Water clusters: Untangling the mysteries of the liquid, one molecule at a time
Материалы собраны Мосиным Олегом.
Frank N. Keutsch* and Richard J. Saykally†
Department of Chemistry, University of California, Berkeley, CA 94720-1460
This contribution is part of the special series of Inaugural Articles by members of the National Academy of Sciences elected on April 27, 1999.
Contributed by Richard J. Saykally, May 29, 2001
Extensive terahertz laser vibration-rotation-tunneling spectra and
mid-IR laser spectra have been compiled for several isotopomers of
small (dimer through hexamer) water clusters. These data, in
conjunction with new theoretical advances, quantify the struc
tures, force ﬁelds, dipole moments, and hydrogen bond rearrange
ment dynamics in these clusters. This new information permits us
to systematically untangle the intricacies associated with cooper
ative hydrogen bonding and promises to lead to a more complete
molecular description of the liquid and solid phases of water,
including an accurate universal force ﬁeld.
The quest to achieve an accurate description of liquid water
has produced major advances in the last t wo decades (1), yet
despite the constr uction of hundreds of model force fields for use
in simulations, the great advances in comput ational technolog y,
and the development of power ful ab initio molecular dynamics
methods, we remain unable to accurately calculate the properties
of liquid water (e.g., heat capacit y, densit y, dielectric const ant,
compressibilit y) over significant ranges in conditions (2). We do
not yet have a satisfactor y molecular description of how a proton
moves in the liquid, we do not fully underst and the molecular
nature of the sur faces of either ice or liquid water (3), nor do we
underst and the origin of the intriguing anomalies and singulari
ties found in the deeply supercooled region (4). Although it is
clear that the hydrogen bond net work and its f luctuations and
rearrangement dynamics deter mine the properties of the liquid,
no experiment al studies ex ist that reveal det ailed infor mation on
a molecular level without considerable interpret ation (5). More
over, the reliabilit y of water models for simulating solvation
phenomena and biological processes remains relatively untested.
A principal obst acle to resolv ing these issues is that of
correctly describing the many-body, or cooperative nature of the
hydrogen bonding interactions among a collection of water
molecules. Theoretical work has shown that the H-bond is
dominated by electrost atic interactions, balanced by the repul
sive electron exchange, but that dispersion makes an appreciable
contribution, whereas induction (polarization) is the dominant
many-body ef fect (6, 7). It has proven notoriously dif ficult to
accurately parameterize these interactions f rom ab initio calcu
lations. Moreover, the ab initio molecular dynamics methods are
based on densit y functional methods that explicitly omit the
dispersion, and its expense mandates rather small sample sizes
(e.g., 64 molecules) in simulations (8). But perhaps the central
obst acle to developing quantit atively accurate and general meth
ods has simply been the lack of a suit ably precise dat a set with
which to test and calibrate theoretical approaches.
The central goal of the research rev iewed below is to advance
the cause for accurately describing water in all its phases over
arbitrarily large ranges of conditions, and the central contribu
tion of our group has been to develop and apply novel methods
of laser spectroscopy for the highly det ailed study of water
clusters to prov ide such a dat a set. Recently, we also have
initiated studies of the hydrogen bond breaking dynamics in
water clusters and comparison of them with mechanisms pro
posed to prevail in liquid water.
Terahertz Laser Vibration-Rotation-Tunneling (VRT) Spectroscopy of Clusters
The first far-IR (FIR) spectra of gaseous water clusters were
measured near 22 cm1 (455 m) by Busarow et al . in 1989 (9).
The spectra consisted of 56 Ka 2 4 1 rot ation-tunneling
transitions of (H2O)2, which complemented the microwave dat a
(10, 11) obt ained by the pioneering work of Dyke et al . (10), in
obt aining an accurate description of the dimer ground st ate.
Zwart et al . (12) subsequently extended these dat a to other
quantum st ates. Af ter some import ant technical developments
that extended the operating range of the spectrometer to higher
f requencies, Pugliano and Saykally (13) first measured an inter
molecular VRT spectr um of a water cluster in 1992, with the
detection of a torsional v ibration of the D2O trimer near 89.5
cm1 (112 m) (Fig. 1) (14 –16). This striking spectr um exhibited
an exact symmetric rotor pattern, and ever y rot ational line was
split into a distinctive quartet pattern that we now know results
f rom quantum tunneling v ia t wo dif ferent hydrogen bond path
ways connecting 48 degenerate minima on the 12-dimensional
inter molecular potential sur face. Pugliano et al . (17) quickly
followed with the first obser vation of a dimer inter molecular
v ibration (acceptor t wist), near 83 cm1 (120 m).
Subsequent work at Berkeley by Liu et al . (18) produced much
more extensive trimer spectra and the first det ailed assignment
of the transitions. Cr uzan et al . (19) discovered VRT spectra of
the tetramer shortly af ter ward, and Liu et al . followed with the
detection of the pent amer (20) and hexamer (21). Recent ef forts
have produced highly det ailed characterizations of both the
dimer and trimer, as well as greatly expanded dat a for the other
clusters (22–27). We describe the current underst anding of the
dimer through hexamer clusters that has been achieved f rom
these dat a, and through the ef forts of many concurrent theo
retical studies, in a later section.
IR Cavity Ringdown Spectroscopy
While mid-IR spectra of water clusters had been obser ved by the
Pimentel group in matrix studies in 1957 (28), the OH stretching
v ibrations of gaseous water clusters were first studied indirectly
in 1982 by Vernon et al . (29) in IR predissociation experiments
in supersonic beams, and shortly af ter that by Page et al . (30).
Vernon et al . assigned the spectra to (H2O)n, n 1–5, and
recorded a narrow transitions (15 cm1) at 3,715 cm1, which
they attributed to the f ree OH stretch in cyclic water clusters, and
a much broader feature (200 cm1) at lower f requency that they
attributed to the bound OH stretch. Page et al . concentrated on
the water dimer, finding four peaks, including the bound OH
stretch, a broad transition at 3,545 cm1, red-shif ted f rom the
f ree monomer OH stretches. Coker et al . (31) found four dimer
OH stretch f requencies identical to those deter mined by Page
Abbreviations: VRT, vibration-rotation-tunneling; FIR, far-IR; ASP, anisotropic site poten
tial; IPS, intermolecular potential surface.
*Present address: Department of Chemistry and Chemical Biology, Harvard University,
Cambridge, MA 02138.
