行政院國家科學委員會補助專題研究計畫 期中進度報告

行政院國家科學委員會補助專題研究計畫 期中進度報告
複雜氫鍵系統之物理化學性質及應用於合成之理論研究
計畫類別: 個別型計畫 □ 整合型計畫
計畫編號:NSC 97-2113-M-007-006-MY3
執行期間: 98 年 8 月 1 日至 99 年 7 月 31 日
計畫主持人:游靜惠
共同主持人:
計畫參與人員: 藍安嚳、黃國韜、劉家源、鄒沛剛、楊昇哲、吳事勳
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執行單位:國立清華大學化學系
中 華 民 國 99 年 5 月 30 日

關鍵字:氫鍵、低能障氫鍵、分子振動、紅外光譜、分子力學、BO 分子
力學、密度泛涵計算
摘要
利用分子力學與量子計算所得之位能面解薛丁格方程式分析氫鍵中之質
子的動力學。研究系統包括八維多項式模型位能與(Dih)2H + (Dih: 4,5-dihydro-
Himidazole) 陽離子。此研究目的在於了解氫鍵中質子之運動特性,並藉以釐清
低能障氫鍵是位能面上的真實低能漲氫鍵或只是平均位置看似位於兩重原子
之中而已,此動力學分析亦可預測氫鍵中質子之紅外光譜,輔助分析實驗光
譜。利用 B3LYP/ D95+(d,p) 計算與簡諧近似法預測(Dih)2H + 陽離子中的連結
氫鍵是低能障氫鍵。量子振動分析顯示此一質子其實是侷限於一端的低能量區
而其能量並不足以穩定類過渡態結構的低能量氫鍵結構。進一步的分子動力學
分析則顯示質子靠近一端的氫鍵結構所佔機率甚大。然而在溫度 300K 時,原
子的運動足以支持振動輔助之質子穿隧機制,紅外光譜可觀測到一典型之侷限
質子具有的寬闊的 NH-延伸振動譜線。此氫鍵之不對稱性亦可預測 NH 的倍頻
譜線。
Keyword: hydrogen bond, low barrier hydrogen bond, molecular vibration, IR spectrum, MD,
BOMD, DFT
Abstract
The harmonic approximation applied to B3LYP/D95+(d,p) quantum chemical computations
predicts the hydrogen bridge (HB) in the (Dih)2H + cation (Dih: 4,5-dihydro-Himidazole) to be a
low barrier hydrogen bond (LBHB). Explicit quantum calculations for the linking proton show,
that proton delocalization does not provide enough energy to stabilize a transition state like
geometry to permanently support a LBHB. The analysis of the (Dih)2H + trajectories shows
further that the majority of the (Dih)2H + conformers supports single well HBs with a localized
proton. However, the increased mobility of the atoms at 300 K supports a vibrationally assisted
proton tunneling mechanism. Furthermore, the increase in atomic mobility results in the wide
broadening of the NH stretch modes typically observed for systems with delocalised protons. The inherent
asymmetry of the N···H + ···N bridge suggests an increased probability to observe overtones of the NH
stretch vibration directly in weakly bond cluster ions with long hydrogen bridges.
2

1 Introduction
The theoretical analysis of the proton mobility in hydrogen bridges (HB) has a long history
[1–4] and biochemical research has joined the discussion last by the analysis of low barrier
hydrogen bonds (LBHBs; HBs where the proton ground state vibrational energy ɛ0 is
higher than the barrier E ‡ for the proton movement) for the energetics [5–9] and proton
tunneling for the kinetics of enzymic reactions [10–14].
The harmonic approximation commonly used to analyze the movement of atoms in
molecules [15, 16] works well for the majority of chemically relevant molecules, but with
proton ground state energies comparable to barrier heights in many HBs, the movement
of the proton cannot be described well with classical models and the quantum nature of
the proton needs to be considered explicitly [17–41].
Two systems are frequently used to probe hydrogen bridges. The Zundel ion H 5 O + 2 and
related species [42–52] and carboxylic acid dimers [53–60]. The focus on proton solvation
stems from the importance of proton transfer processes for chemical reactivity and the
richness of experimental data available, while the clear separation of the stretch mode of
the linking proton from the remaining modes of the carboxylic acid dimer can be used to
simplify the theoretical analysis.
Proton transfer along a hydrogen bridge can be modeled by the movement of a proton
in a double well potential. The potential energy function (PEF) can be described by three
parameters: The distance between the minima, which is linked to the distance between the
heavy atoms spanning the bridge (later called dNN); the height E ‡ of the barrier separating
the minima and finally, the energy difference ∆E between the minima. The value of ∆E
2

