Extended tunnelling states in the benzoic acid crystal: Infrared and Raman spectra of the OH and OD stretching modes

Article abstract of DOI:10.1039/b609078h

We compare Raman and infrared spectra of the νOH/OD modes in benzoic acid crystal powders at 7 K. The extremely sharp Raman bands contrast to the broad infrared profiles and suggest adiabatic separation of hydrogen (deuterium) dynamics from the crystal la

Full text of DOI:10.1039/b609078h

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Extended tunnelling states in the benzoic acid crystal: Infrared and
Raman spectra of the OH and OD stretching modesw
Fran c¸ ois Fillaux, Fran c¸ ois Romain, Marie-Helene Limage and Nadine Leygue
´ `
Received 27th June 2006, Accepted 9th August 2006
First published as an Advance Article on the web 18th August 2006
DOI: 10.1039/b609078h
We compare Raman and infrared spectra of the nOH/OD modes in benzoic acid crystal powders
at 7 K. The extremely sharp Raman bands contrast to the broad infrared profiles and suggest
adiabatic separation of hydrogen (deuterium) dynamics from the crystal lattice. There is no
evidence of any proton–proton coupling term. The assignment scheme is consistent with a
quasisymmetric double-minimum potential, largely temperature independent. Tunnel splitting is a
major band shaping mechanism, in addition to anharmonic coupling with lattice modes. The
proton/deuteron dynamics are rationalized with nonlocal pseudoparticles and extended states. We
propose a symmetry-related damping mechanism to account for the broad infrared profiles, as
opposed to the sharp Raman bands. We assign spectral features to distinct interconversion
mechanisms based on either pseudoparticle transfer or adiabatic pairwise transfer. We establish
close contacts with theoretical models based on first-principles calculations.
1
5
1
. Introduction
isms.
Firstly, coherent phonon-assisted tautomerism
prevailing at low temperatures. Secondly, uncorrelated trans-
fer of bare protons in a quasi-symmetrical double minimum
potential with a high barrier leading to a two-step tunnelling
process at elevated temperatures. On the time-scale of
Proton transfer across hydrogen bonds is of fundamental
importance to many physical, chemical and biophysical pro-
1
–7
cesses. There is a general agreement that dynamics can be
modelled as light particles moving along local reaction co-
ordinates coupled to the motion of heavy atoms. These
dynamics are dominated by strong quantum effects with
pronounced sensitivity to isotope substitution.
ꢀ6
ꢀ11
1
0
–10
s probed with NMR and QENS the time-averaged
two-step interconversion is apparently consistent with a single-
step pairwise mechanism.
The double minimum potential for bare protons has been
questioned by calculations carried out with advanced quan-
In the crystalline state, the benzoic acid (C H COOH) form
6
5
centrosymmetric dimers linked by moderately strong hydro-
gen bonds. Single-crystal neutron diffraction studies show that
protons are largely ordered at low temperatures in the same
crystallographic configuration (say T1 in Fig. 1). As the
temperature is increased, protons are progressively transferred
to T2 and the population degrees are almost equal at room
16–20
21
and NMR experiments. We
tum chemistry methods
present below further investigations of the infrared and Ra-
man spectra of the OH/OD stretching modes of benzoic acid, a
15
follow up of a previous work. The motivation was twofold:
(
i) to unravel the band-shaping mechanisms for the nOH/OD
spectral profiles and (ii) to reach a deeper view of quantum
dynamics for proton transfer.
8
temperature. This thermally activated interconversion, or
tautomerism, is monitored by proton transfer dynamics.
–12
A key information to determine the double minimum
potential is the tunnel splitting of the nOH/OD bands. In the
infrared and Raman spectra, this splitting is embedded in a
series of band shaping mechanisms such as: strong coupling
with low frequency modes; fast relaxation; Davydov (u ꢀ g)
9
13
NMR
and quasi-elastic neutron scattering (QENS)
have confirmed that T1 and T2 are quasi-degenerate. The
8
11 ꢀ1
s between
proton transfer rate varies from B10 to 10
0 and 300 K. The non-vanishing rate at low temperature
1
1
4
suggests phonon-assisted tunnelling. The interconversion
mechanism was at first modelled as a stochastic single-step
pairwise proton transfer, through-barrier tunnelling at low
temperature and over-barrier jumping at elevated tempera-
tures.
splitting; Fermi or multiphonon resonances; hot bands; elec-
3,18,22
trical anharmonicity; breakdown of the selection rules.
It
is therefore difficult, if not impossible, to establish unques-
tionable assignment schemes.
Some of these mechanisms yield noticeable differences in the
infrared and Raman spectra and can be therefore distin-
guished (for example, u–g splitting, electrical anharmonicity,
ꢀ12
ꢀ14
At the time-scale of 10 –10 s probed with vibrational
spectroscopy (infrared, Raman, and inelastic neutron scatter-
ing-INS), it is possible to distinguish two competing mechan-
LADIR-CNRS, UMR 7075 Universite´ Pierre et Marie Curie, 2 rue
Henry Dunant, 94320 Thiais, France. E-mail: fillaux@glvt-cnrs.fr
http://ulysse.glvt-cnrs.fr/ladir/pagefillaux.htm
w The HTML version of this article has been enhanced with additional
colour images.
Fig. 1 Sketch of the tautomeric equilibrium in centrosymmetric
hydrogen bonded dimers.
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selection rules). However, whereas there are published works
1
5,18,23
on the infrared profiles of benzoic acid,
the Raman
spectra have been seemingly overlooked. The prevailing idea
is that the nOH/OD bands should be broad and too weak to be
confidently measured. In marked contrast to this prejudice, we
show below that these bands are extremely sharp and clearly
visible. Consequently, some dubious assignments in the infra-
red are straightforward in Raman and lead to the conclusion
that the double minimum potential itself is a major band
shaping mechanism. Thus, we can determine the effective
potentials for proton/deuteron transfer and correlate the ob-
served eigen states with interconversion energetics.
