of 7
State-resolved infrared spectrum of the protonated
water dimer: revisiting the characteristic proton
transfer doublet peak
Henrik R. Larsson,
*
ab
Markus Schr
̈
oder,
c
Richard Beckmann,
d
Fabien Brieuc,
d
Christoph Schran,
§
d
Dominik Marx
d
and Oriol Vendrell
c
The infrared (IR) spectra of protonated water clusters encode precise information on the dynamics and
structure of the hydrated proton. However, the strong anharmonic coupling and quantum e
ff
ects of
these elusive species remain puzzling up to the present day. Here, we report unequivocal evidence that
the interplay between the proton transfer and the water wagging motions in the protonated water dimer
(Zundel ion) giving rise to the characteristic doublet peak is both more complex and more sensitive to
subtle energetic changes than previously thought. In particular, hitherto overlooked low-intensity
satellite peaks in the experimental spectrum are now unveiled and mechanistically assigned. Our
fi
ndings
rely on the comparison of IR spectra obtained using two highly accurate potential energy surfaces in
conjunction with highly accurate state-resolved quantum simulations. We demonstrate that these high-
accuracy simulations are important for providing de
fi
nite assignments of the complex IR signals of
fl
uxional molecules.
1 Introduction
Protonated water clusters play a key role in chemistry and
biology, yet their quantum nature continues to puzzle scien-
tists.
1
5
One of the smallest clusters, the protonated water dimer
or Zundel ion, H(H
2
O)
2
+
, is particularly important and inter-
esting as it can be understood as one of the building blocks of
acidic water,
6,7
where its infrared (IR) spectral signatures
appear.
8
10
Its structure of a proton being
sandwiched
by two
water units leads to an unconventional bonding,
7,11
serving as
a limiting case of a characteristic pattern
6,12
that is observed in
a variety of protonated clusters.
7
Its IR spectrum is of particular
importance, as it reveals information about chemical bonding
and the quantum nature of molecular vibrations. However, the
IR spectrum of the Zundel ion is complicated
as such and due
to tagging e
ff
ects.
3,13
16
Unexpectedly, its complexity even
increases upon deuteration.
17,18
Most dominant in the spectrum
is the famous doublet near 1000 cm

1
, whose vibrational
character had not been understood initially.
13
Quantum
dynamics simulations of the Zundel ion allow us to understand
the spectrum in full detail. However, they are di
ffi
cult to
perform for the following three reasons: (i) its dimensionality
with 15 vibrational degrees of freedom, (ii) the non-obvious
quantum entanglement of the vibrational zero-order states,
and (iii) the
uxional nature of the protons in the Zundel ion
with several equivalent minima, connected by only shallow
potential barriers. When combined as in case of the Zundel ion,
these di
ffi
culties severely challenge quantum dynamics simu-
lations even today.
To date, the multicon
guration time-dependent Hartree
(MCTDH) method has been the only one that could fully
reproduce the experimental IR spectrum (save for tagging
e
ff
ects) and it revealed for the
rst time the characteristics of
the doublet.
14,15,19
According to MCTDH analysis, it consists of
a Fermi resonance, where low-frequency, low-intensity wagging
(pyramidalization) motions of the water units entangle with the
proton transfer motion. The analysis has later been con
rmed
by other simulations.
16,20,21
While MCTDH wavefunction propagation enables the
understanding of the Zundel IR spectrum, the analysis of the
vibrational transitions is cumbersome because information on
the vibrational structure is only indirectly accessible through
observables and low-dimensional probability densities. Further,
the spectral resolution of wavefunction propagations is limited
a
Department of Chemistry and Biochemistry, University of California, Merced, CA
95343, USA. E-mail: azundel_22@larsson-research.de
b
Division of Chemistry and Chemical Engineering, California Institute of Technology,
Pasadena, CA 91125, USA
c
Theoretische Chemie, Physikalisch-Chemisches Institut, Universit
̈
at Heidelberg, Im
Neuenheimer Feld 229, D
69120 Heidelberg, Germany
d
Lehrstuhl f
̈
ur Theoretische Chemie, Ruhr-Universit
̈
at Bochum, 44780 Bochum,
Germany
Electronic
supplementary
information
(ESI)
available.
See
https://doi.org/10.1039/d2sc03189b
Current address: Laboratoire Mati
`
ere en Conditions Extr
ˆ
emes, Universit
́
e
Paris-Saclay, CEA, DAM, DIF, 91297 Arpajon, France.
§
Current address: Yusuf Hamied Department of Chemistry, University of
Cambridge, Lens
eld Road, Cambridge, CB2 1EW, UK.
Cite this:
Chem. Sci.
, 2022,
13
, 11119
All publication charges for this article
have been paid for by the Royal Society
of Chemistry
Received 7th June 2022
Accepted 26th August 2022
DOI: 10.1039/d2sc03189b
rsc.li/chemical-science
© 2022 The Author(s). Published by the Royal Society of Chemistry
Chem. Sci.
, 2022,
13
, 11119
11125 |
11119
Chemical
Science
EDGE ARTICLE
Open Access Article. Published on 30 August 2022. Downloaded on 7/6/2023 9:10:03 PM.
This article is licensed under a
Creative Commons Attribution-NonCommercial 3.0 Unported Licence.
View Article Online
View Journal
| View Issue
by the propagation time and the longer the propagation time,
the more complicated the propagated wavefunction becomes. It
is desirable to being able to obtain the actual vibrational excited
wavefunction that is responsible for a particular peak in the IR
spectrum, but this is di
ffi
cult due to the sheer number of excited
eigenstates that are present in such a system of high anhar-
monicity and dimensionality.
Here, we present new simulations that not only give more
accurate vibrational spectra
via
wavefunction propagations but
also reveal the full eigenstate spectrum with up to 900 eigen-
states to high accuracy for a vibrational energy up to