can be linked to the difference in basicity ∆PA of the linked bases [61–64] and/or to the
environment of the hydrogen bonded cluster [32, 65–69].
The influence of the environment on the properties of the hydrogen bridge can be
readily observed in the infrared (IR) spectra of the cluster. Hayd and Zundel showed that
thermal fluctuations in the environment manifest themselves in a varying electric field,
which again broadens IR peaks linked to the movement of the linking proton [70]. These
interactions between the HB and its environment become important as the HB length
increases, since short HBs often have a large polarizability [71].
Cluster with a N···H + ···N bridge (later abbreviated as NHN + ) tend to have long
nitrogen-nitrogen distances and thereby tend to form weak hydrogen bridges [72, 73].
The harmonic analysis of the proton movement in these cluster ions suggests, that some of
them can be labeled as LBHBs and that they fall thereby into the group of highly polar-
izable hydrogen bridges [74]. Moreover, the harmonic analysis showed a clear separation
of the NH stretch mode of the HB from the remaining modes of the cluster, which can be
used to simplify the theoretical analysis.
The harmonic analysis [74] suggests a LBHB in (Dih)2H + (Dih: 4,5-Dihydro-1H -
imdazole, C3H6N2) and a ∆E value very close to zero as expected for a homo-molecular
hydrogen bridged system.
HN NH N NH HN N HN NH (1)
The Mulliken [75–78], NBO [79–82] and APT [83, 84] charges in the ground (+0.4, +0.5,
+0.9) and in the transition state (+0.4, +0.5, +1.1) indicate that a proton is moving
between the two Dih molecules as suggested by the cation’s Lewis structure (Eq. 1). The
3

small changes in the proton charge during the transfer indicates that any reduction in
electron density at the proton caused by the cleavage of a NH bond is compensated by
the formation of the new one [85] and is typical property of highly polarizable hydrogen
bridges [1].
These properties makes the (Dih)2H + cation a suitable candidate for the analysis of the
coupling [86, 87] between the proton movement in a HB and its environment. However, a
complete quantum treatment of the 23 atom (Dih)2H + cation by VSCF [88, 89] or quantum
Monte Carlo [90, 91] is beyond the scope of this work. Moreover, the recent discussion
about proton storage in bacteriorhodopsin [92, 93] showed that a ”diffuse vibrational
band is not unique to protonated water cluster” [93] and that the discussion of LBHB
type hydrogen bridges may be reduced to that of the PEF which describes the proton
movement [33, 34, 94, 95]. Hence, our study of NHN + bridges started with a numerical
analysis of model PEFs (Section 3.1). A subsequent set of PEFs was obtained from the
optimized geometry of (Dih)2H + at the B3LYP/D95+(d,p) level and related structures
to investigate whether the delocalization of the proton between the two bases provides
enough energy to stabilize a LBHB with a transition state like geometry (Section 3.2).
Thermal effects were included into the model by Born-Oppenheimer molecular dynamics
(BOMD) simulations (Section 3.3). The BOMD simulations were done at the same level
as electronic structure calculations to maintain a smooth transition between the static
and the dynamic model. Potential energy scans of the proton movement in (Dih)2H +
conformers observed during the BOMD simulations were used to generate PEFs for the
quantum analysis of the proton transfer in these clusters. Hence the data triple (dNN, E ‡ ,
∆E) for the quantum model calculations shows the same thermal fluctuations as expected
4

for a real (Dih)2H + in vacuo.
Section 4 focuses on comparing our data with published data on LBHB formation and
stabilization, hydrogen tunneling and assisted proton transfer reactions and published IR
spectra with an emphasis on the over tones of the NH stretch vibration.
2 Computational Setup
2.1 The wave function of the proton
The proton is assumed to move on a straight line between the heavy atoms of the hydrogen
bridge and the the Hamilton Operator of the proton movement is given by
ˆH = ˆ T + ˆ
V = − 0.047818 kcal ˚A 2
d2 +
mol dx2 8
i=0
ai x i
where the coefficients ai of the PEF originate from different sources as discussed below.
The eigenfunction of the harmonic oscillator [1, 2, 34] or gaussian functions [36, 40]
are commonly used to expand the proton’s wave function. Hermite polynomials as basis
seem to be the natural choice, as the PEF of a double well hydrogen bridge (DWHB)
can be regarded as the interaction of two harmonic oscillators [1] or as the perturbation
of a harmonic potential [96]. Nevertheless, intergral evaluation is cumbersome and the
quantum calculations tend to converge slowly [1]. On the other hand, integral evaluation
with gaussians is straight forward and therefore, the wave function ψ of the proton was
expanded into a set of evenly spaced (∆xi = 0.05 ˚A) gaussians χi with a common exponent
α
ψ =
N
i=0
ci χi =
1
2 α 4
π
5
N
i=0
−α(x−xi) 2
ci e
(2)
(3)

while the eigenvalue problem was solved by combining the Cholesky decomposition scheme
for the overlap matrix with the Jacobi algorithm [97, 98]. The orbital exponent α was
optimized for every calculation (αopt) to minimize the value ɛ0. This optimization was
done with a linear search algorithm with variable step size at the beginning of the search
and a quadratic search algorithm at the end.
Tests with harmonic potentials reproduced the energy eigenvalues with an error smaller
than 10 −6 kcal/mol and tests with general PEFs showed that the maximum error for the
energy eigenvalues is typically much better than 0.001 kcal/mol.
2.2 Proton Position
LBHB formation has been described as a resonance effect [67, 99–101] of the proton despite
principle limitations of the model [102–104]. Hence, special effort was put into the quantum
chemical analysis of the proton position in this work.
The expectation value xi provides the average value for the position of the proton in
the i-th state. Highly polarizable hydrogen bridges tend to have broad PEFs with a flat
bottom region and are therefore expected to have energetically narrowly spaced states. To
account for possible excitations, the expectation values xi for the individual states were
weighted by their thermal population possibility.
x = 1
Z
N
i=0
xi e −ɛ i/R T = 1
Z
N
i=0
The position xψ of the maximum in ψ 2 0
〈ψi|x|ψi〉 e −ɛ i/R T
and Z =
N
i=0
e −ɛ i/R T
(4)
describes the place with greatest possibility to
observe the proton in a single experiment and can be used to distinguish between a LBHB
(xψ = 0) and a symmetric double well HB (xψ = 0) where x = 0 marks the geometric
center of the HB.
6