To whom reprint requests should be addressed. E-mail: email@example.com.
and also identified larger clusters in supersonic expansions carr ying increasing concentrations of water. Huang and Miller
(32, 33) reported the first rot ationally resolved spectr um of
(H2O)2 and obser ved the four OH stretch v ibrations, and
recently Frochtenicht et al . (34) used a size selection technique
in which a He beam is used to eject clusters f rom a molecular
beam as a function of their size. They were able to measure the
f ree and bound OH stretching f requencies for clusters up to the
The wide tuning range of our IR cav it y ringdown laser
absorption spectrometer recently per mitted the first det ailed
studies of both the covalent bending v ibrations of H2O clusters
(35), which occur near 1,600 cm1, and the stretching v ibrations
of D2O clusters (36, 37), which fall near 2,700 cm1 (Fig. 2). All
of the obser ved clusters except the dimer exhibit strong v ibra
tional predissociation broadening of their OD stretch spectra
that obscures rot ation-tunneling features. For the D2O dimer,
however, the ac ceptor antisy mmetric stretch exhibits well
resolved acceptor switching doublets for each rot ational line,
whereas the donor stretch exhibits rot ational lines that are
broadened, but by about 30 times less than found for the H2O
isotopomer (36, 37). All bands obser ved for the cluster HOH
bending v ibrations are severely broadened, implying a stronger
coupling with the dissociation coordinate (35). The sharp
rot ation-tunneling str ucture measured for (D2O)2 (Fig. 2b) was
import ant for the deter mination of the dimer potential sur face
(38, 39), because the acceptor switching splittings cannot be
deter mined directly in the FIR experiments because of prohib
itive selection r ules. With the use of theoretical integrated band
intensities, these cav it y ringdown measurements per mitted the
first deter mination of the absolute water cluster concentrations
in a supersonic beam (40). Interestingly, the trimer dominates
the cluster distribution for both H2O and D2O. This domination
is probably caused by the discontinuous increase in the per
monomer binding energ y (D0), which jumps f rom 12 D0 to D0
f rom dimer to trimer, while increasing much more slowly for
The Evolution of Hydrogen Bonding in Small Water Clusters
The archet ype of the H-bond in water is the water dimer (Fig.
3). The OOO dist ance, ref lecting the length of the H-bond is
Fig. 1. The 89.5 cm1 torsional hot band of (D2O)3 (Left) was the ﬁrst
intermolecular vibrational band observed for a water cluster in the gas phase
(13). The spectrum shows splitting of each vibration-rotation transition into a
characteristic quartet by the bifurcation tunneling motion (see Fig. 6b). (Right)
Shown is this quartet for the 41.1 cm1 torsional band (103), the most intense
water trimer band observed to date. The intensities of the quartet compo
nents are determined by nuclear spin weights. SN, Signal to noise.
Fig. 2. The IR-cavity ringdown laser absorption spectrometer spectrum of
the OD stretch region of D2O is shown (a). The spectrum shows vibrational
bands due to the stretch of the free OD at the highest frequencies. Below 2,700
cm1 the OD stretch frequencies of the bound OD groups are observed on top
of a weak broad absorption from an amorphous ice-like phase. The bound OD
stretch frequencies decrease with increasing cluster size (40). The IR-cavity
ringdown laser absorption spectrometer spectrum of the (D2O)2 acceptor
antisymmetric OD stretch (b) shows the clearly resolved acceptor switching
splitting. This splitting is most readily observable in the intense Q-branch
transitions (36). This observation has allowed the determination of the accep
tor switching splitting in the vibrational ground state, which was not possible
with the previously existing data.
Fig. 3. The equilibrium structure of the water dimer as determined by
calculations on the VRT(ASP-W)-II potential surface (R. S. Fellers, M. G. Brown,
L. B. Braly, M. Colvin, C. Leforestier, and R.J.S., unpublished work). The
hydrogen bond deviates 2.3° from linearity, the OOO distance is 2.952 Å, and
the bond strength, D0, is 3.40 kcalmol. The highly nonrigid dimer has six
ﬂoppy intermolecular vibrations.
Fig. 4. The water dimer exhibits three distinct low barrier tunneling path
ways that rearrange the hydrogen bonding pattern. Acceptor switching (AS),
having the lowest barrier of all tunneling motions estimated at 157 cm1 by
VRT(ASP-W), is the most facile tunneling motion. This tunneling pathway
exchanges the two protons in the hydrogen bond acceptor monomer and has
been determined to begin with a ﬂip of the acceptor monomer followed by
a rotation of the donor monomer around its donating OOH bond, and
completed by a 180° rotation of the complex about the OOO bond. The
tunneling motion splits each rovibrational energy level into two. Interchange
tunneling (I) exchanges the roles of the hydrogen bond donating and accept
ing water monomers. Several possible pathways exist for this exchange, the
lowest barrier path being the geared interchange motion. This pathway
begins with a rotation of the donor in the D angle and rotation of the
acceptor about its C2 axis to form a trans transition state structure. This is
followed by a rotation of the initial donor about its C2 axis and a rotation of
the initial acceptor in the A angle such that it becomes the donor. The
pathway is completed by a 180° end-over-end rotation of the complex.
Calculations with the VRT(ASP-W) potential determine the barrier to be 207
cm1. The anti-geared interchange pathway also has been determined to be
important and is similar to the geared pathway except that it has a cis
transition state. The tunneling motion splits each energy level by a much
smaller amount than the acceptor switching resulting into two sets of three
energy levels. The bifurcation tunneling motion B, wherein the hydrogen
bond donor exchanges its protons, consists of the simultaneous in plane
librational motion of the donor with the ﬂip of the acceptor monomer. This
is the highest barrier tunneling pathway [394 cm1 with VRT(ASP-W)] result
ing in a small shift of the energy levels.
2.952 Å, and the hydrogen bond strength (dissociation energ y)
of (H2O)2 is 3.09 kcalmol, corresponding to the zero-point
corrected binding energ y (De) of 4.85 kcalmol. The dimer
equilibrium str ucture was deter mined in the potential sur face fit
described below, wherein a ver y extensive dat a set encompassing
five of the six fundament al inter molecular v ibrations with com
plete resolution of rot ation and hydrogen bond tunneling ef fects,
have been fit to Stone’s highly det ailed anisotropic site potential
(ASP) potential for m (38). Three distinct quantum tunneling
processes (Fig. 4), for which the potential barriers all have been
deter mined, rearrange the H-bond on time scales ranging f rom
about 1 s to 1 ps (38). The tunneling motions connect eight
degenerate minima on the inter molecular potential sur face
(IPS), splitting each rov ibrational transition into six subbands.
The highest barrier (1.13 kcalmol, zero point corrected) process
corresponds to the exchange of the bound and f ree hydrogen
atoms on the donor molecule (bifurcation) and turns out to be
the most facile means of breaking the H-bond, which has
interesting implications with respect to bond-breaking dynamics
in bulk water (5, 41). All six fundament al inter molecular v ibra
tions except for the out-of-plane libration have now been mea
sured for both (H2O)2 and (D2O)2.
The water trimer is a much more rigid structure than the dimer
(42), bound by three strained H-bonds (Fig. 5). The OOO distance
in the trimer is 2.85 Å, significantly shorter than in the dimer, a
result of the increased hydrogen bond strength caused by the
cooperative effect of three-body forces. The appearance of three
body forces make inclusion of trimer VRT data into a fit of existing
water pair potentials like VRT(ASP-W) the next logical step toward
developing an accurate liquid water potential. The initial fits of
potentials to the torsional energy levels below 100 cm1, as recently
explored by Groenenboom et al. (39), will be followed by inclusion
of the higher energy translational and librational vibrations, pend
ing development of theoretical methods for treating such high
dimensional dynamics in clusters. The trimer VRT data therefore
will allow explicit quantification of the three-body forces, the
leading many-body term in the liquid force field.