1900 cm

1
. Two highly accurate potential energy surfaces
(PESs) and dipole moment surfaces (DMSs) are used, namely
the pioneering permutationally-invariant-polynomial-based
HBB surfaces
22
and the more recent BBSM neural-network-
based surfaces,
23,24
which so far has not been used for fully,
rigorous quantum simulations. In the following, by referring to
one of the used PESs we imply the usage of the corresponding
DMS as well.
Our full-dimensional quantum dynamics simulations
consist of recently introduced vibrational tree tensor network
states (TTNS) methodology,
25
as well as new, more accurate and
more compact representations of the PES.
26,27
The TTNS method
is based on the density matrix renormalization group
(DMRG),
28,29
shares the same wavefunction ansatz as the
multilayer (ML) MCTDH method, enables both time-
independent and time-dependent simulations
25,30,31
(denoted
as ti-TTNS and td-TTNS), and, most importantly here, the time-
independent simulations are computationally much less
demanding than in ML-MCTDH.
25
This allows us to fully iden-
tify the nature of each peak in the spectrum by systematically
characterising the eigenstates.
The key
nding of this study is that the interplay between the
wagging and the proton motions is more complex than previ-
ously thought in a critical way. In particular, we disclose that
astonishingly subtle changes in the energetics of the Zundel ion
can lead to very di
ff
erent features in the IR spectrum. This not
only has important implications to PES development and the
quantum dynamics of
uxional molecules in general, but it will
also open the door toward understanding the chemistry of
(micro-)solvated species.
Previous MCTDH simulations on the HBB PES revealed that
the characteristic doublet contains contributions of two states,
one with one quantum in the proton transfer motion, and
another one with one quantum in the water
water stretch
motion and two quanta in the wagging motions. Here, in
contrast, our simulations on the same PES reveal that there are
not two but three dominant eigenstates that are responsible for
the
doublet
in the IR spectrum. Besides the known contri-
butions from a state with two quanta in the wagging motions,
there is an additional dominant contribution from a state
having four quanta
only discovered with our novel TTNS-
based quantum dynamics method
giving overall three domi-
nant peaks in the IR spectrum. In contrast, and in accord with
the established knowledge from the MCTDH simulations, the
newer BBSM PES displays again only two peaks for the doublet
whereas the state with four quanta in the wagging motion is
blueshi
ed and can indeed be identi
ed as satellite peak of the
doublet in the experimental spectrum, in support of our
assignment.
2 Setup
Our benchmark-quality simulations are based on the Hamilto-
nian in polyspherical coordinates set up by Vendrell
et al.
15,19,32
We
tted both the HBB and the BBSM PESs into a form more
suitable for (ML-)MCTDH and TTNS simulations. We use the
recently developed Monte Carlo canonical polyadic decompo-
sition (MCCPD) approach for re
tting the PESs.
26,27
The re
tting
introduced a mean/root-mean-square error of