The dynamical model presented below is largely inspired by
2
4
a work on potassium hydrogen carbonate (KHCO3), an-
other system composed of centrosymmetric hydrogen bonded
ꢀ
dimers (HCO3 )2. Accordingly, we shall introduce nonlocal
Fig. 2 Comparison of the spectral profiles for crystal powders of the
fully hydrogenated benzoic acid at 7 K. (a) Infrared transmittance. (b)
Raman intensity. Numbers are Raman frequency shifts.
pseudoparticles and macroscopic tunnelling states. However,
it should be emphasized from the very beginning that the
interpretation of the Raman spectra presented below is dis-
couragingly difficult and it is unrealistic to pretend that any
reasonable assignment scheme, necessarily tentative, can prove
5
and disappear upon ring deuteration (BA-d h, Fig. 3). They
(
or disprove) any model in its own right. The purpose is more
can be assigned to CH-containing modes.
modestly to establish whether spectra are not inconsistent with
the preferred model, already based on totally independent
experiments and plausible theoretical background.
The most intense Raman bands between 2600 and 2950
ꢀ
1
cm
BA-h
have a pronounced nOH character. The spectra of
d presented in Fig. 4 show the residual nOH bands
5
In section 2, we compare the infrared and Raman nOH/OD
profiles of three isotope derivatives of benzoic acid at 7 K: the
fully hydrogenated (BA-h ); the ring-deuterated (BA-d h); the
corresponding to protons (E15%) surrounded by deuterium
atoms. The bands labelled with stars (*) are barely visible in
Fig. 2, aside the nOH bands. They are not due to traces of OH
groups. Presumably, they are overtones and combinations, but
a more specific assignment is not possible.
6
5
5 5
O-deuterated (BA-h d). We also report spectra of BA-d h at
various temperatures up to 300 K. In section 3, we propose an
assignment scheme and we determine the double-minimum
potential functions for the stretching modes. In section 4, we
elaborate a purely quantum mechanical model based on
nonlocal pseudoparticles and extended tunnelling states. We
show that ‘‘zero-phonon’’ transitions can be related to the
pairwise transfer. Finally, we show that potential functions
derived from vibrational spectra are in excellent agreement
with those calculated with first-principles methods. Experi-
mental details are given in section 5.
ꢀ
1
Bands between 2600 and 2675 cm are virtually unaffected
by ring deuteration and can be therefore attributed to pure
ꢀ1
nOH. Bands between 2700 and 2950 cm in Fig. 2 are also
dominantly nOH, although small frequency shifts by a few
ꢀ
1
cm upon ring deuteration (compare Fig. 2 and 3) suggest
that weak interaction with CH-containing modes should not
be totally excluded.
ꢀ
1
For BA-d h (Fig. 3) the bands at 3134, 3193 and 3260 cm
5
are free of CH modes and can be logically attributed to nOH
2
. Infrared and Raman spectra
The benzoic acid crystal is monoclinic at any temperature up
5
to 300 K. The space group is P2
lographically equivalent C COOH entities in the unit cell
forming nearly planar centrosymmetric dimers (C ) linked by
1
/c (C2h) with 4 crystal-
6 5
H
i
8
,25
˚
hydrogen bonds of length R
E 2.61 A at 20 K.
There are 177 vibrational modes that can be decomposed
Oꢂ ꢂ ꢂO
2
3
into 156 internal and 21 external vibrations. A full assign-
ment scheme would require theoretical modelling, beyond the
scope of the present work. Our purpose is therefore limited to
assigning nOH and nCH bands on the basis of isotope
frequency shifts.
The intense nCH Raman bands of BA-h
between 3000 and 3200 cm (Fig. 2). For BA-d
6
are observed
h, the nCD’s
are at 2250–2300 cm (Fig. 3). The very weak decoupled CH
ꢀ1
5
ꢀ
1
bonds due to traces of undeuterated aromatic sites (E2%)
peak at 3073 cm . The band at 3168 and the shoulder at 3210
Fig. 3 Comparison of the spectral profiles for crystal powders of the
ring-deuterated benzoic acid at 7 K. (a) Infrared transmittance. (b)
Raman intensity. Numbers are Raman frequency shifts.
ꢀ
1
ꢀ
1
cm (BA-h ) are unaffected by O-deuteration (BA-h d, Fig. 4)
6
5
4
328 | Phys. Chem. Chem. Phys., 2006, 8, 4327–4336
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Fig. 4 Comparison of the spectral profiles of traces of OH species
Fig. 5 Comparison of the spectral profiles for crystal powders of the
O-deuterated benzoic acid at 7 K. (a) Infrared transmittance. (b)
Raman intensity. Numbers are Raman frequency shifts.
(
r15%) in the O-deuterated benzoic acid crystal at 7 K. (a) Infrared
transmittance. (b) Raman intensity. Numbers are Raman frequency
shifts. *: OH-free Raman bands.
Fermi resonance is widely recognized as an important band
shaping mechanism. Such anharmonic coupling with multi-
phonons can be modelled as coupled multi-level systems.
However, Raman bands are so narrow that resonance should
be more seldom seen than in the infrared. In addition, since
nOH/OD band intensities are extremely weak, multiphonon
transitions may appear even if there is no resonance. For
modes. For BA-h
6
in Fig. 2, these OH bands are partially
superposed to the CH-containing bands at 3168 and
1
ꢀ
3
210 cm
.
