modes below 2800 cm⫺1 increases by about 1000
cm⫺1 in the range of structures studied, while the modes above 2800 cm⫺1
have nearly constant average frequency. The linear least-squares fit to the
data below 2800 cm⫺1 is 具 ␻ 共cm⫺1 ) 典 ⫽778⫹5366f .
具␻典⫽
兺 i I i␻ i
兺i Ii
共3兲
of the two groups separately calculated. 共In this equation, I i
is the IR intensity of vibrational mode with frequency ␻ i .兲
The modes above 2800 cm⫺1 have nearly unchanging average frequency, while those below 2800 cm⫺1 increase by
⫹
about 1000 cm⫺1 going from the H5O⫹
2 to the H3O limit.
The average frequency of the latter group is roughly fit by
具 ␻ (cm⫺1) 典 ⫽778⫹5366f . The degree of symmetry in the
environment of the excess proton can therefore be extracted
from vibrational spectra below ⬃2800 cm⫺1.
5329
tween two oxygens in two equivalent directions. Hence the
proton transfer coordinate is effectively a symmetric double
well. If the zero-point energy along this coordinate is in excess of the barrier height separating the two wells, similar to
a dynamic Jahn–Teller effect, then the ground state wave
function will have significant density in the symmetrical region. This effect was observed by Ojamäe et al. in reduced5
dimenional model calculations on the H5O⫹
2 ion. At a
slightly stretched oxygen–oxygen separation of 2.6 Å the
proton transfer potential has a small barrier, yet the vibrational wave function has a single maximum.
Chang and co-workers55 have pointed out another factor
which may tend to drive the excess proton away from a
nonsymmetrical potential energy minimum and possibly toward a more symmetric average position. Viewing the proton
transfer coordinate in an adiabatic picture, they calculated
the zero-point energy of several protonated dimethyl ether–
water 关 H⫹关共CH3兲2O兴共H2O兲n 兴 clusters as a function of the
proton transfer coordinate. They found that, in some cases,
the zero-point energy changes significantly along the coordinate, producing a significant shift in the average position of
the excess proton.
The results presented here and the complementary results of thermal simulations using the OSS2 model17 indicate
that care must be taken in the interpretation of vibrational
spectra of H⫹共H2O兲8 in the currently accessible temperature
range.10 The data of Fig. 7 show that there is no clear signa⫺1
ture of an H5O⫹
2 -like excess proton above 2800 cm , unlike
⫺1
the marker at ⬃3180 cm that Jiang et al. could exploit for
H⫹共H2O兲6. In Fig. 7 one finds vibrational peaks in this fre⫹
quency range in both the top 共H5O⫹
2 limit兲 and bottom 共H3O
limit兲 of the figure. The clear signature of symmetrical or
nonsymmetrical excess proton is instead found at lower frequency.
VI. DISCUSSION
ACKNOWLEDGMENTS
In recent work9,10,37 there has perhaps been too much
emphasis on categorizing the excess proton in H⫹共H2O兲n as
either H3O⫹- or H5O⫹
2 -like. This article shows that, at least
for H⫹共H2O兲8, a continuum of structures between these two
limits is possible, and where one draws the boundary between the two types is somewhat arbitrary. At the level of
theory we employed we did not find a single truly symmetri⫹
cal H5O⫹
2 -like excess proton in H 共H2O兲8, although there
were examples in which the deviation from such symmetry
was small. The wide variety of cluster isomers we had studied has enabled us to identify qualitative trends which will
aid in the interpretation of experiments: The degree of asymmetry of the excess proton’s environment is correlated with
the bond distance to the nearest water molecule, and, most
important, there is a clear upward shift in the vibrations be⫹
low 2800 cm⫺1 from the H5O⫹
2 to the H3O limit.
Even though we never found the minimum of the potential energy surface where the excess proton is precisely halfway between two water molecules, the possibility remains
that zero-point vibrational effects may cause some isomers of
H⫹共H2O兲8 to be effectively symmetric.55 In isomers like
⫹ ⫺
兩 ⫹ 其 the excess proton can depart from the midpoint be兵⫺
The calculations reported here were made possible by
resource grants from the EMSL Molecular Science Computing Facility at Pacific Northwest National Laboratory, the
Swedish National Supercomputer Center 共NSC兲, and the
Ohio Supercomputer Center. The authors wish to thank Robert Harrison and David Bernholdt for useful discussions and
technical assistance. This research was partially supported by
the Swedish Natural Science Research Council 共NFR兲. We
are grateful to Huan-Cheng Chang for bringing Ref. 55 to
our attention, and for alerting us to the possibility that zeropoint vibrational effects may cause some isomers to be effectively symmetric.
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