0.2 cm

1
/

11 cm

1
. For comparison, the mean/root-mean-square error
of the original cluster expansion of the HBB PES used in ref. 14,
15, 19 and 32 is about

24 cm

1
/

4

10
3
cm

1
.
27
To validate
the accuracy of both the two PESs and, as additional validation,
of the two MCCPDs of the respective PESs, we compare the
energies of the PESs from samples of

2000 con
gurations with
direct,
gold standard
basis-set-extrapolated coupled cluster
singles and doubles with perturbative triples (CCSD(T)) elec-
tronic structure energies, see ESI
for details. With these

2000
con
gurations, for the HBB PES we obtain a mean absolute
error (MAE) of 203 cm

1
, whereas for the BBSM PES we obtain
an improved accuracy with an MAE of 75 cm

1
. The MCCPDs
have essentially the same accuracy.
We represent the wavefunction as TTNS, which has the same
mathematical structure as the wavefunction used in the ML-
MCTDH method.
25,33
It di
ff
ers, however, by the use of more
e
ffi
cient algorithms that are based on a generalization of the
density matrix renormalization group (DMRG).
25,28,29
For the
HBB /BBSM PES we computed up to

900/550 of the lowest
eigenstates, despite the large density of states that is evident in
this
uxional cluster. We verify the eigenstate
stick
spectrum
by comparing with the spectrum obtained from wavefunction
propagation using the time-dependent DMRG for TTNSs.
30,31
The Fourier-transformed propagation-based spectrum allows us
to obtain the full spectrum with broadened peaks, while the
stick spectrum reveals every single detail. Details of the
employed methodology, the numerical parameters, conver-
gence tests, as well as additional results are presented in the
ESI.
3 Results and discussion
Fig. 1 displays the experimental spectrum
13,17
and the spectra
simulated using either of the two used PESs. For the HBB /
BBSM PES we show in Fig. 1 up to

900/550 states up to

1900 cm

1
/

1700 cm

1
; note that most of these states are not
IR-active so that much fewer lines are visible in the spectra. The
total spectrum, including IR inactive states, is shown as small
lines on the abscissa. The higher the energy, the higher the
density of states, thus indicating the large spectral complexity of
the Zundel ion. The energies of the IR inactive states di
ff
er
substantially between the two PESs, which will be discussed in
more detail elsewhere. Compared to the propagation-based
spectrum (td-TTNS), the stick spectrum (ti-TTNS) has slightly
11120
|
Chem. Sci.
, 2022,
13
, 11119
11125
© 2022 The Author(s). Published by the Royal Society of Chemistry
Chemical Science
Edge Article
Open Access Article. Published on 30 August 2022. Downloaded on 7/6/2023 9:10:03 PM.
This article is licensed under a
Creative Commons Attribution-NonCommercial 3.0 Unported Licence.
View Article Online
di
ff
erent intensities and the transitions are centered at slightly
lower wavenumbers. This is because the stick spectrum is not
convoluted to account for
nite resolution, and because we
computed the eigenstate spectrum with higher accuracy than
the wavefunction propagation. Convergence tests show that the
eigenstates for levels up to

1030 cm

1
are converged to less
than 1.5 cm

1
, much less than the errors of the PES represen-
tation (see ESI
for further details). This is the
rst time that so
many states have been computed to such a high accuracy for
such a high-dimensional,
uxional cluster.
While both PESs, in particular the newer BBSM one, yield
spectra that overall are in very good agreement with the exper-
imental spectrum, a closer look reveals a striking di
ff
erence
between the simulated spectra: the most characteristic signal of
the IR spectrum, the doublet around 1000 cm

