Molecular Reorientation and Self-Diffusion in Solid Cubane by Deuterium and Proton NMR

Article abstract of DOI:10.1021/jp953432i

Deuterium and proton NMR line shape and relaxation measurements are reported for the sym-cubane-d₂ isotopomer (D_3d symmetry) in the temperature range from 200 K to the melting point of the solid.The results confirm the polymorphic phase sequence: solid II <*> solid I <*> liquid, reported earlier by White et al.The deuterium spectrum below 215 K exhibits a Pake doublet corresponding to an axially symmetric quadrupole coupling tensor with a coupling comstant of Q_c = 178 kHz.Above this temperature the spectrum develops features characteristic of the onset of dynamic processes leading to a single line at about room temperature.Longitudinal relaxation time measurements show a T₁ minimum at 344 K.On transition to solid I there is a discontinuous narrowing of the line and a concomitant increase in T₁.These results are quantitatively interpreted in terms of reorientational jumps of the cubane molecules between their various equivalent orientations.The jump rate k in solid II follows an Arrhenius behavior over the range of almost seven decades with an activation energy E = 62 kJ mol^-1.Transition to solid I results in a discontinuous increase of k.Proton line width measurements in solid II show two motional narrowing steps.The first, at around 240 K, is due to the cubic jumps while the second, at around 375 K, corresponds to molecular self-diffusion between the lattice sites.The activation energy for this process is E_d = 83 kJ mol^-1.On transition to solid I there is also a discontinuous increase in the rate of this process.The proton T₁ values are predominantly affected by the reorientation process and are consistent with the deuteron data.At temperatures above 340 K, where the deuterium NMR spectrum is expected to be a single Lorentzian with a width of less than 400 Hz, it actually exhibits, in both phases II and I, a Pake doublet corresponding to an axially symmetric quadrupole coupling tensor with a very small coupling constant Q'_c = 0.48 kHz.This indicates that the reaorientation process is not perfectly cubic.Possible reasons for this surprising effect are discussed.

Full text of DOI:10.1021/jp953432i

9598
J. Phys. Chem. 1996, 100, 9598-9604
Molecular Reorientation and Self-Diffusion in Solid Cubane by Deuterium and Proton
NMR
A. Detken, H. Zimmermann, and U. Haeberlen*
Max-Planck-Institut fu¨r Medizinische Forschung, Jahnstrasse 29, 69120 Heidelberg, Germany
R. Poupko and Z. Luz
The Weizmann Institute of Science, RehoVot 76100, Israel
ReceiVed: NoVember 22, 1995; In Final Form: March 21, 1996X
Deuterium and proton NMR line shape and relaxation measurements are reported for the sym-cubane-d2
isotopomer (D3d symmetry) in the temperature range from 200 K to the melting point of the solid. The
results confirm the polymorphic phase sequence: solid II 394K8 solid I 405K8 liquid, reported earlier by White
et al. The deuterium spectrum below 215 K exhibits a Pake doublet corresponding to an axially symmetric
quadrupole coupling tensor with a coupling constant of Qc ) 178 kHz. Above this temperature the spectrum
develops features characteristic of the onset of dynamic processes leading to a single line at about room
temperature. Longitudinal relaxation time measurements show a T1 minimum at 344 K. On transition to
solid I there is a discontinuous narrowing of the line and a concomitant increase in T1. These results are
quantitatively interpreted in terms of reorientational jumps of the cubane molecules between their various
equivalent orientations. The jump rate k in solid II follows an Arrhenius behavior over a range of almost
seven decades with an activation energy E ) 62 kJ mol-1. Transition to solid I results in a discontinuous
increase of k. Proton line width measurements in solid II show two motional narrowing steps. The first, at
around 240 K, is due to the cubic jumps while the second, at around 375 K, corresponds to molecular self-
diffusion between the lattice sites. The activation energy for this process is Ed ) 83 kJ mol-1. On transition
to solid I there is also a discontinuous increase in the rate of this process. The proton T1 values are
predominantly affected by the reorientation process and are consistent with the deuteron data. At temperatures
above 340 K, where the deuterium NMR spectrum is expected to be a single Lorentzian with a width of less
than 400 Hz, it actually exhibits, in both phases II and I, a Pake doublet corresponding to an axially symmetric
quadrupole coupling tensor with a very small coupling constant Q′c ) 0.48 kHz. This indicates that the
reorientation process is not perfectly cubic. Possible reasons for this surprising effect are discussed.