A direct measure for the proton delocalization is given by the fraction η, which is part
of the CEDE classification scheme for HBs proposed by Karingithi et al. [37].
η = ψ2 0 (x‡ )
ψ 2 0 (xψ)
The expression ψ 2 0 (x‡ ) is the value ψ 2 0 at the position x‡ of the maximum in the barrier
separating the minima in the PEF. A delocalized proton has an η value of 1 while η equal
to 0 indicates a localized proton, which does not percolate the barrier. Hence, a single-well
HB (SWHB) has a default η value equal to zero.
2.3 Relative Transition Probabilities
The relative transition probability between the initial state i and the final state f scales
with the square of the transition dipole moment µf←i [2, 15]. The proton is the only
particle carrying a charge in the model calculations and hence the dipole moment µ of the
system can be written as e x [16], which can be used to simplify the expression for µf←i.
µf←i = 〈ψf |µ|ψi〉 = 〈ψf |e x|ψi〉 = e 〈ψf |x|ψi〉 = e xf←i
The relative transition probabilities are defined here as the product of the energy difference
between the two states, the square of xf←i and the Maxwell-Boltzman factor of the initial
state.
Af←i = (ɛf − ɛi) × 349.75
cm−1
× 〈ψf |x|ψi〉
kcal/mol
2 × e −ɛ
N
i/RT
e
i=0
−ɛ −1
i/RT
Equation 7 was used to estimate the IR spectrum of the linking proton in thermally
fluctuating PEFs by summing the values of Af←i in a given frequency interval (∆˜νf←i =
125 cm −1 ) and constructing a smooth envelope function between 62.5 and 5562.5 cm −1
7
(5)
(6)
(7)

with cubic splines [105]. The final curve was divided by the number of data points used
for its construction to account for different sample sizes.
2.4 The potential energy function
The interaction of a proton with one base can be described with a Lennard-Jones type
interaction potential
VB(x) =
−A
+ ZA |x − xN|
B
|x − xN| ZB
where the parameters A and B were calculated from the targeted well depth Vmin and
position xmin of the minimum relative to the origin of the potential |xN − xmin| = 1 ˚A.
A = Vmin |xN − xmin| ZA
− 1
ZA
ZB
and B = A ZA
ZB
(8)
|xN − xmin| ZA−ZB (9)
The final PEF for the proton movement along the hydrogen bridge is the sum of two
monomer potentials VB(x) each designated by its own subsript.
VHB(x) =
−A1
|x − xN1 |ZA 1
+
B1
|x − xN1 |ZB 1
−
A2
|x − xN2 |ZA 2
+
B2
|x − xN2 |ZB 2
Initial tests showed that choosing V (1)
min = −242 kcal/mol, ZA1 = ZA2 = 1 and ZB1 = ZB2
= 6 yields PEFs similar to those observed for NHN + bridges between amines [74] and
hence these parameters were used as a starting point for the model calculations.
The data points generated by equation 10 were fitted to an 8 th order polynomial
(Equation 2) for the quantum calculations and shifted in energy to ensure V (x1) = 0 at
the global minimum x1 of the PEF.
8
(10)

2.5 Electronic structure calculations
Geometry optimizations and electronic structure calculations for the (Dih)2H + cation were
done at the D95+(d,p)/B3LYP level [106–109] as defined in Gaussian 03 [110]. The nature
of the localized stationary points was tested with subsequent frequency calculations. The
potential energy scans for the generation of the PEF for the proton movement were done
with frozen geometries and the proton moving on a straight line between bases with the
origin of the coordinate system in the geometric centre of the hydrogen bridge and a step
size of 0.01 ˚A.
2.6 Born-Oppenheimer molecular dynamics
Born-Oppenheimer molecular dynamics (BOMD) simulations of the (Dih)2H + cation were
done with GAMESS US [111] at the B3LYP/D95+(d,p) level. The initial geometries and
velocities for the final 4 ps NVE production runs were taken from the last 1.6 ps of a
4.6 ps NVT equilibration run with 1 fs integration steps and a target temperature of 300
K. The points chosen as seeds had to be close to the average total harmonic energy of
the aforementioned last 1.6 ps of the equilibration run and match three different target
temperatures close to 300 K. Computational details are provided within the supplementary
material.
Time series decomposition of data from the BOMD trajectories was done with a moving
average digital filter [112] with a symmetric averaging interval. Automatic peak detection
in these data streams was done with a simple algorithm to locate local extrema in the
filtered signal after additional bezier smoothing [105] followed by a visual verification of
the located points.
9