1
, consists of two
dominant peaks on the BBSM PES whereas on the HBB PES it
consists of three peaks! This is in contrast to the pioneering
MCTDH simulations on the HBB PES.
14,15,18,32
These early
MCTDH simulations of such a complicated system were less
accurate and thus had simply missed the additional peak on the
HBB PES. Indeed, our modern ML-MCTDH simulations with
improved convergence parameters also yield three peaks on the
HBB PES and overall are in full agreement with our TTNS
simulations. Comparing the overlaps between the states on
each PES reveals that the third peak observed for the HBB PES is
not missing in the BBSM PES, but it is blueshi
ed and has
a much smaller intensity.
We focus now on the nature of the three dominating vibra-
tional transitions and pinpoint the di
ff
erences resulting from
the quality of the PESs. According to Fig. 1, we label these states
as
J
a
,
J
b
, and
J
c
, respectively, see also Fig. 1. When needed,
we will use superscripts to display which PES has been used for
their optimization. There are two additional, satellite peaks
between
J
BBSM
b
and
J
BBSM
c
that have a similar intensity than
J
BBSM
c
and contain a combination of various excitations. More
details on these and other eigenstates that do not dominate the
doublet will be presented elsewhere.
Our main
ndings, namely (i) the nature of the signals
dominant in the doublet, (ii) the decomposition of the corre-
sponding wavefunctions into coupling zero-order states, and
(iii) their critical dependence on subtle energetic shi
s, are
summarised in Fig. 2. Panel (a) displays the coupling
a Fermi
resonance
identi
ed in the previous MCTDH studies on the
HBB PES.
14,19
There, the doublet consists of two states that can
be described as linear combination (entanglement) of two zero-
order states: one zero-order state with one quantum in the
water
water stretch motion (
R
) and two quanta in the two
wagging modes (
j
02
i
j
20
i
), and another zero-order state with
one quantum in the proton transfer motion (
z
). These two zero-
order states are labeled
j
1
R
,02
20
i
and
j
1
z
i
respectively. Panel
(b) displays our revised coupling scheme on the HBB PES.
There, an additional zero-order state
j
04
40
i
, consisting of four
quanta in the wagging motion, has a low enough energy that it
couples with the two other zero-order states, leading to a more
complex quantum resonance and a signi
cant intensity sharing
of the
j
J
HBB
b
i
and
j
J
HBB
c
i
. Panel (c) displays the coupling
scheme on the BBSM PES, which, ironically, is much more
similar to that of panel (a) than to (b). Here, the
j
04
40
i
state is
Fig. 1
Infrared spectrum of the Zundel ion: the experimental predissociation spectrum using neon tagging (gray)
13,17
and computed spectra using
either the BBSM (red) or the HBB (blue, negative intensities) potential energy surfaces; the inset magni
fi
ed the spectral region in the gray box. The
computed spectra are either based on time-dependent wavefunction propagations (td-TTNS: lines) or based on eigenstate optimizations (ti-
TTNS: sticks). The wavefunction labels mark the states as analyzed in the text. All eigenstates computed, regardless of intensities, are shown as
small lines on the abscissa.
© 2022 The Author(s). Published by the Royal Society of Chemistry
Chem. Sci.
, 2022,
13
, 11119
11125 |
11121
Edge Article
Chemical Science
Open Access Article. Published on 30 August 2022. Downloaded on 7/6/2023 9:10:03 PM.
This article is licensed under a
Creative Commons Attribution-NonCommercial 3.0 Unported Licence.
View Article Online
blueshi
ed and, therefore, does not contribute anymore to the
doublet. There is no intensity-sharing and
j
J
BBSM
c
i
only appears
in the spectrum as one of two satellite peaks on the blue wing of
j
J
BBSM
b
i
in good agreement with the experimental IR spec-
trum as depicted in the upper panel of Fig. 1. (The other satellite
peak is more complicated and will be discussed elsewhere.)
Let us now explain our
ndings in more detail. Our analysis
is based on the time-independent eigenstate simulations, which
enable us to directly analyze the wavefunctions responsible for
each peak, thus providing rigorous assignment of IR peaks to
the structural dynamics at the level of vibrational motion.
Moreover, we con
rm this analysis by computing overlaps of
the wavefunctions with constructed zero-order states (see ESI
).
Fig. 3 shows cuts of the three most relevant eigenstates
J
a
,
J
b
,
and
J
c
on the two PESs along one of the wagging motions and
the proton transfer motion. Each node (zero-crossing) corre-
sponds to one quantum in a particular coordinate. The plots of
J
a
clearly indicate two quanta (nodes) for the wagging motion
(
j
02
20
i
) and one quantum for the proton transfer motion
(
j
1
z
i
). Further inspections reveal an additional quantum for the
water
water stretch motion (
j
1
R
i
). This con
rms the assign-
ment shown in Fig. 2:
j
J
a
i
consists of two entangled zero-order
states, namely
j
02
20; 1
R
i
and
j
1
z
i
. There is no signi
cant
di
ff
erence between
J
BBSM
a
and
J
HBB
a
.
In stark contrast to
J
a
, for
J
b
and
J
c
we
nd signi
cant
di
ff
erences between the BBSM PES and the HBB PES. At
rst
sight,
J
BBSM
b
looks much more similar to
J
HBB
c
than to
J
HBB
b
.We
rst analyze the states on the BBSM PES and then
compare them with the states on the HBB PES.
J
BBSM
b
has
a similar nodal pattern as
J
a
, as it also displays two quanta
along the wagging motion and one quantum along the proton
transfer motion. These similarities are the characteristics of
a Fermi resonance.
J
BBSM
c
, however, shows a dominant
contribution of not two, but four quanta in the wagging motion.
The rich vibrational structure along the two wagging coordi-
nates is displayed in Fig. 4. This enables us to clearly assign
J
BBSM
c
as
j
04
40
i
wagging state.
We now turn to the states on the HBB PES. For
J
HBB
b
and
unlike
J
BBSM
b
, in addition to one quantum in the proton
transfer motion, there are not two but four quanta along the
wagging motion. Fig. 4 clearly shows the similarities to
J
BBSM
c
. Here, however, Fig. 3 and additional analysis reveals
a more dominant excitation along the proton transfer motion,
and we assign
J
HBB
b
as linear combination of the
j
04
40
i
wagging and the
j
1
z
i
proton transfer zero-order states. Finally,
similar to
J
a
, we identify in
J
HBB
c
not only dominating
contributions of the
j
02
20; 1
R
i
and the
j
1
z
i
states, but also
contributions of the
j
04
40
i
wagging state. Thus, on the
HBB PES next to the resonance of the
j
02
20; 1
R
i
and the
j
1
z
i
state, which alone would lead to two dominant peaks, the
j
1
z
i
state also resonates with the
j
04
40
i
wagging state. This
coupling creates an additional splitting, leading to overall three
peaks in the spectrum, a clear example of intensity sharing.
Why do the three particular zero-order states,
j
1
R
,02
20
i
,
j
04
40
i
, and
j
1
z
i
, couple on the HBB PES and why is the
j
04
40
i
not involved in the coupling on the BBSM PES? Estimating the
energetics of the zero-order states as shown in Fig. 2 is the clue.
The
j
02
20
i
and the
j
1
R
i
states alone do not have enough energy
(374 and 546 cm

