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Do Heavy Nuclei See Light at the End of the Tunnel.

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A Tunneling Event
Do Heavy Nuclei See Light at the End of the Tunnel?
Robert Berger*
carbenes · density functional calculations ·
IR spectroscopy · matrix isolation · tunneling
The most direct way to the valley on the
other side of a ridge passes right through
the mountain. This unsurprising insight,
however, does not open up new possibilities for classical particles, since they
have typically only a single option for
surmounting a ridge, namely, to try to
climb a suitable mountain pass. Any
such attempt will be crowned with
success only if the particles are equipped
with sufficient energy for overcoming
the highest point of the pass. As early as
in 1927 Friedrich Hund pointed out,[1]
however, that quantum mechanics is in
this respect more generous to its particles: Just as if there were horizontal
tunnels right through the mountain,
quantum-mechanical particles have a
nonvanishing probability for reaching
the other side of a finite barrier even if
their energy is below the threshold
energy of the mountain pass. In return,
however, particles in the quantum world
are plagued by nonclassical reflection at
the barrier even at energies above the
threshold. Chances for tunneling are
usually considered good for light particles like, for instance, electrons or
protons, but the probability of a tunneling event is commonly believed to be
almost negligible for heavier nuclei. The
evidence for significant C nucleus tunneling in a chemical reaction with a
sizable barrier reported recently by
Zuev et al.[2] contradicts this widely
accepted notion and is thus of particular
[*] Dr. R. Berger
Institut fr Chemie
Technische Universit#t Berlin
Strasse des 17. Juni 135, 10623 Berlin
Fax: (+ 49) 30-314-21102
importance for our understanding of molecules (see ref. [4] for a review and,
quantum effects in chemical processes. for instance, ref. [5] for recent calculaIt is due to their wavelike properties tions on parity-violating effects in chiral
that quantum-mechanical particles are molecules).
The concept of tunneling received
able to penetrate finite barriers and
have therefore a nonvanishing proba- particular interest in the nuclear physics
bility of reaching classical forbidden community, when Gamov[6] as well as
regions of the coordinate space. This Gurney and Condon[7] proposed quandynamic passage through the barrier is tum-mechanical tunneling to be crucial
already reflected on the level of the for the nuclear a-decay process. Gamov
stationary energy eigenstates. If we demonstrated that the energy-dependconsider, for instance, a one-dimension- ent transmission coefficient T(E) in a
al problem, a particle in a bound eigen- one-dimensional system can for suffistate with a corresponding energy below ciently small values of T(E) be approxia finite barrier separating two potential mated with the proportionality relawells will not be detected exclusively on tion (1).
one and the same side of the barrier.[3] In
the special case of a symmetric potenZb pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 m ðVðxÞEÞdx ð1Þ
tial, moreover, a particle prepared in an TðEÞ / exp½ð4p=hÞ
energy eigenstate will even be measured
with equal probability in each of the two
Here, h is Planck's constant, V(x) the
wells, regardless of the particular finite
height of the barrier. In the limit of a coordinate-dependent potential energy,
high barrier, these eigenstates in a E the total energy, and m the tunneling
symmetric potential come in almost mass. For x = a and x = b, E = V(a) =
degenerate pairs of a symmetric and an V(b). Therefore, the probability for
antisymmetric state, which give rise to tunneling depends strongly on the width
the well-known tunneling splitting for and the height of the barrier as well as
instance in the microwave and infrared on the particle's mass, which underlines
spectra of the prototype system ammo- the pronounced importance of tunneling
nia. Hund realized[3] that tunneling leads for light rather than heavy particles.
In the early stages of the developapparently to a paradoxical situation for
chiral compounds, for which such sym- ment of the transition-state theory for
metric (parity-conserving) potentials chemical reactions by Eyring, Evans and
were assumed. But he argued that once Polanyi, and others,[8] (see also ref. [9]),
a chiral state of a typical optically active which provides an avenue to the ab
compound has been generated, its inter- initio calculation of absolute rate conconversion to its mirror-image as a result stants of chemical reactions, it was alof tunneling will take a million years or ready recognized that tunneling through
more. Nowadays it is assumed, however, the potential barrier should be accountthat the fundamental parity-violating ed for. In this context Wigner[10, 11] eminteractions destroy this symmetry for phasized that tunneling plays a particchiral systems and induce a slight energy ular role in two distinct cases: Firstly, if
difference between the enantiomers molecules have almost enough energy to
which would have substantial conse- overcome the potential barrier, and
quences for the dynamics of chiral secondly, if the potential barrier is thin
2004 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
DOI: 10.1002/anie.200301675
Angew. Chem. Int. Ed. 2004, 43, 398 –401
and high and the temperature is very
low. In the former situation, tunneling
should lead, according to Wigner, only
to a correction of the classically calculated rate constants (see, however, for
example also ref. [12]). It was concluded
that apart from reactions involving hydrogen tunneling, corrections can be
neglected at ordinary temperatures.[11]
Indeed, for numerous proton-, hydrogen-atom-, and hydride-transfer reactions significant tunneling contributions
have been identified.[13] . Experimental
evidence for tunneling in these reactions
is, apart from measurements of tunneling splittings in infrared or microwave
spectra discussed above, generally obtained from nonlinearities in Arrhenius
plots (ln k vs T1 plots), from an unusual
ratio of the proton vs the deuterium or
the tritium Arrhenius prefactor, or from
pronounced and temperature-dependent kinetic isotope effects (see, however,
recent results on kinetic isotope effects
in multiple-proton-transfer reactions[14])
to name but a few. While in principle all
nuclei of a system may contribute to
some extent to the tunneling motion and
therefore it is often not well justified to
attribute the entire process to the tunneling of a single nucleus, reactions for
which evidence for significant heavynucleus tunneling exist are scarce and
even those suggested as potential candidates are contested (see refs. [15–19]
for a discussion on heavy-nucleus tunneling in the cyclobutadiene automerization reaction; ref. [20] reports tunneling in isonitrosyl chloride, while ref. [21]
gives a more sceptical view and [22–25]
discuss other proposals of heavy-nucleus
tunneling). Typically, for these systems
the second mechanism, which Wigner
identified as the prevailing contribution
at very low temperatures, is relevant.