Each monomer in the water trimer acts both as a single donor
and single acceptor of an H-bond, and each has one bound and
one f ree hydrogen. Because of the alternation of the f ree
hydrogen atoms above and below the plane of the ox ygens, this
str ucture is chiral, as are those of all the odd-membered rings.
Two distinct tunneling processes operate to rearrange the H
bond net work, here connecting 48 degenerate minima on the IPS
(43, 44). The first is f lipping, which is essentially barrierless (Fig.
6a) (45), and the same bifurcation process described above for
the dimer (Fig. 6b). This latter motion turns out to be a highly
local one, with an uncorrected barrier near 2 kcalmol(43, 44).
These rearrangement pathways were systematically described in
Fig. 5. The water trimer has a chiral cyclic equilibrium structure with each
water monomer acting as a single hydrogen bond donor and acceptor (75, 98).
It is homodromic in the sense that the donor OH bonds all are directed in a
clockwise or anticlockwise pattern. The free hydrogens lie alternatingly above
and below the plane of the oxygen atoms. This results in two adjacent free
hydrogens being on the same side of the ring, making the trimer a frustrated
structure, which gives rise to very facile torsional motions. These vibrationally
average the structure to that of an oblate symmetric top on the experimental
time scale. The average OOO distance of 2.85 Å [2.80-Å equilibrium ab initio
value (105)] is signiﬁcantly shorter than that of the water dimer, which can
largely be attributed to the effect of three-body forces.
Fig. 6. Two distinct tunneling pathways rearrange the hydrogen bond
pattern in the cyclic water trimer. The torsional (ﬂipping) motion (a) of the
free hydrogens atoms from one side of the plane determined by the oxygen
atoms to the opposing side connects two degenerate minima on the IPS. The
barrier for this tunneling motion is lower than the vibrational zero-point
energy for (H2O)3 and close to the vibrational zero-point energy for (D2O)3 (43,
44, 104). Inclusion of ﬂipping of all free hydrogens splits each torsional energy
level into a manifold of six states. This ﬂipping motion is symmetrically
equivalent to rotation around the axis and hence is a pseudorotational motion
coupling strongly to the overall rotation of the cluster, which results in severe
Coriolis perturbations that can be readily observed in all torsional bands.
Development of a detailed Hamiltonian accounting for this coupling was
necessary for a complete understanding of the torsional states and analysis of
the torsional bands (42). The bifurcation tunneling motion (b) in the water
trimer consists of the exchange of a free and a bound hydrogen together with
the ﬂipping motion of the free hydrogens on the two neighboring water
monomers. The bifurcation tunneling pathway is the lowest energy hydrogen
bond breaking motion observed in water clusters, and, in the trimer, connects
eight degenerate minima on the IPS, splitting each rovibrational transition
into a quartet with relative intensities determined by the nuclear spin statis
tics. The barrier for this tunneling motion is about 2 kcalmol and thus results
in much smaller splittings than does the torsional tunneling motion (43, 45).
Fig. 7. The FIR spectrum of liquid D2O and H2O are shown together with the
D2O and H2O cluster data in the translational (100 –200 cm1) and librational
(300 –1,000 cm1) band region. (Center Left) A stick spectrum of the 142.8
cm1 degenerate antisymmetric stretch band of (D2O)3. (Top Left) A scan of
the RR2(2) transition, representative of the strongest observed rovibrational
transitions. The 142.8 cm1 band lies well within the translational band of the
liquid. No bifurcation tunneling splittings are observed, indicating that they
are unchanged with respect to the ground state. (Center Right) The out-of
plane librational band of (H2O)3. Three parallel bands of (H2O)3 centered at
517.2, 523.9, and 525.3 cm1 were assigned. Theory predicts only one parallel
trimer band for the whole librational band region. The subbands were ex
plained by a dramatic (1,000) increase of tunneling splitting through the
bifurcation pathway. (Top Right) A scan of the Q-branch of the 523.9 cm1
subband is shown and from the observed relative intensities the rotational
temperature can be estimated at 5 K.
elegant work by Walsh and Wales (44). The water trimer VRT
dat a set is the most extensive ex isting for any water cluster, with
seven, four, and six complete inter molecular v ibrational bands
obser ved for (D2O)3, (H2O)3, and mixed isotopomers, respec
tively. We have now achieved a ver y complete description of the
low-f requency (100 cm1) torsional modes in the trimer (42,
46, 47), and have measured extensive VRT spectra in the
translational (26) (about 150 cm1, Fig. 7) and librational (27)
(about 520 cm1, Fig. 7) band regions as well. Thus, all of the
features of liquid water that appear in the inter molecular
v ibrational region ( 10 m) also ex ist for the trimer and have
been quite well characterized.
The H-bonding motif of the water tetramer is similar to that
of the trimer, with each monomer acting as a single donor and
acceptor, and hav ing one f ree and one bound H (Fig. 8). The
average OOO dist ance is further shortened to 2.79 Å. Interest
ingly, the water tetramer turns out to be more dif ficult to
characterize than the trimer, as a result of its higher (S4)
symmetr y (19, 24, 48, 49). Whereas the torsional motion of the
f ree hydrogens in the trimer is ver y facile and results in a large
number of low-f requency v ibrational bands (100 cm1), the
only inter molecular v ibration of the tetramer obser ved below
100 cm1 both in (D2O)4 and (H2O)4 corresponds to the in-plane
ring defor mation. The lowest f requency v ibrational band involv
ing torsional st ates was obser ved at 137.8 cm1 in (D2O)4
compared with the lowest torsional st ate of (D2O)3 at 8.5 cm1.
The high symmetr y also enforces much more cooperative tun
neling motions, and the bifurcation rearrangement has not been
obser ved. Rather, each v ibration-rot ation line is split into a
doublet in a complicated tunneling process connecting only t wo
degenerate minima on the tetramer IPS (49, 50), probably
involv ing a second-order saddle point (Fig. 9).
The pent amer continues this str uctural evolution, being ver y
similar to the trimer in both str ucture and dynamics (Fig. 10) (20,
22, 51, 52). Both torsional (f lipping) and bifurcation tunneling
are obser ved, connecting 320 degenerate minima on the IPS, but
the time scale of both tunneling motions is slower than in the
trimer. In contrast to the trimer, splittings due to bifurcation
tunneling have been obser ved only for (H2O)5 and not (D2O)5,
due to both the stronger H-bond and stronger coupling of
f lipping and ring puckering motion in the latter (52, 53). This
coupling also requires heavy atom motion for the torsional
motions and results in a much denser torsional manifold and a
large number of dipole-allowed transitions at low f requencies
(54). Many such transitions have been measured for the pen
t amer (Fig. 11), and the characterization of this torsional man
ifold is nearing completion (20, 22, 51, 55, 56).
The H-bond in the pentamer is nearly linear and the OOO
distance of 2.76 Å is close to the value found for liquid water and
especially ice, as the desired tetrahedral hydrogen bonding geom
etr y of the monomers is ver y nearly realized. Moreover, molecular
dynamics simulations have shown that five-membered rings are a
dominant topology in liquid water, and pentamer-like patterns have
Fig. 8. The water tetramer has a highly symmetric homodromic S4 equilib
rium structure resulting in oblate symmetric top spectra with no vibrational
averaging required. The symmetric structure requires highly concerted tun
neling motions and results in a signiﬁcantly more rigid structure than for the
water trimer. The effect of many-body forces reduces the vibrationally aver
aged OOO distance from that of the trimer to 2.79 Å (2.74-Å ab initio
equilibrium value) (105). The hydrogen bond is only 12° from linearity and the
structure is very nearly planar.