1
, respectively) to couple with
j
1
z
i
. Only
a combination of these vibrational excitations leads to a state
with similar energy than the
j
1
z
i
state, which is required for
e
ffi
cient coupling. Next to the
j
02
20; 1
R
i
state, on the HBB PES
the
j
04
40
i
state has an energy that is very close to that of
j
1
z
i
.
This leads to additional coupling, resulting in
J
HBB
b
and
J
HBB
c
. While
j
04
40
i
and
j
02
20; 1
R
i
alone are barely IR active,
large components of the
j
1
z
i
lead to intensity sharing and thus
Fig. 2
Schematics of the coupling scheme of the zero-order states around the doublet in the Zundel infrared spectrum. Panel (a) displays the
previously identi
fi
ed coupling on the pioneering HBB PES.
14,19
Panel (b) displays our revised coupling on the same PES. Panel (c) displays the
coupling on the recent BBSM PES. The insets visualize schematically the vibrational excitations of the zero-order states. The energies of the zero-
order states are estimated either from the exact
j
1
R
i
and
j
02
20
i
/
j
04
40
i
states, or from the
j
1
z
,1
a
i
combination state, where
a
represents the
torsion motion.
11122
|
Chem. Sci.
, 2022,
13
, 11119
11125
© 2022 The Author(s). Published by the Royal Society of Chemistry
Chemical Science
Edge Article
Open Access Article. Published on 30 August 2022. Downloaded on 7/6/2023 9:10:03 PM.
This article is licensed under a
Creative Commons Attribution-NonCommercial 3.0 Unported Licence.
View Article Online
large IR intensities in the spectrum for all three entangled
states.
In contrast, on the BBSM PES the
j
04
40
i
state is blueshi
ed
by 40 cm