ꢀ1
The full width at half maximum (FWHM) of E5 cm for
the main nOH bands at E2600 cm is in dramatic contrast
with the broad infrared profiles. The Raman width is similar to
ꢀ1
ꢀ
1
example, weak bands between 2200 and 2600 cm in Fig. 2
and 3 could be multiphonons. Similarly, some bands in the
5
that of the decoupled nCH bands of BA-d h and close to the
spectrometer resolution. (The intrinsic width could therefore
ꢀ1
1700–2000 cm range (Fig. 2) disappear upon ring deutera-
tion (Fig. 3) and some others are shifted downward for BA-h
be less than observed.) Similar sharp bands were reported for
5
d
2
6
dimers of gaseous formic acid. They were assigned to over-
tones or combinations of internal modes, whereas the stretch-
ing profile was attributed to a weak underlying continuum
(Fig. 5). There are probably multiphonons as well as nOH/OD
bands analogous to the intense infrared bands in the same
range. In any case, the very large number of internal modes
hampers a precise assignment scheme.
ꢀ1
with a breadth of E400 cm . For benzoic acid a similar
assignment scheme is untenable. First, there is no convincing
evidence of a broad continuum of intensity analogous to that
observed in the infrared. Second, combinations are unlikely
By contrast to Raman, anharmonic coupling of the infrared
continuum with narrow multiphonons may yield transmit-
2
7,28
29
or even more complicated profiles. For
tance windows
ꢀ1
since all internal modes are shifted by more than 10 cm upon
ring deuteration and they are much broader than the OH
example, among the Raman bands at 2822, 2853, and 2890
ꢀ
1
cm (Fig. 3), the central component could be nOH, for it
corresponds to a minimum of the infrared transmittance, while
the two side-bands corresponding to transmittance windows
could be multiphonons. However, this rule of thumb should be
used with caution. For example, nOH bands at 2603 and 2639
ꢀ1 15
.
bands at E2600 cm
Third, infrared transmittance win-
dows, possibly due to multiphonon transitions, show more
pronounced frequency shifts than the Raman bands and there
is no apparent correlation between the spectra. The sharp
bands are therefore confidently attributed to nOH modes.
ꢀ
1
6
cm in BA-h (Fig. 2) are close to infrared transmission
For BA-h
ꢀ
5
d in Fig. 5, Raman nOD bands at 2041 and 2198
windows, while their analogues in Fig. 3 do not correspond
to such windows. In fact, the isotope mixture in Fig. 4
1
cm are at nearly the same frequencies as infrared bands at
ꢀ1
ꢀ1
2
052 and 2202 cm , and relative intensities are comparable.
enhances multiphonon bands at 2596 and 2628 cm , corre-
Here again, Raman bands are much narrower (FWHM
sponding to the transmission windows, which were not re-
solved in the fully hydrogenated derivative. Presumably, these
bands and their transmission windows are shifted downward
upon ring deuteration, while the nOH is little affected.
Raman spectra at various temperatures (Fig. 6), in line with
ꢀ1
E7 cm ) than their infrared counterparts (FWHM E40
and 80 cm , respectively), although with less marked contrast
ꢀ1
than for nOH bands.
In Fig. 4, the residual nOH bands correspond essentially to
mixed HD dimers (E30%), whereas HH pairs (E2%) can be
ignored. Visual comparison with Fig. 2 shows that nOH bands
are neither shifted, nor broadened, by statistical surroundings
of deuterons. Clearly, intra- and inter-dimer coupling terms
are rigorously negligible. This is at variance with theoretical
1
5
infrared spectra, do not provide conclusive evidence for any
significant change of proton dynamics. The nOH bands are
marginally broadened at 300 K, frequencies are almost un-
affected and the overall profile is little changed. The potential
function for protons is therefore largely temperature indepen-
dent. This is remarkable since between 7 and 300 K protons
ꢀ
1 17,18
estimates of the Davydov splitting, up to E100–200 cm
.
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expansion of the potential around the minima at ꢃx
0
. Then,
tan 2f = n0t/(n01 ꢀ n0t) gives: cos f E 1 and sin f = e E
ꢀ
2 15
1
.8 ꢄ 10
e) measures the degree of symmetry for the double well: e =
/2 in the symmetric case and e = 0 for the totally asymmetric
potential. By definition, the potential is quasi-symmetric for
.
The delocalization degree of the wave function
2
(
1
2
0
o e o 1/2. The key parameter e determines the intensity of
the 0 2 1 transition. This transition has never been observed
with infrared and Raman, presumably because it is too weak
to emerge from other bands in the same range (for example
nOꢂ ꢂ ꢂO). However, it was observed with INS, thanks to the
large incoherent scattering cross-section of protons.
The second feature is the adiabatic separation of the proton/
deuteron dynamics from heavy atoms. Raman band-widths
emphasize very long lifetimes for nOH/OD excited states, on
the time-scale of heavy atom dynamics. Any relaxation me-
2
2
chanism, in particular nonadiabatic terms, can be safely
ignored. The nOH dynamics is therefore amenable to bare
protons/deuterons and non-crossing adiabatic potentials for
heavy atoms. As the nonadiabatic coupling terms do not
depend on the spectroscopy technique, the very broad infrared
profiles suggest homogeneous broadening by a very fast
relaxation mechanism of a different origin. We suggest below
a symmetry-related damping mechanism (holding for u modes
in the infrared but not for g species in Raman) specific to
double minimum potentials.
Fig. 6 Raman spectral profiles for a crystal powder of the ring-
deuterated benzoic acid at various temperatures.
The third feature is the strong coupling between nOH/OD
3
distances. Since the seminal paper by Stepanov,
3
34
and R
Oꢂ ꢂ ꢂO
strong coupling and adiabatic separation are cornerstones of
8
are progressively transferred to unoccupied sites and there are
1
5
spectacular changes of the lattice modes at low frequency.
the dynamical models for moderately strong hydrogen
3
,15,22,35–38
Clearly, there is no evidence for stochastic disorder, or any
transition from the quantal to the classical regime. This is in
marked contrast with conclusions presented in other
bonds.
nOꢂ ꢂ ꢂO in nOH/OD excited states is shifted toward short
distances, in the range characteristic of short symmetric
The minimum of the adiabatic potential for
R
Oꢂ ꢂ ꢂO
1
4,20,30,31
works.
hydrogen bonds. Simultaneously, the nOꢂ ꢂ ꢂO frequency in-
creases and the nOH decreases, both dramatically.