Thin, high barriers result in only a weak
temperature dependence of the reaction
rate in the low-temperature regime.
Wigner concluded, however, that then
the reaction rate is too low to be of
practical importance.
Contradicting Wigner's conclusion,
Zuev et al.[2] presented recently chemical reactions in matrices at cryogenic
temperatures for which the authors
ascribed the significant reaction rate to
carbon-nucleus tunneling. These authors reported that ultraviolet irradiation (l = 334 nm) of 3-(1-methylcyclobutyl)-3-fluorodiazirine (1, Scheme 1),
which was isolated in an N2 matrix at
8 K, produced a compound with a strong
IR absorption at 2027 cm1, which was
assigned to 1-(diazofluoromethyl)-1methylcyclopropane (2). Upon subsequent irradiation with visible light (l >
550 nm), this compound was transformed partly into 1-fluoro-2-methylcyclopentene (3), identified by comparison with the IR spectrum of an authentic sample, and partly into a compound
with several intense IR signals around
1100 cm1. In view of its further chemical reactions as well as by comparison
with computed IR and UV/Vis properties this was assigned to (1-methylcyclopropyl)fluorocarbene (4), which according to the accompanying quantumchemical calculations should exist as
two isomers, an exo and an endo form.
When 4 was exposed to visible light (l =
436 nm), Zuev et al. observed rapid
conversion of the fluorocarbene 4 into
the cyclopentene 3. But even at a
temperature of 8 K and in the dark,
slow formation of 3 was detected. The
kinetics of this interconversion were
then studied with the aid of IR spectroscopy by monitoring the change in
intensity of two intense IR signals for
exo- and endo-4 and the increase of the
signal corresponding to the C=C stretching fundamental of 3.
The results of this study were interpreted as follows: At 8 K in the nitrogen
matrix, the endo form of 4 was essentially unreactive while the exo conformer of 4, which appears to be properly
oriented for the intramolecular insertion
of a carbene into the CC s bond,
underwent slow reaction with approximately first-order kinetics and a corre-
sponding rate constant of k(8 K) 4 D
106 s1. At 16 K the exo conformer
disappeared initially 12 times faster than
at 8 K but due to deviations from firstorder kinetics (a phenomenon that may
be caused by varying reaction barriers at
different matrix sites[26]) the corresponding rate constant dropped to a value that
was only about two times larger than at
8 K. Disappearance of the slower reacting endo conformer was also observed at
this temperature with an initial rate
constant of 6 D 106 s1, which dropped
to 1 D 106 s1 after 65 % conversion. At
25 K both conformers of 4 disappeared
ten times as fast as at 16 K.
In an argon matrix both conformers
of 4 reacted already at 8 K with a rate
constant for the disappearance of exo-4
of 4 D 105 s1 at 8 K, and the reaction
was about twice as fast at 16 K. This
relatively modest rise of the reaction
constant with doubling the absolute
temperature was considered to be inconsistent with a thermally activated
process. It was therefore concluded that
the ring-expansion reaction of fluorocarbene 4 proceeds by means of carbonnucleus tunneling. The experimentally
observed increase of the rate constant
with rising temperature in this lowtemperature regime was construed as
an environmental effect on the carbene
lifetime, for which in particular matrix
softening was suggested to be responsible.
The supposition of heavy-nucleus
tunneling was supported by accompanying quantum-chemical calculations. The
reaction enthalpy of the ring-expansion
reaction was computed with density
functional theory (modified Perdew–
Wang hybrid functional with one parameter
(MPW1K), 6-31 + G(d,p) basis set) to
amount to DrH00 = 328 kJ mol1, and
the activation enthalpy was obtained as
D°H00 = 27 kJ mol1. This barrier appears to be too high to be surmounted
by thermal activation in the temperature
range studied (< 25 K). Indeed, direct
Scheme 1. The photochemical conversion of matrix-isolated 1 leads to exo- and endo-(1-methylcyclobutyl)fluorocarbene (exo- and endo-4).