Introduction
cubane. A subsequent temperature dependence Raman study
by Dalterio and Owens8 indicated the onset of order-disorder
fluctuations above room temperature and a solid-solid phase
transition (by DSC) at about 10 K below the melting point. They
suggested that the high temperature solid is an orientationally
disordered plastic phase.
More recently, the thermodynamic parameters as well as the
reorientation kinetics in solid cubane were investigated by White
et al.9 using calorimetry and 13C NMR. Their results clearly
show that cubane is polymorphic with the following phase
sequence
Cubane (C8H8) is one of the three molecules with Platonic
structure that can be chemically synthesized.1 It was first
prepared by Eaton and Cole in 19642 and has since been the
subject of extensive studies with regard to its physical and
chemical properties and as an intermediate moiety in organic
synthesis.3,4 Unsubstituted cubane crystallizes at room tem-
perature in the space group R3h with the cell constants a ) 5.340
Å, R ) 72.26° and one molecule per unit cell.5 Projections of
the crystal structure and of a single cubane molecule are shown
in Figure 1. Within the experimental accuracy of the X-ray
measurements,5 the molecules have cubic (Oh) symmetry. In
the crystal they align with one of their main body diagonals
along the crystallographic S6 axis. A detailed analysis of the
vibrational spectrum of (isotopically) normal cubane and of four
of its deuterated isotopomers, including cubane-d1 (with C3v
symmetry), sym-cubane-d2, sym-cubane-d6 (D3d), and cubane-
d8 (Oh) in the solid state and in solution, was carried out by
Della et al. using infrared and Raman spectroscopy.6 It is
noteworthy that although the spectra of the d0 and d8 isoto-
pomers in solution follow the selection rules for Oh symmetry,
in the solid they correspond to S6, reflecting the local symmetry
in the crystal rather than that of an isolated molecule. The
Raman results (for C8H8) were confirmed by Cole et al.,7 who
have also investigated the vapor phase infrared spectrum of
394 K
(5.94)
405 K
(8.7)
solid II
f solid I
f liquid
where the numbers in parentheses correspond to the transition
enthalpy in kJ mol-1. The crystal structure quoted above (R3h)
corresponds to solid II; that of solid I has so far not been
determined. The 13C T1 measurements indicate rapid molecular
reorientations already well below room temperature in solid II.
For this phase, White et al. found an Arrhenius behavior for
the rate k of this process, with an activation energy E ) 58 (
6 kJ mol-1. This result agrees with that obtained by Raman
scattering.8
In the present work we extend the investigation of the
dynamic properties of cubane, using proton and deuterium NMR
of a sym-cubane-d2 sample. There are several reasons for
extending the 13C results. Firstly, since the earlier NMR
X Abstract published in AdVance ACS Abstracts, May 15, 1996.
S0022-3654(95)03432-0 CCC: $12.00 © 1996 American Chemical Society

Molecular Reorientation and Self-Diffusion in Solid Cubane
J. Phys. Chem., Vol. 100, No. 23, 1996 9599
increasing temperature, the second step being clearly related to
self-diffusion.
a
An unexpected feature in the deuterium NMR spectrum was
observed above ∼340 K in both solid phases. In this temper-
ature range, instead of a sharp Lorentzian line, the deuterium
spectrum exhibits a Pake doublet with a very small quadrupole
coupling constant of 0.48 kHz. Very likely, this residual
interaction also exists at lower temperatures, but is masked by
the larger line width. For solid II this behavior can be explained
in terms of an isotope effect on the orientation of the main
diagonal D-D axis relative to the S6 crystallographic axis. If
solid I is cubic, as we suspect, the effect in this phase must
correspond to a remarkable cooperative alignment of the D-D
axes in the otherwise cubic lattice.
b
Experimental Section
Material. Sym-cubane-d2 was prepared from the correspond-
ing sym-cubanedicarboxylic acid dimethyl ester (I) by a series
of steps involving saponification of the ester, bromination, and
debromination by deuterium as described in the scheme below.
The starting compound (I) was obtained from Drs. U. Romano
and C. Neri (EniChem Synthesis, Milano). Five grams of I
were refluxed and stirred with 100 mL of a 10% aqueous NaOH
solution for 1.5 h. After cooling, the mixture was acidified with
concd HCl, and the resulting precipitate, sym-cubanedicarboxylic
acid (II), was filtered and dried under vacuum; yield 3.6 g.