In Fig. 7, the upper adiabatic potentials are semi-quantita-
3
. Proton dynamics
tively sketched with Morse functions. We label |n
i
OH g/u
In this section, Raman spectra are tentatively rationalized with
the dynamical model proposed in ref. 15. This model is based
on three features of moderately strong OHꢂ ꢂ ꢂO hydrogen
bonds. First, the stretching mode experiences a quasi-symme-
trical double minimum potential (see Fig. 13 in ref. 15). In
contrast to the symmetric case, energy levels have no parti-
cular symmetry and can be labelled in order of their increasing
energy, n = 0, 1, 2, 3. . . The ground-state splitting has been
|N
i
OO ng/u
the state vectors in the nth ng/u OH and Nth nOꢂ ꢂ ꢂO
state. Transitions from the ground state |0
i
|0
OH g/u OO 0g/u
i
give
Franck–Condon-like progression bands whose relative inten-
sities depend on the overlap integral for the wave functions
along DR. The most intense bands correspond to ‘‘N-pho-
nons’’ vertical transitions (DNOO Z 1) while the ‘‘zero-
3
,37
phonon’’ transitions (DNOO = 0) are much weaker.
that infrared-active zero-phonon transitions, namely
|0 i |0 |2 i |0 and |0 i |0
Note
ꢀ
1
measured with INS for BA-d
5
h (|0i2|1i E 172 cm ) and
i
2
i
i
2
OH u OO 0u
OH u OO 2u
OH u OO 0u
ꢀ1
upper states were derived from infrared profiles (|0i2|2i E
|3 i |0 i , are at 1787 and 1907 cm , respectively, for
OH u OO 2u
ꢀ
1
ꢀ1
ꢀ1
BA-h , 1788 and 1902 cm , respectively, for BA-d h. In
6 5
15
2
570 cm and |0i2|3i E 2840 cm ). The ground state
splitting, largely dominated by the potential asymmetry,
should not be confused with tunnelling. For the symmetrized
contrast to the broad profiles observed for higher transitions,
the zero-phonon bands are quite sharp at low temperature.
These long-life states are virtually stationary.
ꢀ
1
potential the estimated tunnel splitting is hn0t E 6 cm
Within the two-level system approximation,
.
1
5,32
wave func-
It should be needless to emphasize that Fig. 7 is over-
simplified. There may be additional coupling between nOH
and low frequency modes, other than nOꢂ ꢂ ꢂO. In addition,
overtones and combinations can interact with OH states
through anharmonic coupling terms (for example, Fermi
resonance).
tions for the |0i and |1i levels can be written as:
C0 ¼ cos fc ðx ꢀ x0Þ þ sin fc ðx þ x0Þ
0
0
ð1Þ
C1 ¼ ꢀ sin fc ðx ꢀ x0Þ þ cos fc ðx þ x0Þ
0
0
The harmonic wave functions c (x ꢀ x ) and c (x + x ) are
For OD derivatives, the overall sketch is similar but the
upper adiabatic potentials are flattened and the potential slope
0
0
0
0
eigen functions of the Hamiltonian obtained by second-order
4
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ꢀ1
Table 1 Energy levels in cm units derived from nOH/OD Raman
bands
|2i
g
|1OO
i
2g
|2i |1OO
g
i
2g + |1i
g
|0OO
i
1g
|3i
g
|1OO
3g
i
BA-h
BA-d
BA-h
6
5
5
2603
2602
2041
2781
2773
2198
2910
2853
—
h
d
ꢀ
1
15
5
cm for BA-d h. Although strong nonlinear coupling terms
5
cannot be excluded for sure, it is possible to assign the BA-d h
ꢀ
1
band at 2773 cm to a combination with the |0i 2 |1i
ꢀ
1 15
.
transition previously observed at 172 cm
Although the
analogue transition for the fully hydrogenated derivative is
unknown, it is not unrealistic to suppose, for the sake
of consistency, that the |0i 2 |1i splitting is shifted to
ꢀ1
E178 cm for that derivative (Table 1).
ꢀ1
For the nOH band of BA-h at 2910 cm , the frequency
6
ꢀ
1
shift to 2853 cm upon ring-deuteration is much greater than
anticipated for any combination with lattice or internal modes.
In accordance with infrared spectra, these bands are assigned
to transitions to the upper nOH state: |0OH
|1OO 3g. The bands at 3134, 3193 and 3260 cm for
h are tentatively attributed to further Franck–Condon-
like transitions to upper nOꢂ ꢂ ꢂO states: |0OH
i
g
|0OO
i
0g
2
ꢀ
1
|
3
OH
i
g
i
BA-d
5
i
g
i
0g
|0OO 2
|
2
i |2
i
and |3 i |2 i . These states are probably
OH g OO 2g
OH g OO 3g
close to the upper potential thresholds.
For BA-h d (Fig. 5), we assign the nOD Raman bands at
5
Fig. 7 Schematic view of the effective adiabatic potentials along the
nOꢂ ꢂ ꢂO coordinate DR for the nOH states in a quasisymmetric double
minimum potential function.
ꢀ
1
2
041 and 2198 cm by analogy with the nOH (Table 1). Then,
the ground state splitting should be 157 cm . However, there
ꢀ
1
is no distinct band to be assigned to the upper nOD state
|
3
OD
i
g
|1OOi3g (see below, section 3.2) and there is no clear
interpretation for the weak bands around 1880, 2013 and
ꢀ1
in the relevant DR range is decreased. In principle, the minima
of the upper states, and therefore the zero-phonon states, are
little affected by deuteration.
2
115 cm , which could be multiphonons.
