Heavy-atom tunneling in the ring opening of a strained cyclopropene at very low temperatures

Article abstract of DOI:10.1002/chem.201303792

The highly strained 1H-bicyclo[3.1.0]-hexa-3,5-dien-2-one 1 is metastable, and rearranges to 4-oxacyclohexa-2,5-dienylidene 2 in inert gas matrices (neon, argon, krypton, xenon, and nitrogen) at temperatures as low as 3 K. The kinetics for this rearrangement show pronounced matrix effects, but in a given matrix, the reaction rate is independent of temperature between 3 and 20 K. This temperature independence means that the activation energy is zero in this temperature range, indicating that the reaction proceeds through quantum mechanical tunneling from the lowest vibrational level of the reactant. At temperatures above 20 K, the rate increases, resulting in curved Arrhenius plots that are also indicative of thermally activated tunneling. These experimental findings are supported by calculations performed at the CASSCF and CASPT2 levels by using the small-curvature tunneling (SCT) approximation. Quantum mechanical tunneling: Despite an estimated activation barrier of more than 6 kcal mol ^-1, the strained cyclopropene 1 rearranges at temperatures as low as 3 K to the carbene 2 in its triplet ground state (see figure). Experiments and theory provide clear evidence that the rearrangement proceeds through heavy-atom tunneling.

Full text of DOI:10.1002/chem.201303792

DOI: 10.1002/chem.201303792
Full Paper
&
Rearrangement Reactions
Heavy-Atom Tunneling in the Ring Opening of a Strained
Cyclopropene at Very Low Temperatures
Melanie Ertelt,[a] David A. Hrovat,[b] Weston Thatcher Borden,[b] and Wolfram Sander*[a]
Abstract: The highly strained 1H-bicyclo[3.1.0]-hexa-3,5-
dien-2-one 1 is metastable, and rearranges to 4-oxacyclo-
hexa-2,5-dienylidene 2 in inert gas matrices (neon, argon,
krypton, xenon, and nitrogen) at temperatures as low as 3 K.
The kinetics for this rearrangement show pronounced matrix
effects, but in a given matrix, the reaction rate is independ-
ent of temperature between 3 and 20 K. This temperature
independence means that the activation energy is zero in
this temperature range, indicating that the reaction pro-
ceeds through quantum mechanical tunneling from the
lowest vibrational level of the reactant. At temperatures
above 20 K, the rate increases, resulting in curved Arrhenius
plots that are also indicative of thermally activated tunnel-
ing. These experimental findings are supported by calcula-
tions performed at the CASSCF and CASPT2 levels by using
the small-curvature tunneling (SCT) approximation.
Introduction
for tunneling involving heavier atoms or groups of atoms is
still comparatively rare.[15–22] The bond shift in antiaromatic an-
nulenes such as cyclobutadiene is one example for which cal-
culations indicate that tunneling makes a considerable contri-
bution to the rate constants even at higher temperatures.[23–25]
Kinetic studies in low-temperature matrices reveal that the ring
closure of some cyclobutanediyls is rather insensitive to tem-
perature, which suggests a tunneling reaction mechanism.[16,26]
Another example of a reaction in which heavy-atom tunnel-
ing is implicated was published by us more than 20 years
ago.[27] This is the rearrangement of the highly strained 1H-
bicyclo[3.1.0]-hexa-3,5-dien-2-one 1 to 4-oxacyclohexa-2,5-dien-
ylidene 2. Cyclopropene 1 is formed through visible light pho-
tolysis at 10 K of the triplet carbene T-2, which is obtained
easily through the photolysis of the matrix-isolated quinone di-
azide 3 (Scheme 1). The kinetics of the rearrangement were de-
Quantum mechanical tunneling (QMT) is a fundamental phe-
nomenon that is present in all chemical reactions.[1,2] It influen-
ces the kinetics of chemical reactions under conditions in
which the QMT rates can compete with the rates of passage
over, rather than through, the reaction barrier. Tunneling can
become the dominant reaction mode at temperatures low
enough that there is insufficient thermal energy to make pas-
sage over the barrier rapid. At very low temperatures, at which
QMT occurs from the lowest vibrational level of the reactants,
the reaction rate becomes independent of temperature, and if
the probability of tunneling is small, the reaction rate has
a very small Arrhenius pre-exponential factor. The tunneling
probabilities depend strongly on the height and width of the
barriers, so QMT is usually rapid only in reactions with small
and narrow activation barriers. Matrix isolation spectroscopy is
ideally suited for the investigation of tunneling reactions. Tem-
peratures as low as 3 K can be obtained easily, precluding any
thermally activated reactions, and kinetic studies can be run
for long times of up to several weeks to monitor very slow re-
actions at extremely low temperatures.