1
and the
j
1
z
i
state is redshi
ed by 115 cm

1
. This
disfavors additional coupling between these states. Hence, only
the
j
02
20; 1
R
i
couples with
j
1
z
i
, leading to the pair of states,
J
BBSM
a
and
J
BBSM
b
, which form a classical Fermi resonance.
J
BBSM
c
is almost a pure
j
04
40
i
state with only minor contri-
butions from
j
1
z
i
and thus a weak IR intensity.
Which of these two scenarios re
ects now the actual situa-
tion in experiment? While the experimental resolution is not
high enough to fully reject the possibility of two states lying
under one of the peaks of the doublet, there are two indications
that the scenario on the BBSM PES is more realistic: (1) the peak
positions obtained from the newer BBSM PES are closer to the
experimental spectrum and the PES is much closer to basis-set-
extrapolated CCSD(T) energies as shown in the le
panels of
Fig. S3.
(2)
J
BBSM
c
can be attributed to one of two satellite
peaks in the experimental spectrum, and (3) another state
between
J
BBSM
b
and
J
BBSM
c
likely corresponds to the second
satellite peak. We thus conclude that the additional coupling
seen in the older HBB PES is most likely an artifact that has
previously been missed due to less accurate computation of the
IR spectrum, which is only now disclosed in view of much
improved methodologies.
Based on these
ndings, it is striking to demonstrate that
such relatively small changes in the energetics lead to such
drastic di
ff
erences in the entanglement of the eigenstates and in
the IR intensities, which serve as the experimental observables
for these intricate phenomena. Moreover, we infer that minute
changes in the environment can lead to even larger energy
di
ff
erences and we anticipate e
ff
ects similar to those shown here
for microsolvated clusters and larger clusters with additional
Fig. 3
Representative cuts of the three relevant wavefunctions
J
a
,
J
b
and
J
c
corresponding either to the doublet and a satellite peak (BBSM,
upper panels), or to the triplet (HBB PES, lower panels) in the IR spectrum. The abscissa shows the wagging (pyramidalization) motion of one of
the water molecules whereas the ordinate shows the proton transfer motion. The red lines denote the zero contours.
Fig. 4
Representative cuts of the two wavefunctions
J
b
and
J
c
along
the two wagging motions computed with the BBSM (upper panels) and
with the HBB (lower panels) PESs. The red lines denote the zero
contours.
© 2022 The Author(s). Published by the Royal Society of Chemistry
Chem. Sci.
, 2022,
13
, 11119
11125 |
11123
Edge Article
Chemical Science
Open Access Article. Published on 30 August 2022. Downloaded on 7/6/2023 9:10:03 PM.
This article is licensed under a
Creative Commons Attribution-NonCommercial 3.0 Unported Licence.
View Article Online
solvation shells. To give three examples: (1) deuteration of the
Zundel ion signi
cantly alters the energetics of the zero-order
states and vastly complicates the IR spectrum.
18
(2) The experi-
mental spectrum of H
5
O
2
+
$
Ar (and that of H
5
O
2
+
$
H
2
)isstrikingly
di
ff
erent to that of H
5
O
2
+
$
Ne, as it displays four and not two
peaks around 1000 cm

1
,
17
indicating a similar complex coupling
situation (the Ar atom attaches to one of the OH units and thus
li
s degeneracy; in contrast, the spectra of H
5
O
2
+
$
Ne and
H
5
O
2
+
$
He do not show signi
cant di
ff
erences and their spectra
should be very close to that of the bare H
5
O
2
+
34
). (3) Recent ML-
MCTDH simulations on the solvated hydronium or Eigen ion
(H
9
O
4
+
) reveal the contribution of dozens of eigenstates that
dominate the IR activation of the hydronium O
H stretch motion
and show that the proton vibrations of the Eigen ion can be
understood in terms of an embedded Zundel subunit.
35
Likewise,
recent experimental studies indicate that the dynamics of
protonated water clusters can be related to
uctuations of local
electrical
elds,
36
which also appear for solvated H
5
O
2
+
.
9,37,38
4 Conclusions
In conclusion, we report high-accuracy, state-resolved simula-
tions of the Zundel ion on two PESs with unprecedented accu-
racy. Using variational DMRG-like algorithms adapted to tree
tensor network states and polyspherical coordinates, we are
able to compute up to 900 eigenstates up to 1900 cm

1
with
hitherto unreached accuracy for such a complex. Obtaining this
large number of eigenstates has so far never been accomplished
for a 15-dimensional,
uxional system using curvilinear coor-
dinates, and this will serve as a benchmark for future quantum
dynamics method developments. Only applying this advanced
methodology allows us to draw de
nite conclusions on the
quantum dynamics of the Zundel ion, in particular for the most
dominant feature of the IR spectrum, namely the doublet
around 1000 cm