To a refluxing mixture containing 3.06 g of II, 7.8 g of red
mercury oxide, and 40 mL of CH2Br2 was dropwise added a
solution of bromine (6.4 g in 20 mL of CH2Br2) under constant
stirring. The mixture was kept at 80-90 °C for 2.5 h, cooled,
and filtered, and the filtrate was evaporated under vacuum at
40 °C. The residue was extracted five times with hexane and
the solvent evaporated, yielding 2.5 g of crude sym-cubane-Br2
(III) (mp 180-182 °C). After sublimation at 80 °C/10-5 torr,
pure III (mp 187-189 °C) was obtained. (Caution: While
attempting sublimation at a temperature higher than 80 °C, a
violent explosion occurred during the present synthesis, probably
due to the catalytic effect of residual traces of mercury salts in
the crude product. Therefore, it is advisable to keep the
temperature during sublimation at or below 80 °C.)
To obtain sym-cubane-d2 (IV), 1 g of III was mixed with 3
g of tri-n-butyltin deuterate (Aldrich) and a few milligrams of
azoisobutyronitrile (2,2′-azobis-2-methylpropiononitrile). The
mixture was stirred in a closed vial and irradiated with UV light
from a mercury high pressure lamp (water filter) placed at a
distance of 40 cm from the reaction vessel. After 30 min of
irradiation, the vial was heated to 80 °C for a few minutes,
followed by several hours of room temperature UV irradiation
as before. The process was discontinuned only after a cooled
sample from the reaction mixture did not show crystals of III
under polarizing microscopy examination. The reaction vessel
was then connected to a manifold, cooled with liquid nitrogen,
and evaporated. The closed system was warmed to room
Figure 1. (a) Perspective view of the cubane molecule. (b) Rhom-
bohedral unit cell of the solid II crystal.
experiments were limited to solid II, we deemed it interesting
to extend the investigation to solid I. Secondly, relaxation
studies alone, although they provide reliable kinetic parameters
for the dynamic processes, are usually not very sensitive to the
detailed mechanism of the reaction. In the case of cubane, it is
natural to assume (as do White et al.) that the process consists
of reorientational jumps between equivalent molecular orienta-
tions. In the present work we have actually verified this
mechanism by a detailed line shape analysis of deuterium NMR
spectra of the cubane-d2 sample. To extend the measurements
of the rate k of this process to a larger temperature range, we
performed deuteron and proton T1 measurements in both solid
phases.
While both 13C and 2H spectra are highly sensitive to
molecular reorientations, they are very little so to translational
1
self-diffusion. On the other hand, the H NMR spectrum is
also affected by the latter process. In a rigid solid, the width
of this spectrum has contributions, of similar order of magnitude,
from intra- and intermolecular dipolar interactions. Molecular
motions will average out these interactions. However, while
the intramolecular interactions are predominantly affected by
reorientations, the intermolecular ones are mainly sensitive to
self-diffusion, thus providing a means to probe it. Therefore,
we have measured the temperature dependence of the width of
the 1H NMR line in cubane, finding a two-step narrowing with

9600 J. Phys. Chem., Vol. 100, No. 23, 1996
Detken et al.
time between jumps, and that the deuterium quadrupole coupling
tensors in the various sites are related by the corresponding
symmetry operations. The convergence to a single line at high
jump rates does not, by itself, constitute a proof for this
mechanism, since, e.g., random isotropic diffusion, or large angle
jumps, will also lead to a single-line spectrum. However, the
line shapes in the intermediate regime (230 K to 250 K), in
particular the appearance of the singularities at the center of
the spectra, provide strong evidence for the former mechanism.11
The isotropic reorientation mechanism can, of course, also be
ruled out on the basis of the X-ray data, since it would lead to
complete orientational disorder, contrary to the crystallographic
results for solid II. We shall see further below that the
reorientational jumps in the sym-cubane-d2 sample deviate
slightly from perfect cubic symmetry, but for now we shall
disregard this small effect.