Finally, some Raman bands of the OH derivatives in the
ꢀ1
1
700–1800 cm region should be the analogues of the infrared
active zero-phonon transitions. However, there are also multi-
phonon bands and it is not possible to propose a precise
3
.1. Assignment scheme
We assign the most intense nOH Raman bands at about 2600
assignment scheme. For BA-h
5
d, nevertheless, there is no
ꢀ
1
ꢀ1
cm to |0OH
i
g
|0OO
i
0g 2 |2OH
i
g
|1OO
i
2g. A previous estimate
objection to attribute the main peaks at 1660 and 1707 cm
to zero-phonons: |0OD
|0OD |0OO |1OO
ꢀ
1
15
of 2570 cm from infrared profiles was much less accurate
since energy levels were poorly defined. As there is no intra-
dimer coupling, the difference between infrared and Raman is
not significant.
i
g
|0OO
3g, respectively, although these
transitions cannot be confirmed in the infrared as they are
i
0g
2
|2OD
i
g
|0OO
i2g and
i
g
i
0g 2 |3OD
i
g
i
1
5
hidden by intense nCO bands.
ꢀ1
In Fig. 2, the bands at 2603, 2639 and 2670 cm suggest a
Franck–Condon-like progression of the nOH involving a low-
ꢀ
1
3.2. Double minimum potentials
frequency mode ꢀh O E 33 cm . A similar pattern is observed
1
ꢀ
1
0
ꢀ1
in Fig. 3 at 2602, 2633 and 2673 cm ( ꢀh O1 = 35 cm ).
Intensity ratios suggest a rather modest coupling, possibly
The assignment scheme in Table 1 leads to the conclusion that
Raman spectra allow us to determine the ground state splitting
that cannot be measured otherwise, except with INS for the
ring deuterated derivative. The effective double minimum
potentials in Table 2 were determined through best fit ex-
ꢀ
1 15
.
with the lattice mode reported at E35 cm
A linear
coupling term could slightly shift the equilibrium position
along the relevant lattice coordinate in the excited state while
leaving the frequency virtually unchanged.
1
5
ercises. Since there are only three energy levels and four
parameters, we have included relative intensities as further
constraints. In addition, distances between minima Dr were
kept in reasonable agreement with neutron diffraction data.
Needless to say, alternative mathematical functions could be
envisaged but they would yield similar potential shapes.
For a given set of energy levels and relative intensities, Dr
depends on the effective oscillator mass that is therefore an
ꢀ1
The nOH bands at 2781 and 2813 cm (Fig. 2) could be
ꢀ1
2
combinations of those at 2603 and 2639 cm with ꢀh O = 178
ꢀ
1
ꢀ1
ꢀ
cm . Similarly, the sharp nOH band at 2773 cm for BA-d
5
h
1
0
(
Fig. 3) could be a combination of that at 2602 cm with ꢀh O
E 171 cm . However, for both systems, ꢀh O and ꢀh O are
2
ꢀ1
0
2
2
quite different from the observed lattice frequencies, for
ꢀ1
instance at 199.0, 133.5 cm
for BA-h , or 189.5, 129.5
6
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ꢀ1
Table 2 Potential functions, energy levels, and potential barriers (H) in cm units derived from the assignment schemes presented in Table 1. The
˚
tunnel splitting of the symmetrized potential function is hn0t. The distances between minima (Dr) and x are in A units
V
Dr
H
hn0t
hn01
hn02
hn03
2
2
BA-h
BA-d
BA-h
6
5
5
281x + 301370x + 172300 exp(ꢀ2.27x )
0.68
0.69
0.68
4990
5006
4488
7.7
6.0
0.2
178
171
156
2603
2602
2041
2910
2853
2181
2
2
h
d
270x + 282880x + 171120 exp(ꢀ2.15x )
2
2
240x + 299010x + 171980 exp(ꢀ2.265x )
adjustable parameter. Within the experimental accuracy for
Dr, the effective mass is 1 (2) amu for OH (OD) derivatives,
and the potential coordinate is along the straight line
joining the minima. Greater masses would give unrealistic
shorter Dr’s.
split the coupled oscillators into uncoupled normal modes.
These normal coordinates are independent of the coupling
strength between oscillators (too weak to be observed in the
present case). They are dictated by the crystal structure.
In quantum mechanics, normal modes define pseudoparti-
The barrier decreases substantially upon O-deuteration
while the effect of ring deuteration is tiny (Table 2). A similar
decrease upon OH/OD substitution was reported for potas-
g u
cles P and P , each with mass m = 1 (H) or 2 (D) amu,
corresponding to symmetric or antisymmetric displacements
of two ‘‘half-pseudoparticles’’. These pseudoprotons are ob-
servables and their mass is dictated by experiments. They are
nonlocal since individual positions and momenta of the ‘‘real’’
particles (x ,P and x ,P ) are not observables.
3
8
sium hydrogen/deuterium carbonate. If the potentials were
mere cuts of the Born–Oppenheimer surface along classical
trajectories, they should be unaffected by H/D substitution.
Alternatively, the observed eigenstates define ‘‘effective’’ po-
tentials depending on the domains spanned by the zero-point
motions of H or D atoms. This effect is important for OH/OD
substitution and tiny for ring deuteration.