Fig. 9. The high symmetry of the tetramer requires highly concerted tun
neling motions and limits the number of degenerate minima that can be
connected on the IPS via feasible tunneling motions to two. The tunneling
pathway connects the udud (up-down-up-down) structure with the dudu
(down-up-down-up) one. Despite the small number of minima involved (tri
mer and pentamer 48 and 320 minima, respectively), the details of the
tunneling pathway have been unclear. The observation of the large increase
in tunneling splitting on exciting the 137.8 cm1 eg torsional vibration to
gether with the analysis of the nuclear displacements derived by theory from
this vibration suggests a complicated pathway involving sequential torsional
motions of the free hydrogendeuterium atoms and tunneling via second
order saddlepoints (49).
Fig. 10. The chiral homodromic equilibrium structure of the water pentamer
is analogous to that of the water trimer. The vibrationally averaged OOO
distance is 2.76 Å [2.72-Å equilibrium value from ab initio (54)], and the ring
is puckered by 15.5°. The OOOOO angles are about 108°, very near the
tetrahedral angle preferred in aqueous hydrogen bonding, and yielding
nearly linear hydrogen bonds. As in the trimer, this structure also allows for
very facile torsional motion and tunneling, connecting ten degenerate min
ima on the IPS. However, the torsional manifold resulting from this tunneling
motion is more closely spaced, as coupling to the ring puckering motions is
required. The hydrogen bond strength is larger than in the trimer and, thus,
the bifurcation tunneling splitting, which connects 32 degenerate minima on
the IPS, is reduced, and is only observable in (H2O)5 (Fig. 11).
Fig. 11. The 89.1 cm1 band of (H2O)5 is typical for a parallel band of an
oblate symmetric top with ﬁrst-order Coriolis perturbations into two sub
bands (A, B). Bifurcation tunneling splits each transition by 4.6 MHz into an
equally spaced multiplet with a characteristic intensity pattern, determined by
nuclear spin statistics. In contrast, the rovibrational transitions of the 81.1
cm1 band of (D2O)5, show no sign of splittings due to bifurcation tunneling
because of the larger mass involved.
been shown to be important in solvation of hydrophobic solutes and
in the structures of clathrate hydrates (95).
The hexamer represents the transition of the H-bond net work
f rom t wo-dimensional to three-dimensional in its most st able
arrangement (Fig. 12).
Here four monomers become triply
H-bonded, whereas the t wo apical monomers remain doubly
bonded in a nearly oct ahedral cage str ucture. These latter
monomers engage in a cooperative t ype of bifurcation rear
rangement, again enforced by the symmetr y of the hydrogen
bond net work, and experiment ally, tunneling bet ween only t wo
degenerate minima on the IPS is obser ved. It is not able that the
most st able str ucture of the water hexamer deter mined in the gas
phase resembles the basic unit in ice VI. Naut a and Miller (57)
recently deter mined a cyclic str ucture of the hexamer in liquid
helium droplets, analogous to those of the smaller clusters.
Interestingly, this str ucture closely resembles the six membered
ring for ms ex isting in cr yst alline ice for ms of water.
Although high-resolution experiments have not yet been
successful for clusters larger than the hexamer (H2O)n, clusters
with n 7, 8, 9, 10 have been obser ved in low-resolution IR
depletion experiments of size-selected clusters (58 – 60). The
str uctures of these clusters were deter mined by comparing the
experiment al v ibrational f requencies with the results of ab initio
calculations for various str uctures. This analysis suggested the
ex istence of both the D2d and S4 str uctures of the oct amer, both
of which correspond to st acked tetramer rings. Two hept amer
str uctures derived f rom the S4 oct amer str ucture by remov ing
one of the monomers also were deter mined, as well as a nonamer
str ucture, expanding the D2d cube str ucture of the oct amer,
which is composed of pent amer and tetramer units, and a
decamer str ucture resembling the D2d oct amer str ucture with the
t wo additional water molecules inserted at opposite edges.
Zwier and coworkers’ (61, 62) resonant t wo-photon ioniza
tion, UV hole-burning, and resonant ion-dip IR spectroscopy
studies also have suggested the ex istence of the t wo isomers of
the oct amer att ached to benzene, which acts as a chromophore.
Similarly, the str ucture of the water nonamer att ached to a
benzene chromophore was proposed to consist of an expanded
oct amer D2d str ucture, whereas the ex istence of additional
nonamer str uctures based on the S4 oct amer str ucture could not
be deter mined definitively (63).
Toward a Universal Water Force Field from VRT Spectra
Theoretical calculations have clearly est ablished the rapid con
vergence of the liquid water force field in ter ms of N-body
interactions. Moreover, the leading nonpair wise additive ter m
has been shown to be the relatively simple polarization (induc
tion) interaction (64). It is therefore apparent that the essential
infor mation needed to deduce a quantit atively accurate liquid
water force field can be extracted f rom appropriately det ailed
measurements of small water clusters, particularly dimer and
trimer. In contrast to the results of water cluster studies in bulk
env ironments, FIR VRT spectroscopy has been shown to pro
v ide such a probe (65), being exquisitely sensitive to the det ailed
topolog y of the cluster potential energ y sur face. Extensive VRT
dat a sets now ex ist for several isotopomers of the dimer (66, 67)
and trimer (25–27, 42, 46), encompassing all three t ypes (torsion,
translation, libration) of inter molecular v ibrations of the trimer,
and excluding only librations in the dimer (66, 67), whereas
somewhat less dat a ex ist for the water tetramer, pent amer, and
hexamer. Hence, given the requisite theoretical methods for
computing cluster VRT spectra f rom global potential sur faces,
we now have the capabilit y to actually constr uct a rigorously
accurate force field for liquid water f rom the spectra of these
The first step in our scheme to accomplish this goal is to
rigorously deter mine the water dimer potential energ y sur face by
explicitly fitting the VRT dat a to a det ailed and physically sound
potential model. This dimer potential sur face will accurately
describe not only the dominant pair wise interactions that occur
in the liquid, but if it properly includes both electric multipoles
and polarizibilit y of the water molecule, then it will also correctly
describe the leading nonadditive ter ms, namely N-body induc
tion. Hence, an approx imate potential sur face of the trimer can
be constr ucted by appropriate summation of the polarizable
dimer potential (treating the induction by iteration or matrix
inversion). The interactions not properly described by this
potential [three-body exchange and the much smaller (64)
three-body dispersion] then can be quantified by comparing
VRT spectra of the trimer with those computed f rom the
potential sur face. This process can be extended and refined by
successively addressing the larger clusters (tetramer, pent amer,
and hexamer) in the same way. We suggest that a potential that
rigorously describes the VRT spectra of the trimer will already
accurately reproduce the measured properties of liquid water
over large ranges of conditions, thereby essentially prov iding the
long-sought universal water force field.