In hydrogen-transfer reactions and CH insertions, QMT has
been observed frequently,[3–14] whereas experimental evidence
[a] M. Ertelt, Prof. Dr. W. Sander
Lehrstuhl fꢀr Organische Chemie II
Ruhr-Universitꢁt Bochum, 44781 Bochum (Germany)
E-mail: wolfram.sander@rub.de
Scheme 1. Tunneling rearrangements of the strained cyclopropenes 1 and 4.
[b] Dr. D. A. Hrovat, Prof. Dr. W. T. Borden
Department of Chemistry and the
Center for Advanced, Scientific Computing and Modeling
University of North Texas
1155 Union Circle, #305070, Denton, Texas 76203-5070 (USA)
termined between 10 and 40 K in argon and several other ma-
trices, and the half-life of 1 at 10 K was found to be of the
order of seven days. Although the temperature range that
could be used for kinetic measurements was limited, a curved
Supporting information for this article is available on the WWW under
http://dx.doi.org/10.1002/chem.201303792.
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Arrhenius plot with a zero activation barrier at temperatures
below 20 K was observed, as expected for a tunneling reaction.
However, it should be noted that curved Arrhenius plots do
not necessarily result from tunneling.[28,29]
strong indication that the rearrangement proceeds through
QMT at these low temperatures. Rate constants that are inde-
pendent of temperature over an absolute temperature range
of more than 600% imply that the Arrhenius activation energy
in this temperature range is essentially zero, as expected for
a tunneling reaction that occurs from the lowest vibrational
level of the reactant. At temperatures above 25 K, the rates in-
crease rapidly, indicating that the tunneling reaction becomes
thermally activated, and between 15 and 25 K, both the ther-
mally activated and nonactivated tunneling reactions contrib-
ute to the overall rate. This observation is in excellent agree-
ment with the conclusion drawn in our first publication, al-
though at that time, we could only measure the rates above
10 K.[27]
The ring opening of 1 to 2 is also a remarkable reaction, be-
cause 1 has a singlet ground state, whereas 2 is a triplet car-
bene. Thus, the 1!2 rearrangement requires an intersystem
crossing (ISC) step. However, the position along the reaction
coordinate connecting 1 to T-2 at which ISC occurs could not
be determined by these experiments.
The ring expansion of 1-methylcyclobutylfluorocarbene also
shows unusual kinetic behavior.[17] The reaction rate increases
only slightly between 8 and 25 K. The calculated activation bar-
rier of 6.4 kcalmolꢀ1 would not allow any thermal reaction in
this temperature range, but the rate constant calculated for
tunneling through this barrier was in excellent agreement with
that measured at 8 K.
Comparison of the kinetic data obtained at 3 K in different
matrices reveals surprisingly strong matrix effects for the 1!2
rearrangement. In argon and neon, the rates are very similar,
whereas in nitrogen, krypton, and xenon, the decay is marked-
ly accelerated (Figure 1). Thus, different matrix hosts (xenon,
krypton, argon, and nitrogen) result in different rates for the
rearrangement of 1 to 2 under otherwise similar conditions. It
appears that the matrix has a strong influence on the barrier
width and height, leading to an acceleration of the tunneling
rates in xenon compared to those in argon. In xenon at 3 K,
the half-life of 1 is only about 5 h (compared with seven days
in argon), which results in a rapid decrease in the concentra-
tion of 1 with time in a Xe matrix.
Recently, purely computational evidence was presented that
the kinetics of the Bergman cyclization of a 10-membered ene-
diyne is also influenced substantially by QMT, even at 378C.[30]
In addition, experimental evidence indicates that the ring ex-
pansion of benzazirine 4 to cyclic ketenimine 5 at low temper-
atures also proceeds through QMT (Scheme 1).[22] Similarly to
1, benzazirine 4 is a highly strained anti-Bredt cyclopropene
that undergoes ring expansion readily. Thus, rearrangements
of this type of compound may generally prove to be fruitful re-
actions for investigation in the search for molecules that react
at low temperatures through heavy-atom tunneling.