3 Results
3.1 Calculations with the model potential
Figure 1 summarises the results from calculations with the model PEF. The first set of
calculations (Figures 1a and 1b) analyses the dissociation of a hydrogen bridge with a
fixed ∆PA value of −0.1 kcal/mol. A ∆PA value not equal to zero was chosen to account
for a small asymmetry typically observed in PEFs for the proton movement in cluster
structures obtained from unconstrained geometry optimisations.
The model calculations suggest that the IR spectrum for clusters with a short heavy
atom distance dNN = |rN1 − rN2 | is dominated by the fundamental 1←0 transition as
indicated by the high values for relative transition probability A1←0 (Figure 1a). As the
value of dNN increases, the ground state energy ɛ0 and that of the first vibrationally excited
state ɛ1 degenerate and consequently the first hot band (A2←1) appears in the predicted
IR spectrum. At the very large heavy atom distances dNN, the predicted IR spectrum is
dominated by the first overtone (A2←0) and the transition between the second and fourth
vibrational level (A3←1). The last two transitions correspond to the fundamental vibration
of the proton trapped in the potential energy well at either side of the hydrogen bridge
and thereby mark the breaking of the bridge.
Figure 1b shows the frequencies ˜νf←i of the aforementioned transitions as a function of
dNN. A frequency ˜νf←i is shown only if the corresponding relative transition probability
Af←i has a value greater or equal to 1 ˚A cm −1 . The value of ˜ν1←0 decreases continuously as
the systems passes from a SWHB to a DWHB reflecting thereby the degeneration of ɛ0 and
ɛ1 with increasing values of dNN. As the system passes from a LBHB to a DWHB, the first
10

hot band appears in the predicted IR spectrum. Its intensity (Figure 1a) and frequency
increase until the monomer transitions ˜ν2←0 and ˜ν3←1 appear in the same frequency region
(Figure 1b) and the first hot band vanishes from the predicted IR spectrum.
The fundamental 1←0 and the 2←3 transition are commonly called the tunneling
transitions [2], as they are not observed in long HBs which do not support proton tunneling.
Janoscheck et al. [2] identified the 2←1 and the 3←0 transition as stretch vibrations. At
a HB length of approximate 3.1 ˚A, both transitions have the same frequency until they
vanish at approximately dNN = 3.3 ˚A as the vibronic proton wave functions do not overlap
anymore. From there on, the proton stretch peak in the predicted IR spectra originates
from the proton vibrations of protons captured by the unbound monomers. In so far it
is possible to take the 1←0 peak as an indicator for a shared proton, the 2←1 peak as
indicator for a proton in a strong HB while the 2←0 transition indicates a weak or broken
HB.
Figures 1c to 1e show the relative transition probabilities of the aforementioned transi-
tion as a function of the asymmetry of the potential energy profile for the proton movement
(∆PA = |V (2)
min | − 242 kcal/mol) for various fixed heavy atom distances dNN. As the value
of dNN increases, the relative transition probability A1←0 for the tunneling peak decrease
rapidly. Moreover, Figure 1c shows that symmetric potentials with ∆PA equal to zero
support a tunneling transition stronger than asymmetric ones. At very long heavy atom
distances and large ∆PA values, the value of A1←0 increases again as both the ground and
the first excited state are again bound in a single well.
The ∆PA dependence is more pronounced for the stretch peak of the linking proton
(Figure 1d). Large values of A2←1 require similar energies for the ground and the first
11

excited as observed in systems with an extended heavy atom distance. At very long dNN
values, the vibronic proton wave functions do not overlap anymore and value of A2←1
decreases. However, at this elongated dNN values, seizable value for A2←1 can be observed
only for very symmetric potentials with ∆PA values close to zero.
The opposite behaviour is observed for the 2←0 transition (Figure 1e). Large values
for A2←0 can be observed only in asymmetric potentials with ∆PA values significantly
differing from zero. Moreover, an asymmetry potential can result in large values of A2←0
at distances much shorter than expected from Figure 1a. The intensities of the 2←1 and
the 2←0 transition in the IR spectrum of a hydrogen bridge are therefore likely to provide
essential information about the symmetry of the potential in which the proton moves.
3.2 Static study of the (Dih)2H + cation
The DFT geometry optimization of the (Dih)2H + cation resulted in an asymmetric struc-
ture with the proton localized at one side of the HB (Figure 2a). The proton binding ring
is nearly planar with an NCCN dihedral angle of 0.93 ◦ as expected for the C2v symmetry
of the free DihH + cation, while the other ring is twisted with a NCCN dihedral angle of
15.55 ◦ similar to that observed in the free Dih molecule. In the transition state with E ‡
= 0.881 kcal/mol, the proton stays in the center of the HB and the NCCN dihedrals have
nearly identical values of −10.6 ◦ and −10.3 ◦ . Furthermore, the NN-distance (dNN = 2.57
˚A) is 0.11 ˚A shorter than in the optimized clusters (2.68 ˚A). The frequency analysis of the
optimized structure predicted the NH stretch frequency of the localized proton to be 2172
cm −1 , which corresponds to an harmonic zero point energy (hZPE) of 3.1 kcal/mol. With
hZPE much larger than E ‡ , the (Dih)2H + ion can be classified as a LBHB.
12