This first step outlined above (deter mination of the dimer
potential) has now been accomplished. A key development in
this process was the Split Wigner Pseudospectral method (68)
and its implement ation for the water dimer (69, 70). This
theoretical advance per mitted the accurate comput ation of
dimer eigenst ates f rom van der Avoird’s rigorously derived
body-fixed six-dimensional scattering Hamiltonian (71) and a
suit able global potential sur face with the ver y high ef ficiency and
economy required for incorporation of this procedure into a
regression routine. Two water dimer potential sur faces of spec
troscopic accuracy have now been published. Fellers et al . (38)
fit the dimer VRT dat a to Millot and Stone’s ASP-W potential
(72), the most det ailed dimer sur face available, and van der
Avoird and colleagues (39, 73, 74) ‘‘tuned’’ the ab initio sym
metr y adapted perturbation theor y (SAP T) sur face of Mas and
Szalewicz (7) to reproduce these same dat a. The qualit y of these
t wo potential sur faces is comparable, as judged by their respec
tive abilities to reproduce the VRT dat a and the temperature
dependence of the second v irial coef ficients. Both are far more
accurate than any of the many water dimer potentials prev iously
available (69, 70). Perhaps their most not able feature is the
significantly reduced dimer binding energ y (4.85–5.00 kcalmol),
in accord with the latest ab initio results (6, 7).
Perhaps surprisingly, we now recognize that the key ingredient
for constr ucting a rigorously accurate liquid water force field is
actually this dimer potential sur face. Accordingly, we must seek
to obt ain the ‘‘per fect’’ dimer potential. Although the t wo new
for ms obt ained f rom VRT dat a are cert ainly dramatic improve
Fig. 12. The water hexamer has been determined to have a cage structure
with the oxygens forming a distorted octahedron. The two apical water
monomers are single donors and acceptors, whereas two of the other water
monomers act as single donors, double acceptors and the remaining two as
double donors, single acceptors. Bifurcation tunneling exchanges the free and
bound hydrogens of the two apical water monomers, connecting four degen
erate minima on the IPS. Calculations indicate that this cage form is the lowest
of ﬁve low energy structures (21). A cyclic hexamer form, similar to that of
the pentamer, recently has been identiﬁed in a liquid He droplet envi
ments, they both have f laws. The VRT(ASP-W) sur face is
weakest in its description of the acceptor switching tunneling
motion, yielding a trimer f lipping barrier that is too high. The
symmetr y adapted perturbation theor y (SAP T) potential is not
explicitly polarizable, and therefore cannot be used in its present
for m to constr uct potentials for either larger clusters or bulk
water. Both potentials are constr ucted with ‘‘f rozen monomers,’’
and thus do not produce the slight elongation of the covalent
OOH bond that is known to occur in the donor of the H-bond.
The relaxation of the monomer rigidit y constraint may be
necessar y to improve these features. However, this necessit ates
rigorously solv ing a 12-dimensional v ibrational dynamics Ham
iltonian, something not considered even remotely possible until
ver y recently.
Leforestier et al . (C. Leforestier, N. Goldman, C. Keoshian,
L. B. Braly, and R.J.S., unpublished work) recently per for med a
least-squares fit of H2O dimer VRT dat a to a 12-dimensional
potential, which uses the H2O monomer potential recently
developed by Polyanski et al . (76) to describe the covalent bond
v ibrations, and a modified Matsuoka Clementi Yoshimine
(MCY) potential for m of Clementi and coworkers (77) for
t he i nt er mol ecul ar i nt er act i ons . The ei gens t at es of t he
12-dimensional potential were computed by the Split Wegner
Pseudospectral (SWPS) method, but incorporating an adiabatic
separation bet ween the covalent and inter molecular motions.
Interestingly, this ver y simple MCY potential (11 f ree parame
ters) fits the VRT dat a quite well, but only when the intramo
lecular nonrigidit y is included (but with no additional f ree
parameters); the modified MCY potential per for med rather
badly when only six-dimensional dynamics (rigid monomers)
were calculated (69, 70, 78).
The BerkeleyNijmegen collaboration has achieved a com
plete characterization of the trimer VRT st ates in the torsional
(pseudorot ational) manifold for both H2O and D2O isotopes
(42, 46), first treated theoretically by Schu¨tz et al . (79). Van der
Avoird et al . (80) have derived a rigorous body-fixed Hamilto
nian for these motions, treating the three torsional motions
explicitly (fully coupled) and the librational (bifurcation) mo
tions perturbatively, while f reezing the translational v ibrations.
The Berkeley group (42, 46) has accomplished a global fit of all
of the torsional VRT dat a to this Hamiltonian. Groenenboom et
al . (39) have subsequently used these st ates to check the ‘‘tuning’’
of their symmetr y adapted perturbation theor y (SAP T)-5st
potential, finding an impressive level of agreement (39). Keutsch
et al . have ver y recently characterized both the translational (26)
and librational (27) v ibrations in the trimer by VRT spectros
copy, and Sabo et al . (81) have developed a four-dimensional
trimer model incorporating the ef fects of the symmetric trans
lational mode on the torsional st ates. Hence, the st age is set for
carr ying out the refinement of the pair potential and the rigorous
deter mination of the three-body corrections, pending some
further development of the requisite theoretical for malism (to
describe the coupling of translation, torsion, and libration in the
rot ating f rame—another dif ficult 12-dimensional dynamics
Dynamics of the Hydrogen Bond in Water Clusters
Although the deter mination of an accurate potential for liquid
water has thus far been the main goal of the FIR-VRT water
cluster studies, we recently have sought to exploit the molecular
det ails of str uctures and hydrogen bond dynamics revealed in our
VRT spectroscopy experiments to elucidate specific aspects of
these dynamics occurring in the liquid (21, 49, 51, 55, 82). The
hydrogen bond net work and its dynamics deter mine the unique
properties of liquid water, and many dif ferent experiments have
addressed these in ter ms of the underlying inter molecular mo
tions (83–91). In contrast to the VRT experiments, these bulk
experiments were either insensitive to the microscopic det ails or
required extensive interpret ation (92). In conjunction with in
terpret ation by theoretical models, the experiment al results have
suggested that both the translational (hydrogen bond stretch
v ibration centered at 180 cm1 in H2O) and the librational
motions (hindered rot ational v ibration about 300 –1,000 cm1
H2O) are directly involved in most dynamical processes (5, 85, 86,
The f ree hydrogens in small water clusters—the main distinc
tion bet ween clusters and the bulk—are predicted not to inf lu
ence these t wo import ant inter molecular motions (translation
and libration) significantly (ref. 97 and A. Luzar, personal
communication). Although water clusters are cert ainly not im
port ant as isolated constituents of liquid water and clearly cannot
act as models of long time dynamics (e.g., dif fusion), we have
investigated by using water clusters to unravel the det ails of
hydrogen bond dynamics on ver y short time scales, specifically
the hydrogen bond breaking dynamics (41). Chandler and Luzar
(5, 92) have studied this aspect of the liquid dynamics v ia
computer simulations, and their results imply that these dynam
ics are primarily local and do not var y significantly with the
hydrogen bond order in the liquid. Their studies, in conjunction
with the above experiments, also suggest that librational motions
play the central role in liquid-st ate dynamics because they are the
dominant motion for the initial breaking of hydrogen bonds in
the extended net work. Whereas translational motions them
selves do not lead to significant bond breaking, they can indi
rectly facilit ate the breaking by weakening the hydrogen bond
(5, 91, 92, 95, 99). The interpret ation of recent dielectric
relaxation measurements has even suggested that water mole
cules making only t wo hydrogen bonds might be of special
import ance for bulk dynamics (88). These results therefore
suggest that water clusters can indeed prov ide a paradigm for
elucidating the molecular det ails of specific local processes
contributing to the liquid-st ate dynamics, namely the hydrogen
bond breaking, despite the obv ious disparities.