To simulate the dynamic deuterium NMR spectra according
to the cubic jump model, we applied known procedures in which
the effect of relaxation and exchange during the quadrupole echo
sequence are included.12 Because of the inversion symmetry
of the cube, the eight-site problem reduces to a four-site one. It
is thus sufficient to consider only four deuterons at the corners
of an enclosed tetrahedron. To set up the dynamic matrix it
may appear on first glance that it would depend on whether the
jumps involve rotation about the C4, C2, or C3 axes of the cubane
molecule. If, however, one assumes that the reorientation
probability about the three C4 axes is the same and likewise
the reorientation probability about the six C2 or the four C3 axes,
the various mechanisms lead to equivalent dynamic matrices,
differing only by scalar coefficients. Analysis of the possible
mechanisms yields the following dynamic matrix
Figure 2. Left: Temperature dependence of experimental quadrupole
echo spectra of the sym-cubane-d2 sample. The 90° pulse width was
2.2 µs, the time interval between the pulses in the quadrupolar echo
sequence 20 µs. Right: Simulated lineshape calculated for Qc ) 178
kHz, 1/T2 ) 400 s-1, and the indicated jump rate constants.
temperature and the solid content sublimed into another flask
which was kept at -196 °C. The sublimation procedure was
repeated several times until white crystallites of sym-cubane-d2
were obtained. Yield 0.33 g; m/e ) 105, 104 [M - H(D)]x
base peak; single proton NMR line at δ ) 4.0 ppm. Differential
scanning calorimetry (DSC) measurements on the cubane-d2
sample gave, within experimental errors, the same transition
temperatures and transition enthalpies as determined by White
et al. by the same method.9 This rules out any significant
deuterium isotope effect on the thermodynamic parameters of
cubane, as found, e.g., by Wasylishen et al. in the case of
norbornane.10
NMR Measurements. Deuterium NMR spectra were re-
corded on a Bruker CXP300 spectrometer at a frequency of
46.07 MHz at the Weizmann Institute and on a home-built 72.1
MHz spectrometer at the Max-Planck-Institute. Both quadru-
pole echo and single pulse FID signals were recorded, followed
by Fourier transformation. The proton measurements were
performed on the CXP300 spectrometer at 300 MHz by single
pulse excitation.
1
1
1
1
1
1
-1
/
/
/
/
/
/
3
3
3
3
3
1
-1
1
/
3
3
R ) k
(1)
1
-1
1
/
1
/
3
3
(
)
1
-1
/
/
/
3
3
3
where k is 3/(4τ3), 1/τ4, 1/(2τ2), or 1/τ′2, depending on whether
the reorientations correspond to C13, C41, C21, or C42 rotations, and
τn is the corresponding correlation time (mean lifetime between
jumps).
With this dynamic matrix, we simulated spectra matching
the experimental ones as closely as possible, taking k and an
exchange independent transverse relaxation rate, 1/T2, as the
only adjustable parameters. Examples of such simulated spectra
are shown in the right column of Figure 2. It may be seen that
they faithfully reproduce the experimental results, including in
particular the singularity at the center frequency. This singular-
ity is typical of cubic jumps11 and corresponds to those
molecules in the powder which lie parallel to the external
magnetic field with one of their C4 axes. For such molecules
the deuterium spectrum of all sites is a singlet falling at the
center of the spectrum, since the unique principal quadrupole
coupling tensor directions (the C-D bond directions) of all
deuterons are inclined by the magic angle, (θ ) (cos-1
Results and Discussion
Deuterium Dynamic NMR Lineshape Measurements.
Examples of deuterium NMR spectra obtained from the cubane-
d2 sample in the temperature range 220 K to 260 K are shown
in Figure 2. Starting from a rigid powder spectrum at 220 K,
the spectra develop typical dynamical features when the
temperature is increased, which gradually evolve into a single
sharp peak above ∼250 K. The low temperature spectrum
corresponds to an axially symmetric quadrupole coupling tensor
with a quadrupole coupling constant Qc ) e2qQ/h ) 178 kHz.
The evolution of the spectrum above 230 K, in particular the
convergence to a single line at higher temperatures, is consistent
with a cubic jump model. By a “cubic jump model” we mean
that, in the course of time, the molecules occupy all symmetry-
related orientations of the cube, with equal average residence
x
(1/ 3), to the external field. Cubic jumps therefore do not
modulate the resonance frequency, hence the signal remains
sharp. By fitting the calculated spectra to the experimental
results visually, the rate constants k for the jump process were
derived. The corresponding correlation times, τ ) k-1, are
plotted in Figure 3 with the symbol 0 vs the reciprocal absolute
temperature.
Deuterium Longitudinal Relaxation Time Measurements.