With the advent of experimental and theoretical methods
that were not available 20 years ago, when we initially discov-
ered the rearrangement of 1 to 2, we decided to use these ad-
vanced methods to reinvestigate this reaction. In particular, we
are now able to perform experiments at temperatures down to
3 K, which, compared to 10 K, is more than a factor of three
lower in kT. In addition, CASSCF and CASPT2 calculations can
now be performed rapidly enough to generate potential
energy surfaces, which can be used to compute tunneling rate
constants for this reaction.
Results and Discussion
Matrix effects at 3 K
Figure 1. Rate constants k for the 1!2 rearrangement measured in various
The kinetics of the 1!2 rearrangement were measured in
solid neon, argon, krypton, xenon, and nitrogen at tempera-
tures from 3 K to an upper temperature limited by the vapor
pressure of the matrix (neon: 8 K; nitrogen: 35 K; argon: 40 K;
krypton: 50 K; xenon: 65 K). The rearrangement is enhanced
by the IR radiation from the light source of the IR spectrome-
ter, so an IR band-pass filter blocking all IR light above
2000 cmꢀ1 must be used to obtain meaningful kinetic data.
The decay and increase in intensity, respectively, of the
strongest IR absorptions of 1 and 2 were monitored by IR
spectroscopy. In all the matrices, the kinetics observed at 3 K
matched those at 20 K very closely (neon matrices are usable
only up to 8 K owing to the high vapor pressure of neon at
higher temperatures). This temperature independence is a very
matrices at 3 K as a function of the measurement time.
The kinetic behavior of the 1!2 rearrangement is compli-
cated, because the rate constant k measured in a given matrix
at 3 K is not constant, but decreases over time (Figure 1). This
is observed frequently in solid-state reactions resulting in dis-
persive kinetics. Some molecules of 1 are formed in “fast”
matrix sites and rearrange much more quickly than others that
occupy “slow” sites. Thus, instead of the clean first-order kinet-
ics expected for unimolecular reactions in the gas phase, the
observed rates depend not only on the temperature, but also
on the time the experiments have been running, because
“fast” sites react first and “slow” sites later.
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The kinetics measurements were started directly after the
formation of 1 through the irradiation of carbene 2, and the
decay of 1 and reformation of 2 were followed for several
hours to several days (Figure 1). In a long-duration experiment
in argon, the rate dropped from 4ꢁ10ꢀ6 sꢀ1 after 5 h to 1ꢁ
10ꢀ6 sꢀ1 after two days, but remained constant after that time
(the kinetics were followed for ten days at 3 K, Figure 1). The
largest rate was observed in xenon, with k>2ꢁ10ꢀ5 at the be-
ginning of the measurements, but this rate also showed the
fastest decline over time. Owing to the rapidly decreasing con-
centration of 1 in xenon, the kinetics could only be followed
for about four days.
the empirical exponent b to match the experimental data
(stretched exponent method), as shown in Equation (1).
b
ð1Þ
½Xꢁt ¼ ½Xꢁ0 ꢂ expðꢀðktÞ Þ
Without environmental effects, b=1 should be observed. In
our experiments, we find that b lies in a range between 0.5
and 1, depending on the matrix, the temperature, and the
time that we start measuring the kinetics after the generation
of 1. We used Equation (1) to fit the kinetic data of the 1!2
rearrangement and to extract meaningful rate constants from
the complete set of data for a given temperature and matrix.
A comparison of the changes in the rates measured in vari-
ous matrices suggests that the rate constants converge to
values close to 10ꢀ6 sꢀ1, although there still seem to be signifi-
cant differences between the different matrices (Figure 1). This
kinetic behavior can be reproduced by using the same matrix
several times. After the decay of 1 and formation of 2, cyclo-
propene 1 is recovered upon subsequent irradiation of the
matrix with visible light (>515 nm), and its decay can be moni-
tored again.