1
1
2
2
Pseudoparticles offer a graphical representation of dynamics
in double wells (Fig. 8). In the ground state, P and P are
g
u
superposed in configuration T1 (grey circles), to order e. In
excited states, wave functions are largely delocalized over the
two wells (see Fig. 13 in ref. 15) and displacement vectors
acquire very large amplitudes (EDr). Raman active transitions
can be thought of as symmetric excitations of the two half-
The tunnelling frequency hn0t decreases substantially for the
OD derivative, in spite of the lower barrier. The states are
quite localized in their respective wells and the intensity of
|
0
ODi 2 |3ODi is depressed. In addition, this transition is
calculated at about the same frequency as [|0ODi 2 |2ODi] +
|0ODi 2 |1ODi], thanks to the weak tunnel splitting for the
pseudoparticles of P
transitions involving P
g
to upper delocalized states. In fact,
are also Raman active because the u
u
[
symmetry for small oscillations with respect to the equilibrium
configuration is superseded by large symmetric displacements
in the excited state. The upper states can be thus written as:
ꢀ
1
upper states (E15 cm for the symmetrized potential). There-
ꢀ
1
fore, the Raman band at 2198 cm in Fig. 5, as well as its
infrared counterpart, are likely to be unresolved superposi-
tions of the two transitions.
|2g/ui|0u/gi|1
i
OO 2g/u OO 0u/g
|0
i
, or |3g/ui|0u/gi|1
i
OO 3g/u OO 0u/g
|0
i
(Fig. 8 g) and the combination states for simultaneous excita-
tions of P and P
g
u
as: |2g/ui|1u/gi|1OO
i
i
1u/g
2g/u|0OO .
In order to sustain this assignment scheme, we can estimate
Raman intensities. They are proportional to
4
. Discussion
It may seem straightforward to suppose that the double-well
potential should monitor uncorrelated transfer of bare pro-
)
2
I0
0$20
¼ jh2gjh0ujh1 j h0 j ½aꢅj0gij0uij0 i j0 i j
OO 2g OO 0u
OO 0g OO 0u
1
5
2
tons or deuterons. However, this interpretation is refuted by
two counterarguments. Firstly, the energy cost for conversion
into dimers composed of ‘‘biprotonated’’ and ‘‘deprotonated’’
entities should be much greater than hn01. Secondly, the center
of symmetry should be destroyed at elevated temperatures and
the g–u selection rules for internal modes should be cancelled.
On the contrary, there is no evidence for symmetry breaking at
I00$21 ¼ jh2gjh1ujh1OOj h0OOj ½aꢅj0gij0uij0OOi j0OOi j :
2g
1u
0g
0u
ð3Þ
In addition, I00202 = I00220, I00212 = I00221. Inten-
sities depend on the transition moment for nOH multiplied by
the overlap integral for nOꢂ ꢂ ꢂO. We limit the expansion of the
polarizability tensor to first order as [a] = [a] x. Numerical
8
15
1
any time-scale (neutron diffraction or spectroscopy ). Con-
sequently, we have to conceive a dynamical model preserving
the center of symmetry. We present below a simplified version
integration of the transition moment for nOH gives I00202
/
I
00212 E 25. On the other hand the overlap integral
2
|
h0OO
|
1u/g|0OO
i
0u/g| for the 0 2 1 transition is close to unity,
of a theory elaborated for KHCO
3
and presented in details in
2
while that for 0 2 2, namely |h0OO
|
2g|0OO
|
0g| , is much less.
ref. 24,39. See also ref. 40 for isolated dimers in the gas phase.
4
.1. Pseudoparticle dynamics
Within the adiabatic separation, the stretching modes of a
dimer are modelled with two collinear harmonic oscillators x1
and x , each with mass m. The centrosymmetric normal
2
coordinates and conjugated momenta
Fig. 8 Schematic representation of the displacement vectors for
pseudoparticles in the stretching excited states. Grey and opened
circles correspond to occupied and unoccupied sites, respectively, in
the ground state.
1
2
1
1
2
1
2
xg ¼ pffiffi ðx1 ꢀ x2Þ; Pg ¼ pffiffi ðP1 ꢀ P2Þ;
ð2Þ
xu ¼ pffiffi ðx1 þ x2Þ; Pu ¼ pffiffi ðP1 þ P2Þ;
2
4
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Upon visual examination of Fig. 7 we estimate that this factor is
2
roughly |0.5| , hence I00202/I00212 E 6. The weak transition
moment for the combination band is therefore counterbalanced
by the favorable Franck–Condon factor. This is in reasonable
agreement with the observed intensity ratios of E5 for I2603
/
I2781 and I2602/I2773 in Fig. 2 and 3, respectively.
Furthermore, the double minimum potential offers a tenta-
tive explanation for the dramatic contrast between the sharp
Raman bands and the broad infrared profiles that cannot be
attributed to nonadiabatic coupling terms, or any alternative
coupling terms. In contrast to the totally symmetric oscilla-
tions observed in Raman, the infrared active antisymmetric
displacements with large amplitudes (Fig. 8 u) are incompa-
tible with stationary motions of pseudoprotons. Excited states
are therefore overdamped and relax down along the upper
Oꢂ ꢂ ꢂO adiabatic potential. Homogeneously broaden profiles
can be thought of as complicated interference patterns includ-
Fig. 9 Schematic representation of the single-pseudoparticle two-step
interconversion mechanism, projected onto the (a, c) plane. In the
intermediate virtual state with twofold degeneracy all sites are equally
occupied. For the sake of clarity, H atoms bound to the benzoic ring
are omitted.
3
8
ing, eventually, interaction with multiphonons.
The same conclusions hold for nOD. The Raman bands are
much narrower than their infrared counterparts. Moreover,
the narrowing of the infrared profile, compared to nOH,
accords with a decrease of the upper adiabatic potential slope
for nOꢂ ꢂ ꢂO upon deuteration.
observables. Normal coordinates (2), pseudoparticles and
potential functions are therefore extended on the same scale.
The |0i and |1i states of the double minimum potentials are
largely localized (to order e) in each well. The ground state is
representative of I (to order e) and can be thought of as a
vector state |Ii (Fig. 9). State |1i corresponding to the transfer
of a pseudoproton is sketched as |Vi in Fig. 9. All proton sites
are equally occupied. This state is twofold degenerate. Finally,
4
.2. Nonlocal tunnelling states
Dynamical models for stochastic interconversion of uncorre-
1
lated dimers
0,12–14,21,30,31
accord with our intuition based on
the everyday classical world. A proton is either here or there
and a dimer is in form T1 or T2. However, this view is in
conflict with some observations.