To extend the temperature range of the kinetic results presented

Molecular Reorientation and Self-Diffusion in Solid Cubane
J. Phys. Chem., Vol. 100, No. 23, 1996 9601
) 2.17 ms (at ω0/2π ) 46.07 MHz) a quadrupole coupling
constant of Qc ) 177 kHz can be calculated, in excellent
agreement with the value derived from the low temperature
spectra. The correlation times derived from the T1 results in
solid II are also plotted in Figure 3 (symbols ) for 46.07 MHz
and × for 72.1 MHz). They agree perfectly well with those
obtained from the line shape analysis, yielding an overall
Arrhenius equation for the reorientation rate
k ) k0 exp(-E/RT),
(3)
where E ) 62 kJ mol-1 and k0 ) 1.1 × 1018 s-1, in reasonable
agreement with the values obtained earlier by White et al. from
13C data.9 Note that the value of k0 is too large by orders of
magnitude for identifying k0 with an “attempt frequency” of
any kind.
Figure 3. Arrhenius plot of the correlation times for the cubane
reorientation, τ, and self-diffusion, τd, in solid II, obtained from different
experiments. The symbols indicate results from: 0 deuterium lineshape
analysis; × deuterium T1 at 72.1 MHz; ) deuterium T1 at 46.07 MHz;
+ proton T1 at 300 MHz, 4 proton linewidth.
As indicated before, the experimental results do not provide
information on whether the cubane molecules reorient by two-,
three-, or four-fold jumps. Theoretical calculations of the
barriers for the various single-particle reorientations of cubane
in solid II were carried out by Fyfe and Harold-Smith,15 who
found the lowest value for the C4 jumps. If this mechanism is
indeed the one which nature prefers, τ in eq 2 should be
identified with τ4 of the previous section. The discrepancy
between the calculated barrier height, 35 kJ mol-1, and the
experimental value has been discussed by White et al.9
The temperature range of solid II is too narrow to derive the
corresponding kinetic parameters for this phase from T1 data.
However, from the discontinuous increase in T1 at the transition
to solid I an increase by a factor of 10 in the reorientation rate
may be estimated, with k ≈ 8 × 1010 s-1 at the midpoint of
this phase. Transition to the liquid phase yields another
discontinuous increase in k to about 4 × 1011 s-1 at the melting
temperature.
Proton Line Width and T1 Measurements. The deuterium
results described in the previous two sections provide informa-
tion on the reorientation rate of the cubane molecules in both
solid phases. Deuterium NMR is, however, usually not suitable
for measurements of translational self-diffusion, since its
spectrum is almost unaffected by intermolecular interactions.
The situation is quite different for proton NMR, where the
dominant anisotropic term in the spin Hamiltonian is the dipolar
coupling between all protons in the system, including intra- and
intermolecular interactions with comparable contributions to the
static line width (or second moment). Pure molecular reorienta-
tions will average out the intramolecular dipolar interactions,
while self-diffusion will in addition also average out the
intermolecular ones.
To check whether self-diffusion can be detected in cubane,
we have measured the proton NMR line width δν (full width
at half maximum height) in the cubane-d2 sample over the
temperature range from 190 K to the melting point. The results,
which clearly show the effects of two dynamic processes, are
summarized in Figure 5. Below 230 K the system is rigid and
the observed line width (47 kHz) reflects the full intra- and
intermolecular dipolar interactions between the protons in the
sample. Above 230 K the molecular reorientation rate becomes
comparable with the intramolecular dipolar interactions resulting
in the latter being averaged out and the line width being
concomitantly reduced, until a new plateau at 11 kHz is reached.
This line width reflects solely the intermolecular dipolar
interactions in a lattice consisting of rapidly reorienting cubane-
d2 molecules. Above 350 K a second averaging process sets
in, which results in a gradual reduction in the line width, down
to 1 kHz just below the solid II f solid I transition temperature.
This process evidently averages out the intermolecular dipolar
Figure 4. Plots of the deuterium longitudinal relaxation time of sym-
cubane-d2 measured at 46.07 MHz (]) and 72.1 MHz (×) vs the
reciprocal absolute temperature.
above, measurements of the longitudinal relaxation time T1 were
carried out between 225 K and the melting point of the cubane-
d2 sample, at both 72.1 MHz (between 225 K and 308 K, using
saturation recovery) and at 46.07 MHz (between 300 K and
the melting point, using inversion recovery). Within the
experimental accuracy the build-up of the magnetization was
exponential at all temperatures. The derived relaxation times
are plotted vs the reciprocal absolute temperature in Figure 4.