Using these data, we obtain curved Arrhenius plots in argon,
xenon, and nitrogen, with the onset of temperature depend-
ence in the rate constants occurring at around 20 K. A problem
with the measurements in argon and nitrogen is that at tem-
peratures above 30 and 25 K, respectively, these matrices
become soft and start to sublime off the cold window. There-
fore, the data measured at these higher temperatures are less
reliable because of the deterioration of the matrix. In neon, the
matrix evaporates rapidly even at 8 K. Xenon matrices, on the
other hand, persist to much higher temperatures (up to 60 K).
In xenon, the transition between thermal activation and
temperature-independent tunneling from the lowest vibration-
al level of the reactant is observed at a similar temperature to
that in argon. This observation indicates that the curved Arrhe-
nius plot comes not from a matrix effect, but rather, results
from the intrinsic properties of the matrix-isolated molecules.
The Arrhenius plots in Figure 2 show clearly that the fastest
rate for the rearrangement occurs in xenon, with a slower rate
in nitrogen, and the slowest rates in argon and neon.
Nonexponential decays have been observed for many unim-
olecular reactions that are influenced by their microenviron-
ment. In particular, reactions in solid matrices[31] and also differ-
ent processes such as protein folding[32] or DNA duplex dissoci-
ation[33] often show a decrease in rate constants over time, and
thus, nonexponential decay. However, most of these reactions
proceed thermally, rather than through QMT as in the 1!2 re-
arrangement.
An important question concerns how the environment influ-
ences QMT.[31,34–36] In the 1!2 rearrangement, the slowest tun-
neling rates and the lowest overall drops in the rate constants
are observed for the least interacting rare gases, that is, neon
and argon. The fastest rates and largest drops in rate constants
are both found in xenon, the strongest interacting rare gas. It
is tempting to assume that, depending on the matrix site in
which cyclopropene 1 is trapped, individual matrix atoms in-
teract more or less strongly with 1. Molecules of 1 trapped in
strongly interacting matrix sites rearrange first, leaving unreact-
ed molecules in less interacting sites behind. The weak interac-
tions between the matrix and trapped species might depend
on the polarity, polarizability, and other properties of the
trapped species. ISC is necessary for the interconversion of sin-
glet 1 to triplet 2, so it is tempting to hypothesize that the
heavy-atom effect of xenon increases the ISC rates. This in-
crease could affect the overall rates for the 1!2 rearrange-
ment if, in xenon, 1 tunnels directly to T-2 rather than to S-2.
Figure 2. Arrhenius plots for the 1!2 rearrangement in neon, argon, xenon,
and nitrogen.
Arrhenius plots
Kinetic isotope effect
The matrix effects leading to dispersed kinetics (see discussion
above) make it difficult to extract physically meaningful rate
constants from the kinetic data. Despite it being a unimolecular
reaction, in the solid state, the kinetics of the 1!2 rearrange-
ment cannot be described by a single rate constant k, but
rather, a distribution of rate constants has to be considered to
fit the experimental data. Siebrand and Wildman introduced
In the 1!2 rearrangement, no CH bond is broken, and thus,
only a secondary kinetic isotope effect (KIE) is expected upon
substitution of the H atoms in 1 by deuterium. For measure-
ment of the KIE, the perdeuterated cyclopropene (1-d4) was
generated in argon matrices, and the rearrangement was fol-
lowed at 3 K for up to seven days. The experiments were also
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performed with an approximately 1:1 mixture of 1 and 1-d4 to
ensure identical conditions. Deuteration results in a drop in
the rates from kH =1.2ꢃ0.1ꢁ10ꢀ6 sꢀ1 for 1 to kD <2ꢁ10ꢀ7 sꢀ1
for 1-d4, which corresponds to an increase in half-life in Ar
from 7 to over 45 days. The KIE is determined to be >6, and
within the error limits of our rate measurements, is independ-
ent of temperature between 3 and 20 K.