With neutron diffraction we probe space and time averaged
probability densities at crystal sites, thanks to the translational
symmetry of the lattice. For a random mixture of tautomers,
the crystal configuration II corresponding to the transfer of P
and P
can be viewed as an extended state |IIi at 2 ꢄ hn01. |Ii,
Vi and |IIi are ‘‘single-pseudoproton’’ states, for which each
pseudoparticle is evenly distributed over a macroscopic num-
g
u
|
2
2
say a T1 + (1 ꢀ a )T2, spatial correlations should be partially
destroyed and a significant amount of the coherent scattering
should collapse into off-Bragg peak diffuse scattering. Well-
studied examples, like ice Ih or C60 and many others, are
presented in ref. 41. A decrease of the total probability density
for protons, compared to that anticipated from the chemical
ꢀ
1
ber of sites (N) and the occupation number per site is (2N)
for P or P
Vi is an intermediate in the interconversion process. The
Ii 2 |Vi and |Vi 2 |IIi transitions are first order, while the
direct transition |Ii 2 |IIi is second-order. A similar inter-
g
u
.
|
|
2
2
4
formula, should be observed. At room temperature, a E 0.5,
the average scattering length for protons should be reduced by
mediate state, reported for KHCO
3
,
was termed ‘‘virtual’’
because its population does not obey Boltzmann’s law. How-
ever, as the |0i 2 |1i transition is effectively observed, one can
say that this state does not exist if it is not measured/observed.
4
1
a factor of E2.
This is at variance with experiments. The sum of probability
densities for protons is equal to unity and temperature in-
8
dependent. Clearly, interconversion does not destroy the
3
Quantum interferences have revealed that KHCO in ther-
mal equilibrium is a superposition of macroscopically en-
tangled states. In principle, there is no objection to apply the
same theoretical framework to the benzoic acid crystal, but
this would be rather formal as long as there is no direct
evidence for quantum interferences. The dynamical model
spatial coherence and there is no evidence for stochastic
disorder. Moreover, the center of symmetry for dimers is
preserved at any temperature. The crystal is therefore a
mixture of two centrosymmetric structures: I composed of
T1, prevailing at low temperatures, and the thermally acti-
vated configuration II composed of T2. (From Bragg diffrac-
tion it is not possible to conclude whether the mixture is a
superposition state.)
1
5
elaborated for benzoic acid in a previous work suggests that
the interconversion rate is consistent with quantum beats
arising from superposition states, but this is a rather circum-
stantial evidence.
With infrared and Raman we probe dynamics at the Bril-
louin-zone center corresponding to in-phase oscillations of the
unit cells throughout the crystal. A straightforward conse-
quence of the uncertainty principle is that only spatially
extended states, on the scale of the incident wavelength, are
4
.3. Isotopic mixtures
The Raman spectrum in Fig. 4 conveys a dramatic informa-
ꢀ1
tion: the bands at 2604 and 2637 cm supposedly due to
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mixed (HD) dimers are virtually at the same frequencies as
those for BA-h in Fig. 2. Similarly, the infrared profile is
almost unchanged. We suppose the bands at 2781 and 2910
6
ꢀ
1
cm for BA-h (see Table 1) should be also unaffected but,
6
presumably, they are too weak to be observed.
Beyond the straightforward evidence for the lack of Davy-
dov splitting, it is tempting to conclude that protons diluted in
the deuteron matrix experience the same double minimum
potential as those in the totally hydrogenated derivative. This
suggests that proton dynamics are still those of nonlocal
centrosymmetric pseudoparticles in a mixture of independent
hydrogenated and deuterated sublattices.
Such nonlocal dynamics accord with the continuous change
of the NMR interconversion rate as a function of the ratio
4
2
r = [D]/([H] + [D]). However, these measurements have
2
1
been questioned by another group. Our results are strong
incentives to pursue these experiments.
Fig. 10 Schematic representation of the adiabatic pairwise transfer,
projected onto the (a, c) plane. In the intermediate zero-phonon state
protons are close to the Oꢂ ꢂ ꢂO centers. For the sake of clarity, H atoms
bound to the benzoic ring are omitted.
4
.4. Adiabatic pairwise transfer
Upon the assumption that broad infrared profiles arise from
double minimum potentials, the sharp infrared-active zero-
phonon bands suggest, by contrast, that protons in the relaxed
excited states experience single minimum potentials. This is
quite possible because these states correspond to a dramatic
shortening of ROꢂ ꢂ ꢂO. Since a symmetric single-minimum is
anticipated for ROꢂ ꢂ ꢂO E 2.45 A, the shortening should be of
E0.15 A. This is in qualitative agreement with the weak
intensities for zero-phonon bands, due to the small overlap
1
1
9
ꢀ1
B10 and B10 s at these temperatures. In addition, the
weak frequency shift of the zero-phonon bands upon deutera-
tion contrasts with the large isotope effect on the observed
transfer rate.
3
3
˚
˚
of the nOꢂ ꢂ ꢂO wave functions. The nOꢂ ꢂ ꢂO transition in the
4
.5. Comparison with theoretical works
upper
potential,
namely
|0gi|2ui|0 i |0 i
OO 0g OO 2u
2
ꢀ
1
Among numerous theoretical investigations of benzoic acid
dimers we shall consider only the most recent state-of-the-art
|
0gi|2ui|0 i |1 i , at E800 cm provides support to a
OO 0g OO 2u
dramatic stiffening of the Oꢂ ꢂ ꢂO bond, a normal consequence
1
9,20
works.
They have in common a semiclassical approach to
of its reduced length. Moreover, the nOH transition
correlated pair dynamics across a subspace of the Born–
Oppenheimer potential energy surface.
|
0gi|2ui|0OO
i
0g|0OO
i
2u 2 |0gi|3ui|0OO
i
0g|0OO 3u at E120
i
ꢀ
1
cm is in line with the very flat nOH potential expected for
short symmetric hydrogen bonds.