Water cluster VRT experiments do not cont ain dynamical
infor mation per se, as they measure transitions bet ween st ation
ar y st ates. However, the magnitude of the tunneling splittings
obser ved for a specific v ibrational st ate characterize the feasible
hydrogen bond rearrangements in the cluster. It is straightfor
ward to extract a time scale for a given tunneling motion f rom
the experiment al splittings (100, 101), and thus the time scale for
the hydrogen bond dynamics associated with a tunneling path
way can be deter mined. This time scale calculation allows a
direct study of the ef fect of exciting specific inter molecular
motions (e.g., translation and libration) on the dynamics for
various cluster sizes.
The bifurcation pathway (Fig. 6b) is uniquely suited for the
study of the dynamics of the hydrogen bond in small water
clusters, as it has been obser ved for the dimer, trimer, pent amer,
and hexamer (49, 51), and corresponds to the lowest energ y
pathway for breaking and making a hydrogen bond, precisely the
aspect of the liquid-st ate dynamics we are interested in. Fur
ther more, the potential barrier for the bifurcation pathway is
similar for all water clusters studied so far and it is a local process
(49, 51), hence we can use this specific tunneling process as a
probe of the hydrogen bond breaking dynamics and investigate
the ef fect of both cluster size and excit ation of various inter
molecular v ibrations on these bifurcation dynamics. From the
experiment ally deter mined magnitude of the bifurcation tun
neling splitting for a given v ibrational st ate, we can extract the
time scale for breaking and making a hydrogen bond—i.e., the
hydrogen bond lifetime (H) of that st ate. Presently, the water
trimer is the only water cluster for which all classes of inter mo
lecular v ibrations have been obser ved, allowing the first study of
inter molecular v ibrational mode selectiv it y on the hydrogen
bond lifetime, and the deter mination of the dominant inter mo
lecular hydrogen bond breaking motion.
From a v ibrational ground st ate splitting of about 0.0033 cm1
the hydrogen bond lifetime of the ground st ate was deter mined
to be H(GS) 1–2 ns for (H2O)3 (101). The ver y small variation
of the magnitude of the tunneling splitting among st ates exam
ined in the torsional manifold confir ms that the hydrogen bond
lifetime does not change significantly upon excit ation of tor
sional v ibrations, H(torsion) 1–2 ns for (H2O)3 (101). The first
obser ved translational v ibration of a water cluster, the 142.8
cm1 (D2O)3 band (Fig. 7), showed no ev idence of bifurcation
tunneling splittings within the experiment al uncert aint y (about
2 MHz) (26). Therefore, the magnitude of these tunneling
splitting and, thus, the hydrogen bond lifetime for the excited
st ate has to be nearly identical upon exciting the translational
v ibration, H(translation) H(GS) 1–2 ns for (H2O)3. Our
recent study of the librational band region revealed three parallel
bands of (H2O)3 at 517.2, 523.9, and 525.3 cm1 (Fig. 7), which
are located close to the center of the librational band of liquid
H2O and are assigned to the out-of-plane libration (27). The
splittings for the excited st ate of the libration are so large that the
three indiv idual bands obser ved within 8 cm1 correspond to the
tunneling components, revealing that the bifurcation splittings
are nearly 3 orders of magnitude larger than for the v ibrational
ground st ate. Although the calculation of the hydrogen bond
lifetime is more complicated for the excited st ate of the out-of
plane libration as the high barrier limit for the tunneling pathway
is no longer valid, we can extract a hydrogen bond lifetime of
H(libration) 1– 6 ps.
This time scale is strikingly similar to the accepted average
hydrogen bond lifetime found in liquid water (1 ps), and the
time scales of a number of import ant dynamical processes in bulk
phases of water, (attributed to single molecule reorient ation)
e.g., dielectric relaxation (D 8 –9 ps, D2 1 ps) (93), reori
ent ation relaxation (1 13 ps, 2 0.7 ps) (90) and proton
mobilit y (1 ps) (93). Whereas the molecular det ails of the
tunneling motions and v ibrations we obser ve in small water
clusters are well understood (43, 44, 102), none of the experi
ments used for measuring these bulk time scales are sensitive to
the microscopic det ails, and accordingly any molecular inter
pret ation of these results on the bulk is necessarily strongly
model-dependent. The value of the VRT results is that they
clearly demonstrate the dramatic ef fect of exciting specific
inter molecular motions on the hydrogen bond breaking rate,
with the librations clearly being the dominant inter molecular
hydrogen bond breaking motion. Therefore, rather than the
perhaps coincident al similarities of H(libration) with those of
the bulk relaxation processes, the most import ant result of our
study is that we have confir med the same dependence of the
hydrogen bond breaking dynamics on inter molecular motions in
the water trimer as postulated in theoretical studies of liquid
water. Specifically, we obser ve a dramatic increase (1,000) in
the rate of hydrogen bond breaking compared with that in the
ground st ate of the water trimer upon excit ation of a single
quantum in the librational mode, no change of H upon single
quantum excit ation of a translational mode, and an insignificant
change of H on excit ation of the torsional v ibrations. Hence,
libration is the dominant vehicle for breaking and making the
hydrogen bond, just as postulated to be the case for liquid water.
This work was supported mainly by the Experiment al Physical Chemistr y
Program of the National Science Foundation. The Berkeley cav it y
ringdown laser absorption spectrometr y ef fort also was supported by the
Chemical Dynamics Program of the Air Force Of fice of Scientific
1. Stillinger, F. H. (1980) Science 209, 451– 457.
2. Chen, B., Xing, J. & Siepmann, J. I. (2000) J. Phys. Chem. 104, 2391–2401.
3. Makkonen, L. (1997) J. Phys. Chem. 101, 6196 – 6200.
4. Harrington, S., Zhang, R., Poole, P. H., Sciortino, F. & St anley, H. E. (1997)
Phys. Rev. Lett . 78, 2409 –2412.
5. Luzar, A. & Chandler, D. (1996) Phys. Rev. Lett . 76, 928 –931.
6. Stone, A. J. (1996) The Theor y of Inter molecular Forces (Oxford Univ. Press,
7. Mas, E. M. & Szalewicz, K. (1996) J. Chem. Phys. 104, 7606 –7614.
8. Silvestrelli, P. L. & Parrinello, M. (1999) J. Chem. Phys. 111, 3572–3580.
9. Busarow, K. L., Cohen, R. C., Blake, G. A., Laughlin, K. B., Lee, Y. T. &
Saykally, R. J. (1989) J. Chem. Phys. 90, 3937–3943.