theory (VTST)[41] and the small-curvature tunneling (SCT) ap-
proximation.[42]
VTST+SCT rate calculations, performed in conjunction with
B3LYP energy and derivative evaluations, have given rate con-
stants that are in excellent agreement with those measured
around 8 K for the ring expansion reactions of 1-methylcyclo-
butylfluorocarbene[17] and noradamantylchlorocarbene.[18,43]
Calculations using the same methodology have provided 13C
kinetic isotope effects for the ring opening of the cyclopropyl-
carbinyl radical at low temperatures, which are also in excellent
agreement with the experimental values.[20]
However, owing to the very long half-life (several weeks) of
the rearrangement of 1-d4, only a small fraction of the decay
can be monitored, which introduces a large statistical error
into the measurements. In addition, because the observed
rates are larger at the beginning of the measurements and
become constant only after a substantial fraction of decay has
occurred, we expect that the observed reaction rate for 1-d4 is
dominated by “fast” matrix sites, and is, thus, probably too
large relative to the average rate of rearrangement of undeu-
terated 1. Therefore, the actual KIE in the absence of such
matrix effects in Ar is probably considerably larger than six. In
fact, in xenon, a KIE of approximately 24 is measured, and this
too could be a lower limit.
The open-shell nature of S-2 made it impossible to use DFT
to perform tunneling calculations, because DFT can describe
triplets and closed-shell singlets correctly, but not open-shell
singlets. Therefore, (8/8)CASSCF calculations were performed,
using the 6-31G(d) basis set, to generate the potential energy
surface connecting 1 to S-2 and to calculate the CASSCF tun-
neling rates. In the computation of the CASPT2 tunneling
rates, the (8/8)CASSCF energies of 1, S-2, and the transition
state connecting them were replaced with the CASPT2 ener-
gies of these stationary points. Tunneling calculations were
then performed by using the interpolated single-point energy
(VTST-ISPE) method.[44] The 6-31G(d), cc-pVDZ,[45] and cc-pVTZ
basis sets[46] were used for the CASPT2 energy calculations.
The results of the (8/8)CASSCF and CASPT2 calculations
served as the input for Polyrate,[47] which was used to compute
the rate constants for the ring opening of 1 to S-2 through
QMT. The (8/8)CASSCF calculations were performed with Gaus-
sian09,[48] using Gaussrate[49] as the interface between Gaussi-
an09 and Polyrate. The CASPT2 energy calculations were car-
ried out with MOLCAS 7.4.[50]
Calculations
The final product of the rearrangement of cyclopropene 1 is T-
2, that is, the carbene in its triplet ground state. Thus, T-2
must be lower in energy than 1, and an ISC step has to occur
somewhere along the reaction coordinate connecting 1 to T-2.
Electronic structure calculations at several different levels of
theory have all indicated not only that 2 has a triplet ground
state, but also that, perhaps surprisingly, the lowest singlet
state of 2 (S-2) is the open-shell singlet state (1B1).[37–39] In the
lowest singlet state of most carbenes, a pair of electrons occu-
pies the s MO on the carbene center. However, the open-shell
singlet state S-2 has the same orbital occupancy as the triplet
T-2, with only one electron occupying the s orbital and
a second electron of opposite spin occupying the 2p-p MO.
A reasonable mechanism for
Computational results
Figure 3 shows the (8/8)CASSCF/6-31G(d) optimized geome-
tries of 1, S-2, T-2, and the transition structure connecting 1 to
the formation of T-2 from 1 at
cryogenic temperatures is ring
opening of 1 to S-2 occurring
through QMT, followed by ISC of
S-2 to T-2. This mechanism re-
quires that a) S-2 and T-2 both
be lower in energy than 1, and
b) QMT from 1 to S-2 occurs
with a rate constant of about
Figure 3. (8/8)CASSCF/6-31G(d) bond lengths [ꢂ].
10ꢀ6 sꢀ1. Both of these mechanis-
tic postulates were tested com-
putationally.
S-2. Reoptimization of these geometries at the CASPT2/6-
31G(d) level of theory (see Figure S1 in the Supporting Infor-
mation) resulted in bond-length changes of less than 0.02 ꢂ
and changes in the relative CASPT2 energies of less than
0.5 kcalmolꢀ1.
Computational methodology
We performed (8/8)CASSCF and CASPT2[40] calculations of
a) the relative energies of 1, S-2, and T-2, and b) the rate con-
stant for ring opening of 1 to S-2 through QMT. The tunneling
calculations were performed using variational transition state
Although S-2 and T-2 each have one electron in the carbene
s orbital and another unpaired electron in the 3b1 p MO,
Figure 3 shows that the geometries of these two states are
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rather different. This is because the 3b1 MO in the triplet is
very different from the 3b1 MO in the singlet.