1
9
According to Tautermann et al., the optimal tunnelling
path is rather close to the minimum-energy path and quite
different from the straight line between the minima. The
The zero-phonon states |zpi can be therefore regarded as
extended transition states for the interconversion mechanism
depicted in Fig. 10. In contrast to ‘‘vertical’’ transitions
between tunnelling states (Fig. 9), the electronic structure
should follow adiabatically the pairwise proton transfer,
˚
transition state with ROꢂ ꢂ ꢂO = 2.42 A is consistent with a
symmetric hydrogen bond and a flat single minimum potential
ꢀ1
for nOH. The computed barrier of E2500 cm , if we subtract
the zero-point energy of E800 cm , compares quite nicely
with the zero-phonon transitions observed at E1800 cm
The effective tunnelling mass is rather large and the splitting of
ꢀ1
through the shortening of R
. Dimers in the transition
Oꢂ ꢂ ꢂO
ꢀ
1
.
state could be D2h but there is no direct evidence for this
symmetry. Compared to the scheme in Fig. 9, the longer
reaction path mixing r and R coordinates, the much heavier
effective mass due to large displacements of heavy atoms and
the energy difference 2hn01 between the initial and final states
concur to cancel out tunnelling. The through-barrier inter-
conversion can be ignored.
ꢀ
3
ꢀ1
B10 cm for the isolated dimer is insignificant compared
to the potential asymmetry in the crystal.
2
0
Smedarchina et al. have determined extreme tunnelling
trajectories with the approximate instanton method. The
potential barrier was evaluated with the heavy atoms fixed in
the equilibrium configuration, which sounds similar to the
adiabatic separation of proton/deuteron dynamics. After re-
normalization including transverse modes, the effective bar-
The proton transfer rate should be roughly proportional to
the thermal population of the zero-phonon states, to the
frequency of the low frequency mode (over barrier crossing)
and to the statistical distribution of tautomers:
ꢀ
1
riers of 5520 and 4380 cm for HH and DD pairs are quite
close to those given in Table 2. However, the calculated tunnel
ꢀ
ꢀ
ꢁ
1
1
nzp þ 2n01
splitting of E0.06 cm
for HH pairs is two orders of
kI$II ꢆ hnOOexp ꢀh
:
ð4Þ
2
kT
magnitude less than hn0t in Table 2.
We suspect this discrepancy to be a consequence of an
inappropriate dynamical model for the pair treated as a
semiclassical rigid entity. Extrapolating our potential, the
8
ꢀ1
ꢀ2 ꢀ1
at 100 K. This
This gives B10 s at 300 K and B10
process has no significant impact to the measured rates of
s
4
334 | Phys. Chem. Chem. Phys., 2006, 8, 4327–4336
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ꢀ
1
splitting of E0.06 cm suggests an effective mass in the range
3
contrast between broad infrared profiles and sharp Raman
bands.
–4 amu, in line with the large contribution of Oꢂ ꢂ ꢂO coordi-
nates to the tunnelling mode depicted in Fig. 3 of ref. 20. We
speculate that if proton dynamics were treated as quantum
pseudoparticles adiabatically separated from heavy atoms,
then the tunnel splitting should increase by about two orders
of magnitude, and could be consistent with our own estimates
The double-well potentials are temperature independent.
Incoherent tunnelling, stochastic disorder, coupling with pho-
nons, symmetrization of the double minimum, over-barrier
jumping and transition from quantal to classical regime are
not supported by any evidence. Models based upon these
dynamics should be questioned.
(
Table 2). It could be thus possible to establish excellent
contact between experiments and first-principles calculations.
If it were the case, this would show that, quite surprisingly,
nonlocal proton dynamics, purely quantum in nature, are
monitored by the Born–Oppenheimer potential energy surface
calculated with classical protons. In fact, these calculations are
based upon correlated displacements of rigid proton pairs,
which preserve the center of symmetry, although there is no
strong proton–proton interaction. This can be thought of as a
semiclassical approach to nonlocal dynamics, which may
explain the consistency with observed quantum dynamics.
As there is no stochastic disorder arising from the thermally
activated interconversion, and no coupling between protons
(deuterons), the local transfer of either single H/D atoms, or
correlated pairs, can be ruled out. Consequently, we introduce
extended pseudoparticle states preserving the crystal symmetry
and periodicity. This leads to a two-step interconversion
mechanism between structure states |Ii and |IIi through the
intermediate state |Vi. In this latter state, pseudoprotons are
evenly distributed over all sites.
An alternative interconversion route goes through extended
zero-phonon states. The intermediate configuration with very
short Oꢂ ꢂ ꢂO distances corresponds to single-minimum hydro-
gen bonds, possibly symmetrical. The estimated interconver-
sion rate is negligible by orders of magnitude compared to
measurements.
5
. Experimental
Sample preparation as well as infrared and Raman measure-
ments were described in ref. 15. The fully hydrogenated
The excellent agreement between potentials derived from
experiments and those computed with first-principles methods
suggests that the Born–Oppenheimer potential energy surfaces
for an isolated dimer and for the crystal are similar. It is thus
confirmed that interdimer coupling terms are negligible. The
potential asymmetry in the crystal can be tentatively attributed
to a temperature independent static field.
(
BA-h ) and ring deuterated (BA-d h) 98%, Euriso-Top
6 5
S. A.) benzoic acids were commercial products. They were
sublimated. The (BA-h d) derivative was obtained after three
5
exchanges with commercial methanol CH OD (98%). The
3
ratio of the integrated intensities over the nOH and nOD
Raman bands gives E15% of residual OH groups.
The Raman spectra were obtained with a DILOR XY triple
+
monochromator equipped with an Ar laser emitting at 4880
ꢀ
1
˚
A. The spectral resolution was E2 cm . Powdered samples
were sealed in capillary glass tubes and then loaded into a
liquid helium cryostat. The temperature was measured with a
thermometer just above the sample.
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