The data for 46.07 MHz exhibit a T1 minimum at 344 K in
solid II and clear discontinuities at the temperatures correspond-
ing to the transitions solid II f solid I and solid I f liquid.
Within the solid II range, the T1 results can quantitatively be
interpreted in terms of the BPP equation13
2
1
2 3
τ
4τ
)
πQc
+
(2)
2
2
2 2
T1 15 2
(
)
[
]
1 + ω0τ
1 + 4ω0τ
Although this equation applies strictly to random isotropic
reorientations, it is almost indistinguishable from that expected
for cubic jumps.14 Longitudinal relaxation times at 72.1 MHz
were only measured at low temperatures where ω0τ . 1, so
that no minimum was observed. Where measurements at both
frequencies are available at the same temperatures, the results
are within the expected ratio of T1(72.1)/T1(46.07) ) (72.1/
46.07)2 ) 2.45. From eq 2, the T1 minimum is expected at
ω0τ ) 0.6158, so that with the observed value of T1 (minimum)

9602 J. Phys. Chem., Vol. 100, No. 23, 1996
Detken et al.
(6)
r2 1
6τd
D )
and identifying the jump distance r with the lattice constant a
yields
D ) D0 exp(-Ed/RT)
(7)
where D0 ) 1.9 × 10-3 m2 s-1. This result is well within the
range of values found in other orientationally disordered
compounds of cagelike molecules, similar in size to cubane.16
Again, we do not have sufficient data in the solid I phase for a
similar analysis to be made. However, from the observed line
width an estimated value of D ) 8 × 10-14 m2 s-1 in the
midpoint of this phase may be calculated.
For completion, we have also measured the T1 of the protons
in the cubane-d2 sample. The results, shown in Figure 6, exhibit
a very similar behavior to that shown in Figure 4 for deuterium.
There is a clear T1 minimum and two discontinuities at the solid
II f solid I and solid I f liquid transitions. The proton T1
results are, however, more difficult to interpret quantitatively
because they are affected by many dipolar interactions of similar
magnitude. We may, nevertheless, attempt to fit the data with
the BPP type equation
Figure 5. Proton line width (full width at half maximum height) of
the sym-cubane-d2 sample vs the absolute temperature, recorded at 300
MHz. The insert shows details in the temperature range of the solid II
f solid I and solid I f liquid transitions. The symbols are
experimental points, while the full line is calculated as explained in
the text.
interactions and must therefore correspond to self-diffusion of
the cubane molecules between the lattice sites. At the solid II
f solid I transition a discontinuous reduction of the line width
occurs down to 300 Hz, indicating that the self-diffusion is
considerably faster in solid I. Another small discontinuous
reduction in the line width to less than 100 Hz occurs on melting
to the liquid phase. In this phase the line width is dominated
by sample and field inhomogeneity effects.
1
T1
τ
4τ
) K2
+
(8)
2
2
2 2
[
]
1 + ω0τ
1 + 4ω0τ
where K should be of the order of the dipolar field, expressed
in angular frequency units, experienced by a proton due to its
nearest neighboring protons. Its (experimental) value, as
calculated from the T1 minimum (0.54 s), is K ) 4.95 × 104
s-1. This may be compared with the intramolecular contribution
to the rigid-lattice second moment of the proton resonance line
(powder average)13
An exact quantitative analysis of the proton line width is not
possible, but a good estimate of the correlation times involved
in the two processes can be obtained by analyzing the results
in terms of the equation13
2
2
δν2 ) δν2 tan-1(2πR1δντ) + δν2 tan-1(2πR2δντd) (4)
1π 2π
3
1
∆ω2 ) γ4 p2I(I + 1)
(9)
∑
H
r6jk
5
Here we assume that the square of the observed line width
consists of two contributions. The first, δν21, corresponds to
the intramolecular dipolar interactions, which are averaged out
by the cubic jumps, characterized by the correlation time τ. The
second, δν22, corresponds to the intermolecular interactions
between the rapidly reorienting molecules in their lattice sites.