10. Dyke, T. R., Mack, K. M. & Muenter, J. S. (1977) J. Chem. Phys. 66, 498 –510.
11. Fraser, G. T. (1991) Int . Rev. Phys. Chem. 10, 189 –206.
12. Zwart, E., Ter meulen, J. J., Meerts, W. L. & Coudert, L. H. (1991) J. Mol .
Spectrosc. 147, 27–39.
13. Pugliano, N. & Saykally, R. J. (1992) Science 257, 1937–1940.
14. Blake, G. A., Laughlin, K. B., Cohen, R. C., Busarow, K. L., Gwo, D. H.,
Schmuttenmaer, C. A., Steyert, D. W. & Saykally, R. J. (1991) Rev. Sci . Instr.
15. Blake, G. A., Laughlin, K. B., Cohen, R. C., Busarow, K. L., Gwo, D. H.,
Schmuttenmaer, C. A., Steyert, D. W. & Saykally, R. J. (1991) Rev. Sci . Instr.
16. Liu, K., Fellers, R. S., Viant, M. R., McLaughlin, R. P., Brown, M. G. &
Saykally, R. J. (1996) Rev. Sci . Instr. 67, 410 – 416.
17. Pugliano, N., Cr uzan, J. D., Loeser, J. G. & Saykally, R. J. (1993) J. Chem.
Phys. 98, 6600 – 6617.
18. Liu, K., Elrod, M. J., Loeser, J. G., Cr uzan, J. D., Pugliano, N., Brown, M. G.,
Rzepiela, J. & Saykally, R. J. (1994) Faraday Discuss. Chem. Soc. 94, 35– 41.
19. Cr uzan, J. D., Braly, L. B., Liu, K., Brown, M. G., Loeser, J. G. & Saykally,
R. J. (1996) Science 271, 59 – 62.
20. Liu, K., Brown, M. G., Cr uzan, J. D. & Saykally, R. J. (1996) Science 271,
21. Liu, K., Brown, M. G., Carter, C., Saykally, R. J., Gregor y, J. K. & Clar y, D. C.
(1996) Nature (London) 381, 501–503.
22. Liu, K., Brown, M. G., Cr uzan, J. D. & Saykally, R. J. (1997) J. Phys. Chem.
23. Liu, K., Brown, M. G. & Saykally, R. J. (1997) J. Phys. Chem. 101, 8995–9010.
24. Cr uzan, J. D., Viant, M. R., Brown, M. G. & Saykally, R. J. (1997) J. Phys.
Chem. 101, 9022–9031.
25. Viant, M. R., Cr uzan, J. D., Lucas, D. D., Brown, M. G., Liu, K. & Saykally,
R. J. (1997) J. Phys. Chem. 101, 9032–9041.
26. Keutsch, F. N., Brown, M. G., Petersen, P. B., Saykally, R. J., Geleijns, M. &
van der Avoird, A. (2000) J. Chem. Phys. 114, 3934 – 4004.
27. Keutsch, F. N., Fellers, R., Viant, M. R. & Saykally, R. J. (2000) J. Chem. Phys.
114, 4005– 4015.
28. van Thiel, M., Becker, E. D. & Pimentel, G. C. (1957) J. Chem. Phys. 27,
486 – 490.
29. Vernon, M. F., Lisy, J. M., Krajnov ich, D. J., Tramer, A., Hoi-Sing, K., Ron
Shen, Y. & Lee, Y. T. (1982) Faraday Discuss. Chem. Soc. 73, 387–397.
30. Page, R. H., Frey, J. G., Shen, Y. R. & Lee, Y. T. (1984) Chem. Phys. Lett .
31. Coker, D. F., Miller, R. E. & Watts, R. O. (1985) J. Chem. Phys. 82, 3554 –3562.
32. Huang, Z. S. & Miller, R. E. (1989) J. Chem. Phys. 91, 6613– 6631.
33. Huang, Z. S. & Miller, R. E. (1988) J. Chem. Phys. 88, 8008 – 8009.
34. Frochtenicht, R., Kaloudis, M., Koch, M. & Huisken, F. (1996) J. Chem. Phys.
105, 6128 – 6140.
35. Paul, J. B., Provencal, R. A., Chapo, C., Roth, K., Casaes, R. & Saykally, R. J.
(1999) J. Phys. Chem. 103, 2972–2974.
36. Paul, J. B., Provencal, R. A. & Saykally, R. J. (1998) J. Phys. Chem. 102,
37. Paul, J. B., Provencal, R. A., Chapo, C., Petterson, A. & Saykally, R. J. (1998)
J. Chem. Phys. 109, 10201–10206.
38. Fellers, R. S., Leforestier, C., Braly, L. B. & Brown, M. G. (1999) Science 284,
39. Groenenboom, G. C., Mas, E. M., Bukowski, R., Szalewicz, K., Wor mer,
P. E. S. & van der Avoird, A. (2000) Phys. Rev. Lett . 84, 4072– 4075.
40. Paul, J. B., Collier, C. P., Saykally, R. J., Scherer, J. J. & O’Keefe, A. (1997)
J. Phys. Chem. 101, 5211–5214.
41. Keutsch, F. N., Fellers, R. S., Brown, M. G., Viant, M. R., Petersen, P. B. &
Saykally, R. J. (2001) J. Am. Chem. Soc. 123, 5938 –5941.
42. Viant, M. R., Brown, M. G., Cr uzan, J. D., Saykally, R. J., Geleijns, M. & van
der Avoird, A. (1999) J. Chem. Phys. 110, 4369 – 4381.
43. Wales, D. J. (1993) J. Am. Chem. Soc. 115, 11180 –11190.
44. Walsh, T. R. & Wales, D. J. (1996) J. Chem. Soc. Faraday Trans. 92, 2505–2517.
45. Fowler, J. E. & Schaefer, H. F. (1995) J. Am. Chem. Soc. 117, 446 – 452.
46. Brown, M. G., Viant, M. R., McLaughlin, R. P., Keoshian, C. J., Michael, E.,
Cr uzan, J. D., Saykally, R. J., Geleijns, M. & van der Avoird, A. (1999)
J. Chem. Phys. 111, 7789 –7800.
47. Keutsch, F. N., Kar yakin, E. N., Saykally, R. J. & van der Avoird, A. (2000)
J. Chem. Phys. 114, 3988 –3933.
Keutsch and Saykally PNAS September 11, 2001 vol. 98 no. 19 10539
48. Cr uzan, J. D., Brown, M. G., Liu, K., Braly, L. B. & Saykally, R. J. (1996)
J. Chem. Phys. 105, 6634 – 6644.
49. Brown, M. G., Keutsch, F. N., Braly, L. B. & Saykally, R. J. (1999) J. Chem.
Phys. 111, 7801–7806.