The PEF for the proton moving in the frozen framework of optimized (Dih)2H + cation
is very asymmetric with the second minimum 2.523 kcal/mol above the global one. The
barrier separating the minima is 3.071 kcal/mol high and slightly off center at −0.05 ˚A.
The asymmetry in the PEF is the direct result of the cation’s different dihedral angles of
the Dih rings and it cannot be removed by simple shortening the nitrogen-nitrogen distance
dNN. This principle asymmetry is also visible in the proton’s stationary wave function for
the optimised geometry: The proton is effectively localised in the deeper minimum and
the wave function’s tail penetrates the barrier, but no second maximum is observed at the
other side.
The proton in a LBHB is often compared with the proton in the transition state
[40, 61, 113, 114]. The resulting PEF for the proton movement in the transition state is
symmetric and has a small barrier of 0.135 kcal/mol in the center of the HB. In order to
observe a permanently delocalized proton, the reduction the proton’s ground state energy
ɛ0 resulting the removal of the barrier has to compensate the energetic costs of a transition
state like geometry. The summary of the corresponding energy analysis is presented in
Figure 2b. The horizontal line labeled ZPE is the ground state energy ɛ0 of the proton in
the PEF obtained from the fully optimized (Dih)2H + tautomer with the proton localized
at either of the Dih molecules. The framework energy ∆EFr = E ‡ (dNN, xglob) − Eopt is the
energy E ‡ (dNN, xglob) of the (Dih)2H + cation in a generalized transition state geometry at
a given nitrogen-nitrogen distance dNN and the proton at the position xglob of the global
minimum of the PEF relative to the energy Eopt of the fully optimized (Dih)2H + tautomer
(dNN = 2.68 ˚A, xglob = 0.239 ˚A). ∆EFr provides therefore a measure for the energetic costs
of the framework distortion and has been found to be minimal at dNN equal to 2.63 ˚A.
13

A structure can be labeled as stabilized by proton delocalization, if the sum ΣE of ɛ0
and ∆EFr falls below the value of ZPE. The ΣE curve has its minimum at 2.61 ˚A, which
suggests a very small stabilization energy of 0.639 kcal/mol. The PEF at this point has a
barrier of 0.508 kcal/mol, but the hydrogen bridge is still a LBHB with ɛ0 = 1.15 kcal/mol.
3.3 Dynamic study of the (Dih)2H + cation
The results of the final NVE BOMD simulations are summarized in Table 1 with a focus on
the NH stretch vibration along the hydrogen bridge. The analysis of the proton movement
in run number 2a suggests a periodicity of 15 fs for the NH stretch vibration, which agrees
very well with an estimated value of 15.3 fs from the harmonic frequency analysis. The
time series decomposition of dNN in run number 3 (supplementary material) suggests a
periodicity of 225.4 fs for the NN stretch vibration, which perfectly reproduces a periodicity
of 225.4 fs from the harmonic frequency analysis. A repetition of this analysis with the first
and second run yielded values of 222.5 and 223.6 fs in good agreement with the harmonic
frequency analysis. The proton in the hydrogen bridge vibrates therefore 15 times faster
than its environment, which justifies the adiabatic separation of both movements in the
model.
The value of dNN varies between 2.47 and 2.98 ˚A in run number 2 (Figure 3a): 248
geometries (6 %) have dNN values smaller than the transition state geometry (2.57 ˚A) while
the majority (2227 structure, 55%) has a NN distance larger than 2.68 ˚A. The average
value for dNN (2.70 ˚A) is therefore slightly larger than that observed in the optimised
geometry (2.68 ˚A).
Despite the periodic occurrence of short NN distances in which the proton automati-
14

cally moves towards the centre of the hydrogen bridge, the event of proton crossing appears
to be aperiodic (Figure 3b). One of the reasons for this aperiodicity is the uncorrelated
twisting of the Dih rings as indicated by the large standard deviations for δ1 and δ2. The
dihedral angles of the Dih rings sway between ± 40 ◦ and usually differ strongly in size
resulting in asymmetric potential energy functions which localise the proton at one side
of the bridge.
A small section of points (run number 2a, grey area in Figure 3a) between 2530 and
2977 fs from run number 2 was chosen for the quantum analysis of the proton movement.
Figure 3c shows the projection of the proton position onto the NN line in comparison with
the global maximum of ψ 2 0
of the proton’s ground state wave function. The position (xψ)
of the maximum of ψ2 0 follows the classical proton position (xH +) most times, though the
ability of the quantum particle to tunnel into the barrier moves xψ closer to the centre of
the hydrogen bridge than its classic counter part x H +.
Thermal fluctuations can deepen the potential energy well of the unoccupied hydrogen
position relative to that of the occupied one. Hence, the maximum of ψ 2 0
can be observed
on the opposite site of the hydrogen bridge five times during the BOMD simulation. It
stays there about 10 ± 2 fs 1 , which can be compared with the average periodicity of 15 fs
for the NH stretch vibration and 9.1 fs for the non-bonding NH stretch vibrations (3657
and 3675 cm −1 ). Moreover, a time interval of approximately 10 fs is equal to the period
length of a frequency of 3336 cm −1 which lies in the region NH stretch vibrations typically
observed for amines (3300 to 3500 cm −1 ) [115, 116]. Hence, it is possible to say that the
1 Peak ➀ is only half as wide as the remaining ones. Its exclusion from the average calculation yields a
time of 11 ± 1 fs for the maximum of ψ 2 0 being on the other side of the hydrogen bridge.
15

