464 J. Phys. Chem. A, Vol. 101, No. 4, 1997
Kimura et al.
theory,34, 40 the tunneling rate k can be calculated by use of the
values EB and ∆x
hydrogen shift and 2.18 × 10-2 nm for the deuterium shift.
The parameters used to obtain the theoretical tunneling rates
are listed in Table 2. By using these tunneling parameters and
eq 16, the tunneling rate constants, k1 and k2 for the 1,2-hydrogen
shift at V ) V0 and V ) V1, were obtained as 0.12 and 1.4 × 102
s-1, respectively. For the 1,2-deuterium shift, k1 and k2 at V )
V0 and V ) V1 were obtained to be 1.4 × 10-2 and 0.9 × 102
s-1, respectively.
By use of these tunneling rate constants k1 and k2, the values
of k1,2 can be calculated by eq 8 at various temperatures. The
k1,2 values for the 1,2-hydrogen and deuterium shifts are plotted
in Figure 5 (broken lines). The good agreements between the
experimental and theoretical values for k1,2 show that the 1,2-
hydrogen and deuterium shifts proceed via tunneling processes
at V ) V0 (E ) 0) and V ) V1 (E ) EV (2.9 kcal mol-1 for the
hydrogen shift and 3.3 kcal mol-1 for the deuterium shift))
according to the Boltzmann distribution law.
2π
h
k ) ν exp - [2µ(EB - EV)]1/2∆x
(16)
{
}
where k ) k1 at V ) V0 (i.e., EV ) 0, ∆x ) ∆x) and k ) k2 at
V ) V1 (i.e., EV ) ∆E, ∆x ) ∆x′). Here, ν is the average
frequency for the tunneling and generally taken as 1013 s-1 in
the case of intramolecular reaction.40 However, in the present
system, the frequency is very small since the reaction occurs
only when all the corresponding oscillators vibrate as promoting
modes. Therefore, the frequency of the tunneling reaction was
obtained to be 5 × 1010 s-1 by the best fitting method with use
of eqs 8 and 16.
The value of EV in eq 16 is equal to the energy difference
∆E between two discrete vibrational levels as stated above. Here,
µ is the reduced mass estimated by31,32
Concluding Remarks
1/2
µ1/2 ) µCN1/2 + µCH
(17)
The following concluding remarks are drawn by the direct
measurements of the 1,2-sigmatropic hydrogen (or deuterium)
shift of the PRI of N-acetylpyrrole and also the theoretical
considerations according to the tunnel effect theory proposed
by Formosinho.31-36
(1) The rates for the 1,2-sigmatropic hydrogen (or deuterium)
shift of the PRI of NAH produced by 266-nm laser flash
photolysis are directly measured in several solvents. It is found
that the rate of the 1,2-hydrogen shift is enhanced by the starting
material (NAH or NAD) and the intramolecular rate constant
for the 1,2-sigmatropic hydrogen or deuterium shift is deter-
mined to be 0.27 s-1 for NAH (0.12 s-1 for NAD) in MCH at
293 K. It is noteworthy that the 1,2-sigmatropic hydrogen shift
proceeds via the intramolecular process at a low concentration
of NAH (1.7 × 10-4 M) in dehydrated MCH.
where µCN and µCH are the reduced masses for the CN and CH
vibrations.
The value of ∆R (the displacement between the potential
minima of reactant and product along with the reaction
coordinate) was calculated by means of the intersecting-state
model (ISM) proposed by Formosinho.32,36 According to the
ISM model, the sum of the bond distensions, d, at the transition
state is expressed as
a′ ln 2
nq
d )
(rp + rr)
(18)
where a′ is a constant ()0.156), rp and rr are the equilibrium
bond lengths of product and reactant, respectively, and nq is
the bond order at the transition state. In the present system,
two kinds of bond rearrangements are involved, e.g., H (from
CsH to NsH) and N (from CdN to CsN). In the first place,
for the bond rearrangement of H-migration which includes the
processes of C-H bond-breaking and N-H bond-forming, the
transition-state bond order nq is 0.5. Therefore,
(2) The rate of 1,2-hydrogen shift is remarkably increased
by a basic catalyst, such as triethylamine, alcohols, and water.
The effect of the basic catalyst on the 1,2-hydrogen shift can
be explained by proton exchange between the intermediate (PRI)
and the basic catalysts.
(3) On the basis of the experimental and theoretical results
of temperature and isotope effects, it is shown that the
intramolecular 1,2-hydrogen (or deuterium) shift in MCH
proceeds via quantum mechanical tunneling at two vibrational
energy levels: E ) 0 (V ) V0) and E ) EV ()2.9 kcal mol-1
for the hydrogen shift or 3.3 kcal mol-1 for the deuterium shift)
(V ) V1) under the experimental conditions.
a′ ln 2
0.5
dXH
)
(rC-H + rN-H
)
(19)
is derived.
In the second place, for the CN rearrangement involving a
bond-breaking process from CdN to CsN, the transition bond
order also equals 0.5. Thus,
Acknowledgment. This work was supported by a Grant-
in-Aid on Priority-Area-Research: Photoreaction Dynamics
(06239101) from the Ministry of Education, Science and Culture
of Japan. We thank Professor Susumu Tajima of Gunma
College of Technology for MS measurements of NAH and
NAD.
a′ ln 2
0.5
dCN
)
(rCdN + rC-N
)
(20)
The value of the bond distension (d ) ∆R) is represented by
eq 21 as the average of the two bond distensions dXH (X ) C
or N) for H and dCN for N. The optimized bond lengths are
References and Notes
1
2
(1) This work was presented at The International Symposium on Recent
Progress and Future Prospects of Molecular Electronic Spectroscopy in
honor of Professor Saburo Nagakura, Hayama, Japan, Oct 1995.
(2) Woodward, R. B.; Hoffmann, R. J. Am. Chem. Soc. 1965, 87, 2511.
(3) Fleming, I. Frontier Orbitals and Organic Chemical Reactions;
Wiley: New York, 1976.
(4) Evanseck, J. D.; Houk, K. N. J. Phys. Chem. 1990, 94, 5518.
(5) Dorigo, A. E.; McCarrick, M. A.; Loncharich, R. J.; Houk, K. N.
J. Am. Chem. Soc. 1990, 112, 7508.
∆R ) (dXH + dCN
)
(21)
obtained as rC-H ) 10.87 × 10-2 nm, rN-H ) 9.95 × 10-2 nm,
rCdN )12.73 × 10-2 nm, and rC-N )13.65 × 10-2 nm by ab
initio MO calculations. By substituting these values for rCH
,
rNH, rCdN, and rC-N into eqs 19 and 20, the values of dXH and
dCN are obtained as 4.50 × 10-2 and 5.71 × 10-2 nm,
respectively, and ∆R is determined to be 5.09 × 10-2 nm.
The values of ∆x′ at E ) EV ()∆E) were calculated from
the potential energy curves to be 2.28 × 10-2 nm for the
(6) Bernardi, F.; Olivucci, M.; Robb, M. A.; Tonachini, G. J. Am. Chem.
Soc. 1992, 114, 5805.
(7) Sobolewski, A. L. Chem. Phys. Lett. 1993, 211, 293.
(8) Tapia, O.; Andre´s, J.; Safont, V. S. J. Phys. Chem. 1994, 98, 4821.