50. Wales, D. J. & Walsh, T. R. (1997) J. Chem. Phys. 106, 7193–7207.
51. Brown, M. G., Keutsch, F. N. & Saykally, R. J. (1998) J. Chem. Phys. 109,
52. Wales, D. J. & Walsh, T. R. (1996) J. Chem. Phys. 105, 6957– 6971.
53. Gregor y, J. K. & Clar y, D. C. (1996) J. Chem. Phys. 105, 6626 – 6633.
54. Graf, S., Mohr, W. & Leut wyler, S. (1999) J. Chem. Phys. 110, 7893–7908.
55. Liu, K., Cr uzan, J. D. & Saykally, R. J. (1996) Science 271, 929 –933.
56. Cr uzan, J. D., Viant, M. G., Brown, M. G., Lucas, D. D., Kun, L. & Saykally,
R. J. (1998) Chem. Phys. Lett . 292, 667– 676.
57. Naut a, K. & Miller, R. E. (2000) Science 287, 293–295.
58. Sadlej, J., Buch, V., Kazimirski, J. K. & Buck, U. (1999) J. Phys. Chem. 103,
59. Br uder mann, J., Melzer, M., Buck, U., Kazimirski, J. K., Sadlej, J. & Buch,
V. (1999) J. Chem. Phys. 110, 10649 –10652.
60. Buck, U., Ettischer, I., Melzer, M., Buch, V. & Sadlej, J. (1998) Phys. Rev. Lett .
80, 2578 –2581.
61. Gr uenloh, C. J., Carney, J. R., Hagemeister, F. C., Arrington, C. A., Zwier,
T. S., Fredericks, S. Y., Wood, J. T., III & Jordan, K. D. (1998) J. Chem. Phys.
109, 6601– 6614.
62. Gr uenloh, C., Carney, J., Arrington, C., Zwier, T., Fredericks, S. & Jordan,
K. (1997) Science 276, 1678 –1681.
63. Gr uenloh, C. J., Carney, J. R., Hagemeister, F. C., Zwier, T. S., Wood, J. T.
& Jordan, K. D. (2000) J. Chem. Phys. 113, 2290 –2303.
64. Xantheas, S. S. (1994) J. Chem. Phys. 100, 7523–7534.
65. Saykally, R. J. & Blake, G. A. (1993) Science 259, 1570 –1575.
66. Braly, L. B., Cr uzan, J. D., Liu, K., Fellers, R. S. & Saykally, R. J. (2000)
J. Chem. Phys. 112, 10293–10313.
67. Braly, L. B., Liu, K., Brown, M. G., Keutsch, F. N., Fellers, R. S. & Saykally,
R. J. (2000) J. Chem. Phys. 112, 10314 –10326.
68. Leforestier, C. (1994) J. Chem. Phys. 101, 7357–7363.
69. Leforestier, C., Braly, L. B., Kun, L., Elrod, M. J. & Saykally, R. J. (1997)
J. Chem. Phys. 106, 8527– 8544.
70. Fellers, R. S., Braly, L. B., Saykally, R. J. & Leforestier, C. (1999) J. Chem.
Phys. 110, 6306 – 6318.
71. Brocks, G., van der Avoird, A., Sutclif fe, B. T. & Tennyson, J. (1983) Mol .
Phys. 60, 1025–1043.
72. Millot, C. & Stone, A. J. (1992) Mol . Phys. 77, 439 – 462.
73. Mas, E. M., Bukowski, R., Szalewicz, K., Groenenboom, G. C., Wor mer,
P. E. S. & van der Avoird, A. (2000) J. Chem. Phys. 113, 6687– 6701.
74. Groenenboom, G. C., Wor mer, P. E. S., van der Avoird, A., Mas, E. M.,
Bukowski, R. & Szalewicz, K. (2000) J. Chem. Phys. 113, 6702– 6715.
75. Xantheas, S. S. & Dunning, T. H., Jr. (1993) J. Chem. Phys. 99, 8774 – 8792.
76. Polyansky, O., Jensen, P. & Tennyson, J. (1996) J. Chem. Phys. 105, 6490 –
77. Matsuoka, O., Clementi, E. & Yoshimine, M. (1976) J. Chem. Phys. 64,
78. Fellers, R. S. (1998) Ph.D. Thesis (Univ. of California, Berkeley).
79. Schutz, M., Burgi, T., Leut wyler, S. & Burgi, H. B. (1993) J. Chem. Phys. 99,
80. van der Avoird, A., Olthof, E. H. T. & Wor mer, P. E. S. (1996) J. Chem. Phys.
105, 8034 – 8050.
81. Sabo, D., Bacic, Z., Graf, S. & Leut wyler, S. (1999) J. Chem. Phys. 110,
82. Liu, K., Loeser, J. G., Elrod, M. J., Host, B. C., Rzepiela, J. A., Pugliano, N.
& Saykally, R. J. (1994) J. Am. Chem. Soc. 116, 3507–3512.
83. Draegert, D. A. & Williams, D. (1968) J. Chem. Phys. 48, 401– 407.
84. Ricci, M. A., Signorelli, G. & Mazzacurati, V. (1990) J. Phys. Cond. Matt . 2,
85. Teixeira, J., Bellissent-Funel, M.-C. & Chen, S.-H. (1990) J. Phys. Cond. Matt .
86. Barthel, J., Bachhuber, K., Buchner, R. & Hetzenauer, H. (1990) Chem. Phys.
Lett . 165, 369 –373.
87. Sastr y, S., St anley, H. E. & Sciortino, F. (1994) J. Chem. Phys. 100, 5361–5366.
88. Buchner, R., Barthel, J. & St abuer, J. (1999) Chem. Phys. Lett . 306, 57– 63.
89. Lang, M. J., Jordanides, X. J., Song, X. & Fleming, G. R. (1999) J. Chem. Phys.
110, 5884 –5892.
90. Woutersen, S., Emmerichs, U. & Bakker, H. J. (1997) Science 278, 658 – 660.
91. Bakker, H. J., Woutersen, S. & Nienjuys, H. K. (2000) Chem. Phys. 258,
92. Luzar, A. & Chandler, D. (1996) Nature (London) 379, 55–57.
93. Agmon, N. (1996) J. Phys. Chem. 100, 1072–1080.
94. Chen, S.-H., Gallo, P., Sciortino, F. & Tart aglia, P. (1997) Phys. Rev. E 56,
95. Csajka, F. S. & Chandler, D. (1998) J. Chem. Phys. 109, 1125–1133.
96. Schmitt, U. W. & Voth, G. A. (1999) J. Chem. Phys. 111, 9361–9381.
97. Bosma, W. B., Fried, L. E. & Mukamel, S. (1993) J. Chem. Phys. 98,
98. Xantheas, S. S. & Dunning, T. H. (1993) J. Chem. Phys. 98, 8037– 8040.
99. Luzar, A. & Chandler, D. (1993) J. Chem. Phys. 98, 8160 – 8173.
100. Feynman, R. P., Leighton, R. B. & Sands, M. (1964) in The Feynman Lectures
on Physics, eds. Feynman, R. P., Leighton, R. B. & Sands, M. (Addison–
Wesley, Reading), pp. 8-12– 8-14.
101. Brown, M. G. (1999) Ph.D. Thesis (Univ. of California, Berkeley).
102. Olthof, E. H. T., van der Avoird, A., Wor mer, P. E. S., Liu, K. & Saykally, R. J.
(1996) J. Chem. Phys. 105, 8051– 8063.
103. Suzuki, S. & Blake, G. A. (1994) Chem. Phys. Lett . 229, 499 –505.
104. Klopper, W., Schutz, M., Luthi, H. P. & Leut wyler, S. (1995) J. Chem. Phys.