maximum of ψ2 0 stays on the opposite of the hydrogen bridge for approximately the length
of the period of NH stretch vibration.
Whether these times with the maximum of ψ 2 0
on the opposite site of the hydrogen
bridge provide an opportunity for proton tunneling or not depends on the localisation of
the H + . Figure 3d shows the value of η as a function of time for run number 2a. LBHBs
(marked grey) can have η values as low as 0.8 and a η value of 0.8 was therefore chosen as
a criterion for the effective proton delocalisation in DWHBs. Using this criterion, peaks ➀,
➁, ➃ and ➄ indicate a high possibility for the proton to move to the other side of the HB,
while the low η value for peak ➂ suggests that the proton stays at the position predicted
by the BOMD simulation and thereby represents an excited sate of the proton in the HB.
Figure 3e shows the correlation between ˜ν1←0 and ˜ν2←1 in run number 2a. The left
corner with low values for ˜ν1←0 and medium large ˜ν2←1 values marks cluster geometries,
which support long double well hydrogen bridges (DWHBs). In these hydrogen bridges
ɛ0 and ɛ1 are nearly degenerate and the first hot band has become the dominant transi-
tion. The upper right corner of the triangle is marked by short SWHBs with their nearly
harmonic PEFs. The connecting line between these two points is marked by hydrogen
bridges, which preserve their symmetry while stretching the NN distance. LBHBs can be
found in a small region at the centre of this line. The lower right corner is marked by
long, strongly asymmetric DWHBs with transition state energies for the proton movement
close to the energy of the first excited state. The lower arm of the triangle marks therefore
hydrogen bridges with increasing asymmetry.
The quantum calculations for the proton movement in run number 2a showed that only
6.5% of all hydrogen bridges in Figure 3e can be labelled as LBHBs and that short SWHBs
16

are also rare. Hence, the majority of clusters in run number 2a supports asymmetric
hydrogen bridges. This principal asymmetry becomes visible by looking at the averaged
results (Table 1) from the individual quantum calculation for the proton: The value of η
plunges down to 0.18, the energies ɛ0 and ɛ1 rise while the value of ˜ν1←0 increases to 938
cm −1 .
Initial calculations on the proton movement in the frozen geometry of the optimized
(Dih)2H + cation showed, the ion’s dipole moment can be reasonably well approximated
with a linear fit (supplementary material). Hence, equation 7 can be used to estimate
the IR intensities of the proton vibration in (Dih)2H + . Figure 3f shows the corresponding
simulation of the IR spectrum for a proton in a fluctuating PEF at 300 K. The spectrum is
clearly dominated by the fundamental tunneling peak at 1008 cm −1 . Small contributions
from the first hot band (1155 cm −1 ) blue shift the maximum of the envelop to 1069
cm −1 . The next big contributor is the first overtone with its maximum at 1828 cm −1 .
The dominance of the 2←0 transition over the 2←1 is the direct result of the underlying
asymmetry as a high A2←1 value requires a symmetric hydrogen bridge (Figure 1d), while
asymmetric hydrogen bridges generally result in high values for A2←0 (Figure 1e). The
smallest contribution to the spectrum arises from the 3←0 transition (2445 and 3051
cm −1 ), which has been assigned to the proton stretch in the free monomers (Figure 1a),
but also benefits from the general asymmetry of the PEF.
The average dNN value of 2.680 ˚A at 80 K reproduces that of the optimised geometry
2.681 ˚A very well and the gaussin envelope of the dNN distribution function (Figure 4a)
reaches values close to zero for dNN values close to that observed in the transition state
geometry. Within the time window of observation, no proton transfer was recorded for
17

un number 4.
Figure 4b shows the correlation between ˜ν1←0 and ˜ν2←1 in run number 4a. Areas of the
plot assigned to long DWHBs and short SWHBs are hardly populated while the majority
of points cumulates in the centre with a clear tendency towards asymmetric DWHBs. This
property of the trajectory is reflected in the properties of the dihedral angles δ1 and δ2:
Their average values are close to zero and the corresponding standard deviations similar
to the values observed at 300 K. This uncorrelated movement of the dihedrals prevents
the formation of symmetric cluster geometries which could support proton transfer. The
position (xΨ) of the maximum of the squared ground state wave function (Ψ2 0 ) follows
therefore that of the proton (x H +) in the BOMD trajectory with only one exception as
shown in Figure 4c. This peak overlaps with a region of LBHBs with η values close to 1 and
hence marks an opportunity for a successful tunneling process, while four opportunities
were observed in an interval of similar length at 300 K.
This result contrasts the observation that the average η value in run number 4a (η =
0.28) is significantly larger than that in run number 2a (η = 0.18). The small amplitude
atomic movement in run number 4a generates cluster geometries which resemble that of
the optimised structure and thereby η values close to 0.26. At 300 K, the trajectory
runs through areas of more distorted geometries with η values close to zero. The average
value of η is therefore expected to be much lower than that of the optimised structure.
Meanwhile, the large amplitude of the atomic movement in run number 2a generates more
cluster geometries which support proton transfer either by shortening the NN distance or
by equilibrating both sides of the hydrogen bridge. The thermal motion thereby supports
the proton tunneling process in run number 2a while simultaneously reducing the LBHB
18