This part is averaged out by the self-diffusion process, charac-
terized by the correlation time τd. The first (low temperature)
plateau in Figure 5 thus corresponds to (δν21 + δν22)1/2, and the
second to δν2. The parameters R1 and R2 are numerical factors
of order unity which are, following Abragam,13 rather ill-defined.
k
where j indicates an arbitrary proton in the cubane molecule
and the summation is over all other protons in the same
molecule. The calculated value of ∆ω2 1/2, which is 3.83 ×
104 s-1, perfectly supports our interpretation of K. From the
measured temperature dependence of T1 and the experimental
result for K, the τ values in eq 8 were determined. They are
plotted in Figure 3 with the symbol +. In general they are
consistent with those obtained from the deuterium spectra,
although at around room temperature and below they deviate
somewhat, apparently due to the presence of another relaxation
process.
The Deuterium NMR Spectrum in Solid I and in the High
Temperature Region of Solid II. Up to now, the discussion
of the deuterium spectra of cubane-d2 has implied that in the
fast exchange regime the spectrum can be described by a single
Lorentzian line. The width of this line should gradually narrow
with increasing temperature. This behavior is in fact observed
up to about 340 K, where the line width is around 0.4 kHz.
However, above this temperature the spectrum exhibits features
which gradually evolve into a Pake doublet corresponding to a
very small, axially symmetric quadrupole coupling tensor with
Q′c ) 0.48 kHz. The structure persists on transition to solid I,
but disappears on melting. Examples of spectra in both phases
are shown in Figure 7. The spectra are not always reproducible
and continuously change with time during the measurements.
To analyze the results of Figure 5 in terms of eq 4, we first
fitted the low temperature data with the rate equation for the
reorientation process, eq 3, obtaining R1 ) 0.13 (which
illustrates Abragam’s qualification “ill-defined”), δν21 ) 21 ×
108 s-2 and δν22 ) 1.2 × 108 s-2. The high temperature data
were then used to calculate τd (assuming R2 ) 1). The results
are plotted in Figure 3 with the symbol 4 and yield the
Arrhenius equation
τd ) τ0d exp(Ed/RT)
(5)
with Ed ) 83 kJ mol-1 and τ0d ) 2.5 × 10-17 s. The value of
τ0 is at best a rough estimate since the uncertainty of R2 fully
d
translates into the uncertainty of τ0d. Applying Einstein’s
relation for the self-diffusion coefficient

Molecular Reorientation and Self-Diffusion in Solid Cubane
J. Phys. Chem., Vol. 100, No. 23, 1996 9603
Figure 6. Proton T1 results of the sym-cubane-d2 sample at 300 MHz
vs the reciprocal absolute temperature.
They are sometimes not symmetric and often exhibit sharp
features. We believe that this behavior arises from continuous
sublimation of the cubane within the sample tube, even during
the measurements, and the formation of a coarse-grained texture.
There can, however, be no doubt that the deuterons experience
a finite residual quadrupole coupling interaction (less than 0.3%
of the static value) and that, therefore, the averaging process
due to reorientations cannot be strictly cubic. To check whether
this interaction could already exist at lower temperatures, we
have simulated Pake doublets with varying ratios of the line
width to Q′c. These simulations convinced us that such a Pake
doublet would not be detected in the experimental spectra
recorded below 340 K. We therefore conclude that the small
deviation from perfect cubic jumps most likely exists over the
entire dynamic range, but it becomes observable only at higher
temperatures when the homogeneous lines become sufficiently
narrow.
The small deviation from cubic symmetry has probably a
negligible effect on the quantitative kinetic analysis given in
the previous sections; however, its origin is intriguing. In the
solid II phase with space group R3h it could arise from an isotope
effect on the orientation of the sym-cubane-d2 D-D axis, or
from differences in the static Qc values for different orientations
of the D-D axis in the crystal. In this phase there is a unique
axis with S6 symmetry, and one of the main diagonals of the
cubane molecule is aligned along this direction. If the prob-
ability of the D-D axis to align along S6 is not exactly the
same as for the other (H-H) main diagonals, the statistical
average of the deuterium quadrupole coupling interaction over
all the cubic orientations will not be zero. Likewise, if the value
of Qc, when the D-D axis is along S6, is not identical to that
when the molecule aligns along the other directions, the average
quadrupole coupling interaction will not vanish exactly either.
We cannot tell from the experimental results which of the two
effects is responsible for the deviation from exact cubic
symmetry, nor whether both are present. The effect itself,
however, is not inconsistent with the symmetry of solid II and
the sym-cubane-d2 molecule. It is nevertheless a remarkable
effect and we are not aware of any similar case in other
molecular crystals.