Angew. Chem. Int. Ed. 2004, 43, 398 –401
2004 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
dynamics calculations in the framework
of canonical variational transition-state
theory[27] predict a low-temperature limit for the rate constant of about 9 D
106 s1 when tunneling contributions
are accounted for (here within the
small-curvature tunneling approximation, see ref. [28]). Computations with
exclusion of tunneling, in contrast, predict a rate constant at 8 K that is smaller
by a factor of about 10152. This astounding ratio results, however, plainly as a
consequence of the finite low-temperature-limit for the tunneling process as
opposed to the vanishing classical barrier-hopping in this limit. Tunneling at
8 K originates according to the calculations nearly exclusively from the vibrational ground state. The tunneling
mode's first excited vibrational state
contributes to the rate constant by only
about 0.04 % at this temperature. Figure 1 sketches the vibrational adiabatic
ground-state potential curve along the
reaction path together with representative molecular structures and with the
lowest two vibrational levels in the
tunneling mode. Evidently, significant
movement of the heavy nuclei is involved in the tunneling processes along
this particular path.
As noted by Wigner, due to the
tunneling through the barrier the rate
constant depends only weakly on the
temperature in this temperature range.
According to the calculations, for temperatures below 20 K the rate constant
deviates by less than 10 % from the lowtemperature limit. The Arrhenius activation energy Ea increases according to
the calculations from 0.3 J mol1 at 8 K
to 1.2 kJ mol1 at 40 K to 23.6 kJ mol1 at
150 K. At 216 K the contribution to the
rate constant from the classical over-thebarrier and from the quantum-mechanical through-the-barrier processes are
equal, and at room temperature the
classical process contributes roughly
three times more strongly than the
tunneling process. This clearly demonstrates that tunneling is also a temperature-activated process since in higher
vibrational states the barrier width is
typically smaller and tunneling is therefore more likely.
Unless the underlying potential energy hypersurface fails to properly describe the relevant energetics of the
reaction, the computations indicate that
heavy-nucleus tunneling is the dominating dynamic process of the low-temperature ring expansion of (1-methylcyclopropyl)fluorocarbene. In contrast to
most systems for which heavy-nucleus
tunneling has been proposed, contributions from electronic triplet states may
Figure 1. Representative structures along the reaction path q for the ring-expansion reaction of
(1-methylcyclobutyl)fluorocarbene (4). The graph is based on Figure 3 in ref. [2]. Black: C, gray:
F, white: H.
2004 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
be neglected for fluorocarbene 4, since
typically the triplet states in fluorocarbenes are energetically well removed.
Although a rigorous treatment of tunneling requires in principle the solution
of the full-dimensional rovibratoric
SchrHdinger equation, experience with
reaction-path and -surface Hamiltonian
treatments shows that main features can
often be reproduced successfully (see
for instance ref. [29] for calculations on
torsional tunneling splittings in hydrogen peroxide).
The reasonable agreement between
the computed and the measured orders
of magnitude for the rate constant in the
present reaction studied by Zuev et al.
provides further evidence. A more detailed comparison with the quantumchemical calculations, which neglect the
surrounding medium, requires, however, that experimentally the sources for
deviations from first-order kinetics are
quantified. Here in particular also the
interconversion of endo- to exo-4 should
be included in an improved modelling.
Experiments performed with different
matrices will perhaps further substantiate the supposed matrix-softening effect
made responsible for the observed temperature-dependence of the rate constant. It will then be particularly interesting to experimentally extend the
studied temperature range to above
40 K, where a significant increase of
the Arrhenius activation energy is predicted by theory. Some indirect evidence, however, is obtained from the
study of the rearrangement of the chloro
analogue of 1. Zuev et al. did not detect
the corresponding cyclobutylchlorocarbene. Accompanying computations suggest that in the cyclobutylchlorocarbene
the probability of tunneling through the
barrier is so large that the rate constant
at 8 K is about 1 D 104 s1, and therefore
the reaction to 1-chloro-2-methyl-cyclopentene proceeds too fast for the alkylchlorocarbene to be observable in the
Three-quarters of a century after the
concept of tunneling was introduced by
Friedrich Hund, it is still one of the most
intriguing quantum effects in chemistry.
Quantum phenomena in heavier systems are still at the frontier of current
research, and they are related to important fundamental questions including
the long-term prospect of the preparaAngew. Chem. Int. Ed. 2004, 43, 398 –401
tion of superposition states for heavy or
even macroscopic objects. Zuev et al.
have indicated that even for heavy
nuclei there is light at the end of the
tunnel. This work may therefore be an
important further step in the direction of
a better understanding and control of
quantum phenomena in chemistry.
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