character of the hydrogen bridge.
The same behaviour can be observed in the thermally averaged expectation value x of
the hydrogen position. The proton movement shows a large amplitude sampling both sides
of the hydrogen bridge at 300 K (Figure 3c), while the proton stays at one side at 80 K
(Figure 4c). The values of x stay closer to center of the HB as those of xψ and x H + in both
cases. However, the average of the absolute value of x is to large to classify the system as
a symmetric DWHB or as a LBHB. Moreover, the excited states of the proton movement
are not populated at 80 K and the properties of the proton are solely determined by the
ground state wave function.
Figure 4d shows the IR spectrum estimated from the PEFs gathered from run number
4a. The most dramatic changes can be observed in the low frequency part of the spectrum.
The tunneling peak is red shifted by 193 cm −1 and significantly taller as the result of the
tighter distribution of dNN values (Figure 4a), while the peak for the first hot band is
completely missing. The peaks ˜ν2←0 and ˜ν3←0 are blue shifted by 173 and 343 cm −1 while
the minor peak (2445 cm −1 ) of the bimodal ˜ν3←0 band vanishes. The small decrease in
the intensities of the ˜ν2←0 and ˜ν3←0 peaks can be linked to the tighter distribution of dNN
values at 80 K which does not cover the long distances necessary for large value of A2←0
and A3←0. These intensity changes are small compared to changes in the ˜ν1←0 peak and
the intensities of these peaks appear to be almost constant.
19

4 Discussion
It has been argued that the vanishing barrier for the proton movement in LBHBs and
the associated energy gain play a significant role in enzymic catalysis [5–9, 117]. The
calculations with the static model show that the barrier does not completely vanish and
that proton delocalisation can indeed stabilise the (Dih)2H + ion with transition state
like geometries. Nevertheless, the energy gain is much smaller than the estimated upper
limit of 4 kcal/mol published by Perrin and Nielson [118]. The observed energy gain
of 0.64 kcal/mol is equal to the kinetic kinetic energy of the 23 atoms in (Dih)2H + at
9 K. As all atoms of the (Dih)2H + ion need to be aligned for the necessary symmetric
PEF, the observation of a LBHB becomes increasingly unlikely at temperatures markedly
higher than 9 K. The analysis of the BOMD trajectories underpins this argument as only
14% of all (Dih)2H + conformers at 80 K (Ekin = 5.5 kcal/mol) can support a LBHB
compared with 5.8% at 300 K (Ekin = 21 kcal/mol). In that respect, the definition of
LBHB supporting environments [6, 9, 99, 117–120] has to be amended by the constraint
that symmetry breaking modes are thermally unaccessible in the temperature region of
interest. Moreover, the necessity of such an amendment increases with the number of
atoms necessary to be lined.
The quantum proton represented either by xψ or by x generally follows the movements
of its classic counterpart x H + (Figure 4c), however with a lower frequency. On several
occasions, xψ and x are on the opposite of the hydrogen bridge and high η values suggest
a high tunneling probability for the ground state proton. The only execption from this
general behaviour is peak ➂ in Figure 3c, where the proton would be trapped in the
20

first excited state for a short time [3]. Moreover, the thermal analysis indicates that the
excited states of the proton are sparsely populated and that the dynamics of the proton
in (Dih)2H + are dominated by the properties of the ground state and that the adiabatic
approximation holds [4].
The comparison of Figures 3c and 4c suggests that the possibility of proton tunneling
increases with the temperature. A similar effect has been reported for the protonated
1,8-bis(dimethylamino)naphthalene proton sponge [121]. Its electron density map at 120
K suggests two proton positions in the HB, while a single position in the centre is observed
at 300 K. Two mechanisms are commonly used to explain the temperature dependence
of the proton position in enzymes [10–14, 122, 123]. In the case of vibrationally enhanced
tunneling the vibrations of the environment compress the reaction coordinate and thereby
facilitate the tunneling process. In the second case, vibrationally assisted tunneling, the
normal modes of the environment equilibrate the relative energies of the two proton po-
sitions to support the tunneling process. The peaks ➀, ➁, ➃ and ➄ in Figures 3c and 3d
indicate tunneling possibilities while the reaction coordinate for the proton transfer (dNN)
is markedly stretched in agreement with the second mechanism.
The smallest number of proton transfer reactions was observed in run number 1 and
the analysis of the (Dih)2H + conformers with minimum dNN values in this run show, that
short dNN distances create PESs with a single minimum; however, this minimum is not
necessarily located in the centre of the hydrogen bridge. The asymmetric PESs show a
narrow minimum which locates the proton firmly at one side of the HB [124] and do not
support proton transfer of any kind. Hence, vibrationally assisted tunneling seems to be
more important for the (Dih)2H + ion at 300 K than the vibrationally enhanced process.
21