8.2 kcalmolꢀ1 (7.1 kcalmolꢀ1 with ZPE corrections). Absorption
of a photon of IR around 2000 cmꢀ1 would deposit 5.7 kcal
molꢀ1 in the matrix-isolated molecule. This amount of energy,
although not enough to take 1 over the barrier, could allow
1 to tunnel to S-2 closer to the top of the barrier, where the
energy below the top of the barrier is smaller and the barrier
is narrower than at the bottom of the barrier.
In the triplet state, the Pauli exclusion principle prevents the
unpaired s and p electrons from appearing simultaneously in
the region of space in which the s and 2p-p AOs on the car-
benic carbons overlap. Consequently, in the triplet state, the
3b1 p MO has a large coefficient on the carbene carbon.
Indeed, the bond lengths in T-2 suggest that its dominant res-
onance structure is that shown in Figure 3, with a strong p
bond to oxygen and the unpaired electron in a pentadienyl-
like nonbonding MO, with significant unpaired density in the
carbene 2p-p AO.
In addition to its higher barrier, the ring-opening reaction is
computed to be 6.7 kcalmolꢀ1 less exothermic at the CASPT2
than at the (8/8)CASSCF level of theory. The reason for the
smaller reaction exothermicity at the CASPT2 level of theory is
that CASPT2 correlates the electrons in all three of the strained
s bonds of the cyclopropene ring in 1, whereas (8/8)CASSCF
correlates the electrons only in the s bond that is broken in
the ring-opening reaction.
In contrast, in the open-shell singlet state, the electrons in
the s and 3b1 p MOs have opposite spins, so the motions of
these two electrons are actually anticorrelated in the regions
of space where the s and p AOs on the carbenic carbons over-
lap. Therefore, if the 3b1 p electron appears at the carbene
carbon, there is a large Coulombic repulsion energy between
this p electron and the electron of opposite spin in the s orbi-
tal on this carbon. Consequently, unlike the case in the T-2
state, in the S-2 state, the 3b1 p MO has a very small coefficient
on the carbene carbon. Indeed, the bond lengths in Figure 3
for S-2 suggest a dominant resonance structure with the un-
paired p electron localized largely in a 2p-p AO on the oxygen.
Although CASSCF calculations deal satisfactorily with static
electron correlation, CASPT2 calculations also include the ef-
fects of dynamic correlation between electrons.[40] Consequent-
ly, CASPT2 is expected to give much more reliable relative en-
ergies than CASSCF. Figure 4 shows schematically the relative
energies of 1, S-2, and the transition structure that connects
them at both the (8/8)CASSCF and CASPT2 levels of theory,
using the 6-31G(d) basis set. Also shown is the relative energy
of the triplet ground state of the product of ring opening (T-2)
at both levels of theory.
The lower exothermicity of the ring-opening reaction at the
CASPT2 level of theory should make the height and width of
the CASPT2 barrier greater than those of the CASSCF barrier.
The heights and widths of the CASSCF and CASPT2 barriers are
shown schematically in Figure 5, in which the barrier heights
are plotted as a function of S, the progress (in arbitrary units)
along the reaction coordinate.
Figure 5. (8/8)CASSCF/6-31G(d) and interpolated single-point energy (ISPE)
CASPT2 energy curves. The width of the barrier at its base is given by 3.0
plus the value of S at which each curve crosses the y axis.
Rate constants for tunneling reactions decrease exponential-
ly with the product of the width of the barrier and the square
root of the energy below the top of the reaction barrier at
which tunneling occurs.[51] Consequently, tunneling rates are
more sensitive to barrier width than to barrier height.[52,53]
Thus, the greater height and especially the greater width of
the CASPT2 barrier should both make the calculated rate con-
stants for tunneling smaller at the CASPT2 than at the CASSCF
level of theory.
Figure 4. (8/8)CASSCF (red) and CASPT2 (green)/6-31G(d) relative energies
[kcalmolꢀ1]. Zero-point energy corrected values are in parentheses.