1
. Previous assignments are shown to be
incomplete due to several limitations of the methods available
close to two decades ago. The novel assignment of the doublet
around 1000 cm

1
not only con
rms the Fermi resonance
nature of this proton transfer mode, but also explains so-far
unassigned low-intensity satellite peaks in the experimental
spectrum. In broader terms, our
ndings not only highlight the
striking mechanistic consequences of the strong anharmonicity
and
uxional character of this challenging but important
molecular cluster, but also reveal how subtle e
ff
ects of the
energetics of the zero-order states can lead to qualitative
changes of the IR spectrum, revealing fundamental quantum
e
ff
ects such as resonances, entanglement and intensity sharing.
We are therefore convinced that the same computational
approach as unleashed here for the
rst time will unveil similar
spectral complexity upon assigning high-resolution IR spectra
in a multitude of molecular complexes, notably when it comes
to proton transfer in (micro-)solvation environments.
Data availability
Further data is available from the authors upon reasonable
request.
Author contributions
H. R. L. and O. V. conceived the study; H. R. L. designed the
study, developed the TTNS methodology and performed and
analyzed the TTNS simulations; M. S. and O. V. performed the
MCCPD
ts and additional ML-MCTDH simulations; R. B., F.
B., C. S., and D. M. created the interface for the BBSM PES,
provided samples for the MCCPD procedure, and performed
benchmarks for DMC con
gurations provided by M. S.; H. R. L.
dra
ed the manuscript; all authors contributed to discussing
the results and editing the manuscript.
Con
fl
icts of interest
There are no con
icts to declare.
Acknowledgements
The authors thank Professor Joel Bowman for having made
their potential energy surface available. HRL acknowledges
support from the University of California Merced start-up
funding during the last part of this work. HRL acknowledges
support from a postdoctoral fellowship from the German
Research Foundation (DFG)
via
grant LA 4442/1-1 during the
rst part of this work. HRL acknowledges computational time
both on the Pinnacles cluster at UC Merced (supported by NSF
OAC-2019144) and at the Resnick High Performance
Computing Center, a facility supported by the Resnick
Sustainability Institute at the California Institute of Technology.
The Bochum work has been funded by the Deutsche For-
schungsgemeinscha
(DFG, German Research Foundation)
under Germany's Excellence Strategy
EXC 2033
390677874
RESOLV as well as by the individual DFG grant MA 1547/19 to
DM and supported by the
Center for Solvation Science ZEMOS
funded by the German Federal Ministry of Education and
Research and by the Ministry of Culture and Research of North
Rhine-Westphalia. RB acknowledges funding from the Stud-
iensti
ung des deutschen Volkes and CS acknowledges partial
nancial support from the Alexander von Humboldt-Sti
ung.
MS and OV thank the High Performance Computing Center in
Stuttgart (HLRS) under the grant number HDQM_MCT as well
as the bwHPC project of the state of Baden-W
̈
urttemberg under
grant number bw18K011 for providing computational
resources.
Notes and references
1 K. R. Asmis, N. L. Pivonka, G. Santambrogio, M. Br
̈
ummer,
C. Kaposta, D. M. Neumark and L. W
̈
oste,
Science
, 2003,
299
, 1375
1377.
2 J.-W. Shin, N. I. Hammer, E. G. Diken, M. A. Johnson,
R. S. Walters, T. D. Jaeger, M. A. Duncan, R. A. Christie
and K. D. Jordan,
Science
, 2004,
304
, 1137
1140.
3 J. M. Headrick, E. G. Diken, R. S. Walters, N. I. Hammer,
R. A. Christie, J. Cui, E. M. Myshakin, M. A. Duncan,
M. A. Johnson and K. D. Jordan,
Science
, 2005,
308
, 1765
1769.
11124
|
Chem. Sci.
, 2022,
13
, 11119
11125
© 2022 The Author(s). Published by the Royal Society of Chemistry
Chemical Science
Edge Article
Open Access Article. Published on 30 August 2022. Downloaded on 7/6/2023 9:10:03 PM.
This article is licensed under a
Creative Commons Attribution-NonCommercial 3.0 Unported Licence.
View Article Online