Figure 7. Deuterium NMR spectra in solid I and in the high
temperature region of solid II. The central line in the 406 K and 404
K spectra arises from the liquid (L) coexisting with solid I at these
temperatures. Note the highly expanded scale compared to the spectra
in Figure 2.
solid I is cubic. In fact the rhomboedric unit cell of solid II
with R ) 72.26° is already very close to cubic (for R ) 70.53°
it would be cubic). If solid I is indeed cubic, the explanation
offered for Q′c * 0 in solid II cannot apply to this phase: In a
cubic structure there is no unique direction, and an isotope effect
on the alignment of the D-D axis cannot be invoked.
A possible explanation for Q′c * 0 in solid I could involve
a cooperative alignment of the D-D axes. If the orientation
of this axis in one molecule depends on the orientation of the
D-D axis in a neighboring one, causing both to preferentially
align in parallel, domains will be formed with spontaneous
polarization of the D-D axes in a lattice that is otherwise cubic.
If, in addition, the lifetime of these (hypothetical) domains is
sufficiently long, a residual quadrupole interaction, as we have
observed it, will survive. We are currently attempting to
determine the structure of solid I by powder X-ray diffraction.
Hopefully the method will prove sensitive enough to distinguish
between the R3h structure of solid II and that of solid I. On a
more optimistic level, if the latter turns out to be cubic we may
even hope to find structural differences between normal cubane
and sym-cubane-d2.
Acknowledgment. This work was supported by a grant from
G.I.F., the German-Israeli Foundation for Scientific Research
and Development. One of us (Z.L.) thanks the Humboldt
Foundation for a Humboldt Research Award and the Max-
Planck-Institute for Medical Research (Heidelberg) for hospital-
ity.
The situation is more complicated for solid I. Although the
structure of this phase has not yet been determined, it most
probably belongs to one of the cubic space groups. This
conjecture is based on the assumption that the endothermic
transition of solid II to solid I is associated with a structural
change and that, as is usual, this change is to a more symmetric
phase. Since R3h has only cubic supergroups we anticipate that
References and Notes
(1) The other two are tetrahedrane and dodecahedrane. The analogues
of the icosahedron and octahedron are not accessible to synthesis, at least
not from a carbon skeleton.

9604 J. Phys. Chem., Vol. 100, No. 23, 1996
Detken et al.
(2) Eaton, P E.; Cole, T. W., Jr. J. Am. Chem. Soc. 1964, 86, 962;
3157.
(10) Wasylishen, R. E.; Peiris, S.; Arnold, D. R. Chem. Phys. Lett. 1985,
114, 31.
(3) Griffin, G. W.; Marchand, A. P. Chem. ReV. 1989, 89, 997.
(4) Eaton, P. E. Angew. Chem., Int. Ed. Engl. 1992, 31, 1421.
(5) Fleischer, E. B. J. Am. Chem. Soc. 1964, 86, 3889.
(6) Della, E. W.; Mc Coy, E. F.; Patney, H. K.; Jones, G. L.; Miller,
F. A. J. Am. Chem. Soc. 1979, 101, 7441.
(7) Cole, T. W., Jr.; Perkin, J.; Putman, S.; Pake, P. W.; Strauss, H. L.
J. Phys. Chem. 1991, 85, 2195.
(8) Dalterio, R. A.; Owens, J. F. Solid State Commun. 1988, 67, 673.
(9) White, M. A.; Wasylishen, R. E.; Eaton, P. E.; Xiong, Y.; Pramod,
K.; Nodari, N. J. Phys. Chem. 1992, 96, 421.
(11) Alexander, S.; Baram, A.; Luz, Z. Mol. Phys. 1974, 27, 441.
(12) Vega, J. A.; Luz, Z. J. Chem. Phys. 1987, 86, 1803.
(13) Abragam, A. The Principles of Nuclear Magnetism; Clarendon
Press: Oxford, 1961.
(14) Amoureux, J. P.; Bee, M.; Virlet, J. Mol. Phys. 1980, 41, 313.
(15) Fyfe, C. A.; Harold-Smith, D. J. Chem. Soc., Faraday Trans. 2
1975, 71, 967.
(16) Chezeau, J. M.; Strange, J. H. Phys. Rep. 1979, 53, 2.
JP953432I