(8/8)CASSCF and CASPT2 both predict that the ring opening
of 1 to S-2 is exothermic, and passes over a reaction barrier
that is computed to be 1.9 kcalmolꢀ1 higher at the CASPT2
than at the (8/8)CASSCF level of theory. The CASPT2 barrier is
Our calculations of the rates constants for ring opening of
1 to S-2 show that the rate constants are independent of tem-
perature below about 50 K (the calculated rate constants are
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given as a function of temperature in Table S1 of the Support-
ing Information). Therefore, up to 50 K, tunneling occurs from
the lowest vibrational level of the reactant without any contri-
bution from tunneling from thermally populated, excited vibra-
tional levels.
Table 1 also gives the kH/kD kinetic isotope effects (KIEs) for
the ring opening of 1 to S-2 and 1-d4 to S-2-d4. The tunneling
rates decrease exponentially with the product of the barrier
width, the square root of the energy below the top of the re-
action barrier at which tunneling occurs, and the square root
of the effective tunneling mass.[51] Wide and/or high barriers
are responsible for the smallest rate constants for tunneling of
1 to S-2 in Table 1, so it follows that the smallest calculated
rate constants for the ring opening of undeuterated 1 in
Table 1 should be associated with the largest KIEs. This is gen-
erally found to be the case.
The temperature-independent (8/8)CASSCF and CASPT2 rate
constants, computed with the 6-31G(d) basis set, are given in
Table 1. The rate constants are calculated to be 66 sꢀ1 and
2.3ꢁ10ꢀ8 sꢀ1, respectively. As expected, the CASPT2 rate con-
stant is calculated to be much smaller than the CASSCF rate
constant. However, the 3ꢁ109 ratio of these two rate constants
reveals how sensitive the rate constants for tunneling are to
the differences in barrier heights and barrier widths that are
shown schematically in Figure 5.
Although the CASPT2 calculations give much better agree-
ment with the experimental rate constant for the ring opening
of undeuterated 1 than the (8/8)CASSCF calculations, it ap-
pears that the CASPT2 calculations may overestimate the size
of the experimental KIE. Even the CASPT2/cc-pVDZ calcula-
tions, which give the calculated rate constant for the ring
opening of 1 to S-2 that is in the best agreement with the ex-
perimental result, overestimates by a factor of 40 the effect of
deuterium on the rate of ring opening of 1-d4 to S-2-d4. How-
ever, as noted in the section on the experimental determina-
tion of the kinetic isotope effect, the value of kH/kD ꢄ6 in an
argon matrix is almost certainly a lower limit, and the actual
size of the kinetic isotope effect could be considerably higher.
Table 1. Calculated and measured low-temperature rate constants (sꢀ1
for the ring opening of 1 to S-2 and of 1-d4 to S-2-d4.
)
Calculation
kH (1!S-2)
kD (1-d4!S-2-d4)
kH/kD
CASSCF/G[a]
CASPT2/G[a]
CASPT2/D[b]
CASPT2/T[c]
experiment[d]
6.6ꢁ101
1.3ꢁ101
4.4ꢁ10ꢀ11
1.0ꢁ10ꢀ8
2.2ꢁ10ꢀ11
<2ꢁ10ꢀ7
5.1
2.3ꢁ10ꢀ8
2.4ꢁ10ꢀ6
2.6ꢁ10ꢀ8
1.2ꢁ10ꢀ6
520
240
1200
>6
[a] 6-31G(d) basis set. [b] cc-pVDZ basis set. [c] cc-pVTZ basis set.
[d] Argon matrix.
Conclusion
Figure 5 shows that, not surprisingly, the CASPT2 barrier
heights and widths depend on the basis set with which the
calculations are performed. The cc-pVDZ basis set gives the
lowest and narrowest barrier. The cc-pVTZ basis set gives a bar-
rier that is only 0.1 kcalmolꢀ1 higher than that computed with
cc-pVDZ, but the transformation of 1 to S-2 is calculated to be
1.6 kcalmolꢀ1 less exothermic with cc-pVTZ than with cc-pVDZ.
Consequently, as shown in Figure 5, the cc-pVTZ reaction barri-
er is not only slightly higher, but also wider than the cc-pVDZ
barrier.