The quantum calculations on the proton movement also provide the means to interpret
the IR spectra of hydrogen bonded cluster. A single IR peak below 1700 cm -1 for an
NH or OH stretch mode is an indicator for a symmetric HB in according Johnson and
Rumon [125] and proton tunneling peak is usually indicated by a continuum below 1600
cm -1 with maximum values between 500 and 950 cm -1 [126]. Quantum calculation for the
proton in proton sponges [38] predict frequencies for the tunneling peak between 509 and
788 cm -1 and similar calculations for the proton in H5O + 2
[1] yield frequencies between 4
and 759 cm -1 depending on the oxygen-oxygen distance. A more recent study [37] reports
frequencies for the tunneling peak between 17 and 245 cm -1 for malon-aldehydes and 1378
cm -1 for the maleate anion with a symmetric SWHB. Test calculations with the model
potential show that only if the hydrogen bridges have the same length, the ˜ν1←0 value can
be used to gauge the delocalization of the proton as the effective width of the potential
energy well has a much stronger influence on ˜ν1←0 than the barrier height E ‡ .
Thermal and temporal averaging of cluster geometries tends to result in more sym-
metric structures and elongated heavy atom distances and consequently low ˜ν1←0 values.
In so far, the strong red shift observed for ˜ν1←0 peak for temporal averaged geometries
has to be regarded as a false positive, specially since averaging the individual IR spectra
suggests a significant blue shift. Nevertheless, the predicted ˜ν1←0 values for the optimized
structures are at the upper limit of those expected for a LBHB and that the maxima of
simulated IR still in range or slightly above.
The observation of IR absorption in the first hot band requires a large heavy atom
distance (Figure 1a, dNN > 2.6 ˚A) and a symmetric potential energy function (Figure 1d,
∆PA = 0). As the distance between the heavy atoms increases with the excitation of the
22

NN stretch vibration, thermal excitation is expected to increase the value of A2←1 and
˜ν2←1 to be part of combination adsorption [2]. The harmonic frequency calculation for
the (Dih)2H + ion suggests a NN stretch frequency of 148 cm −1 and the combination peak
for ˜ν2←1 is therefore expected to be blue shifted by a similar value. However, the general
asymmetry of the thermally excited (Dih)2H + ion drastically reduces the value of A2←1
and hot band absorption is expected to play only a minor role at 300 K (Figure 3f) and
to be negligible at 80 K (Figure 4d). The NHN + hydrogen bridges in proton sponges tend
to be highly symmetric, but their short NN distance still prevents the observation of hot
band absorption [38, 121].
A different behavior is observed for the first overtone (˜ν2←0). Though forbidden in a
harmonic potential energy well and to be expected for even longer heavy atom distances
than the 1 st hot band (Figure 1a), the first overtone contributes significantly to the sim-
ulated spectra (Figures 3f and 4d). The asymmetry of the thermally excited (Dih)2H +
ion makes its observation possible. A similar phenomenon has been reported by Roberts
et al. for the proton transfer in hydroxide solutions [46]. During each proton transfer
attempt, the frequency of the first the overtone of the OH stretch vibration drops into the
experimentally probed frequency region. But, calculations by McCoy et al. [90] for the free
H3O − 2
ion suggest the intensity of the the first overtone signal to be very small as expected
for the short OO distance and the high symmetry of the ion. Moreover, the calculations
on the Zundel ions suggest [50, 90], that combinations of the heavy atom stretch and out
of plane bending vibrations can cover the overtone vibration. Weakly bound NHN + ions
therefore seem to be ideal candidates for the direct observation of these overtones.
23

5 Conclsions
The standard quantum chemical analysis predicts the (Dih)2H + cation to form a LBHB.
The quantum analysis of the proton movement in (Dih)2H + shows that the proton wave
function penetrates the barrier, which causes a a low NH stretch frequency of 802 cm −1
and the proton is essentially localized on one side of the hydrogen bridge. This asymmetry
can be traced back to the conformation of the Dih rings. The proton binding Dih ring is
planar as observed in the free DihH + ion while the other ring is twisted as observed in the
free Dih molecule. The twisted ring binds the moving proton weaker than the planar ones
which results in the observed asymmetry of the hydrogen bridge.
Thermal excitation of the cluster ion increases the chance of a proton transfer reac-
tion via a vibrationally assisted tunneling mechanism. However, the ion’s trajectory runs
through conformational regions, which strongly localize the proton. The overall quantum
delocalization of the proton decreases therefore despite an increase in the proton transfer
reactions. At high temperatures, the (Dih)2H + ion resembles a system with the proton
hopping between two basins. At low temperatures, the proton is localized in one basin,
but its wave function penetrates the barrier which causes a low frequency tunneling peak
(˜ν1←0) in the IR spectrum.
The BOMD simulations reproduce correctly the periodicity results from the harmonic
frequency analysis and thereby link seamlessly to standard quantum calculations. The
most dramatic change can be observed in the position of the proton tunneling peak. It is
red shifted from 2170 cm −1 in the harmonic approximation to 1069 cm −1 in the explicit
proton quantum calculation and extremely broadened. The quantum calculation for the
24

proton movement suggests the IR absorption form the first overtone should be observable
at 1828 cm −1 overlapping with Dih ring vibrations and out of plane bending vibrations of
the linking proton (harmonic approximation 1602 cm −1 ). Moreover, the long NN distance
in (Dih)2H + and the thermally induced asymmetry predict a sizeable intensity for the
first overtone, which makes weakly bound NHN + ions suitable candidates for the direct
spectroscopic analysis of overtones from proton stretch vibrations in hydrogen bridges.
25

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