The 6-31G(d) reaction barrier is 0.4 kcalmolꢀ1 higher than
the cc-pVTZ barrier, but the calculated reaction exothermicity
is 0.7 kcalmolꢀ1 larger with the former basis set than with the
latter. Therefore, as shown in Figure 5, the 6-31G(d) reaction
barrier is calculated to be slightly narrower than the cc-pVTZ
barrier.
The ring opening of the highly strained cyclopropene 1 pro-
ceeds through QMT at temperatures as low as 3 K. The final
product is the triplet ground-state carbene T-2, and therefore,
an intersystem crossing (ISC) step is necessary during the rear-
rangement. This suggests that the primary product of the rear-
rangement is the singlet carbene S-2, which undergoes subse-
quent ISC to T-2.
Both (8/8)CASSCF and CASPT2 calculations predict an open-
shell singlet state for the carbene S-2. At the more reliable
CASPT2 level of theory, the activation energy for the ring
opening of 1 to S-2 is calculated to be 8 kcalmolꢀ1, and the
energy of reaction is found to be ꢀ4 kcalmolꢀ1
.
The reaction rates expected for the rearrangement 1 to S-2
by passage over a reaction barrier of 6–8 kcalmolꢀ1 at cryogen-
ic temperatures are effectively zero. The fact that this rear-
rangement occurs at 3 K is therefore a very strong indication
that this reaction proceeds by tunneling through the reaction
barrier.
As already noted, not only the barrier heights but also the
barrier widths affect the calculated temperature-independent
rate constants for tunneling from the lowest vibrational level
of 1 to S-2. As shown in Table 1, the cc-pVDZ basis set, which
gives a reaction barrier that is both lower and narrower than
the cc-pVTZ and 6-31G(d) barriers, gives a CASPT2 rate con-
stant about two orders of magnitude larger than those com-
puted with the cc-pVTZ and 6-31G(d) basis sets. The CASPT2/
cc-pVDZ temperature-independent rate constant of 2.4ꢁ
10ꢀ6 sꢀ1 is only a factor of two larger than the experimental
value of 1.2ꢁ10ꢀ6 sꢀ1 in argon, but this agreement is clearly
fortuitous, because the larger cc-pVTZ basis set gives poorer
agreement with the experimental rate constant.
Between 3 and 20 K, the rate constants for this rearrange-
ment are independent of temperature within the error limits of
our measurements. Thus, the experimental activation energy is
zero, as expected for a tunneling reaction. At higher tempera-
tures, the rates increase, suggesting that thermally activated
tunneling contributes to the rates. Argon and nitrogen matri-
ces begin to soften at these higher temperatures, so the rates
of rearrangement measured above 20 K may have contribu-
tions that involve changes in the matrix (e.g., rearrangements
of the rare gas atoms surrounding matrix-isolated atoms of 1,
rotation of molecules of 1 in the matrix, etc.).
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The results of our tunneling calculations are in reasonable
agreement with our experimental results. Our calculations pre-
dict that tunneling dominates the reaction rates at tempera-
tures below 50 K. Our CASPT2/cc-PVDZ calculations reproduce
very closely the experimental rate of reaction of 1 in argon.
Unfortunately, the calculated CASPT2/cc-PVDZ KIE for the re-
arrangement of 1-d4 is about forty times larger than the exper-
imental value measured in argon. However, the experimental
value may contain large errors because of the extremely slow
rate of rearrangement of 1-d4, and only a lower limit for the
experimental value of kH/kD can actually be estimated. There-
fore, the apparently sizable discrepancy between the experi-
mental and calculated KIEs may reflect errors in both sets of
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Keywords: calculations
rearrangement · tunneling
·
carbene
·
matrix isolation
·
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Received: September 27, 2013
Published online on && &&, 0000
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FULL PAPER
&
Rearrangement Reactions
M. Ertelt, D. A. Hrovat, W. T. Borden,
W. Sander*
&& – &&
Heavy-Atom Tunneling in the Ring
Opening of a Strained Cyclopropene
at Very Low Temperatures
Quantum mechanical tunneling: De-
spite an estimated activation barrier of
more than 6 kcalmolꢀ1, the strained cy-
clopropene 1 rearranges at tempera-
tures as low as 3 K to the carbene 2 in
its triplet ground state (see figure). Ex-
periments and theory provide clear evi-
dence that the rearrangement proceeds
through heavy-atom tunneling.
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