Quantification of nonstatistical dynamics in an intramolecular Diels-Alder cyclization without trajectory computation

Article abstract of DOI:10.1021/jo500035b

Experimental and computational (DFT) investigations reveal that enyne-allenes with an aryl group as probe at the allene terminus follow a dynamic non-IRC Diels-Alder cyclization pathway. Starting from two separate C²-C⁶ (Schmittel) transition states (TS), two distinct reaction paths originate that share a common diradical intermediate, however, without mixing! Because the momentum of the initial TS is transmitted into product formation, we suggest a simple protocol without trajectory computations to estimate the fraction of molecules that follow nonstatistical dynamics: It was calculated from the partitioning at the TSs, as derived from DFT computations, and the experimental ratio. The thus-determined percentage of dynamically reacting molecules only slightly depends on the depth of the intermediate well but rather on ΔΔG^? of the initial and the follow-up transition states.

Full text of DOI:10.1021/jo500035b

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pubs.acs.org/joc
Quantification of Nonstatistical Dynamics in an Intramolecular
Diels−Alder Cyclization without Trajectory Computation
Debabrata Samanta, Anup Rana, and Michael Schmittel*
Department of Chemistry and Biology, Universitat Siegen, Adolf-Reichwein-Strasse, D-57068 Siegen, Germany
̈
S
* Supporting Information
ABSTRACT: Experimental and computational (DFT) investigations
reveal that enyne−allenes with an aryl group as probe at the allene
terminus follow a dynamic non-IRC Diels−Alder cyclization pathway.
Starting from two separate C2−C6 (Schmittel) transition states (TS), two
distinct reaction paths originate that share a common diradical
intermediate, however, without mixing! Because the momentum of the
initial TS is transmitted into product formation, we suggest a simple
protocol without trajectory computations to estimate the fraction of
molecules that follow nonstatistical dynamics: It was calculated from the
partitioning at the TSs, as derived from DFT computations, and the
experimental ratio. The thus-determined percentage of dynamically
reacting molecules only slightly depends on the depth of the intermediate
well but rather on ΔΔG⧧ of the initial and the follow-up transition states.
In the past two decades, the C2−C6 (Schmittel) cyclization9
INTRODUCTION
■
has been a widely investigated reaction because of its utility for
antitumor antibiotics10 and intricate carbocycle11 and hetero-
cycle12 formation. Lately, though, it has additionally moved into
the focus of physical organic chemistry because the reaction
outcome, so far documented for C2−C6/ene5a,13 and C2−C6/
Diels−Alder (DA) cyclizations,14 often is not consistent with
the common TST but needs to be discussed in light of
nonstatistical dynamics.5a,13,14
The epitome of transition-state theory (TST)1 is of central
importance to chemists’ understanding of selectivity in
kinetically controlled chemical reactions. TST allows us to
determine quantitatively the selectivity from the free energy
difference ΔΔG⧧ of competing transition states (TSs). Another
widely used method is RRKM theory,2 which states that the
energy obtained in a bimolecular collision has to be focused in
the proper vibrational mode to cross the TS energy barrier. For
multistep processes, the theory assumes that intramolecular
vibrational energy redistribution (IVR) takes place much faster
in the ensuing intermediate than crossing the next TS barrier.
As a result, molecules usually relax in the intermediate state (if τ
> 300 fs) and have to accumulate sufficient thermal energy one
more time to populate reactive vibrational modes for crossing
the next TS barrier. In contrast, once the reaction of an
intermediate proceeds faster than IVR,3 the selectivity of
product formation is guided by nonstatistical dynamics and not
by TST or RRKM theory.
Undeniably, an increasing amount of reactions involving SN2
ion−dipole complexes,4 organic diradicals,5 radical cations,6
radicals,7 and others8 have lately been identified to escape
classical TST interpretations. In such circumstances, proving
the occurrence of nonstatistical dynamics becomes very tedious
and time-consuming because presently the scientific commun-
ity requires detailed computational studies on the motions of
atoms or groups along with extensive trajectory calculations at
femtosecond-time resolution. While such a rigorous approach
has its obvious merits, the complexity and time demands of
trajectory calculations prevent any wider use within the
community of mechanistic organic chemists.
Recent experimental evidence and DFT computational
results obtained for the C2−C6/DA14 cyclization of various
enyne−allenes suggest that nonstatistical dynamic behavior
depends on the substituents at both the alkyne and allene
terminus. As an important experimental criterion for dynamic
behavior, we utilize the temperature independence of the
product ratio.15 In the case of enyne−allenes 1 with an aryl
group at the alkyne terminus, the reactant molecules prefer the
stepwise over concerted pathway (Scheme 1), passing initially
across the high-energy C2−C6 transition state 2 (ΔG⧧ ≈ 16−19
kcal mol−1 at 25 °C, referenced to 1) into a very shallow
minimum harboring the σ,π-diradical intermediate 3 (ΔG ≈ 12
kcal mol−1 at 25 °C, referenced to 1). From the singlet
intermediate 3 there are two low energy exit channels with free
energy barriers of 0.5−2.2 kcal mol−1 that lead to the products
6 and 7, with a direct continuation of momentum from the
initial TS.
For the present work, we introduced strong donor
substituents in the probing aryl ring in order to fine-tune the
system in a way to merge the C2−C6 and concerted TSs.
Received: January 9, 2014
Published: March 12, 2014
© 2014 American Chemical Society
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Scheme 1. C2−C6/DA Reaction of Enyne−Allenes (Numbering Started from Alkyne Terminus)
Indeed, we found no separate saddle point for a concerted TS,
but two C2−C6 TSs that are separated by a significant energy
barrier. While the product ratios are no longer temperature
independent, the sum of experimental and computational
evidence suggests that the enyne−allenes of this study undergo
thermal DA cyclization via two distinct stepwise reaction
pathways, which share a common intermediate well without
mixing. We analyzed the current situation without trajectory
computations, relying on DFT-computed ΔΔG⧧ and exper-
imental product ratios, by developing a straightforward assay
that allows to quantify the amount of molecules following
nonstatistical dynamics. The results suggest that the percentage
of dynamically reacting molecules only slightly depends on the
depth of the intermediate well but more on ΔΔG⧧ of the C2−
C6 and the follow-up transition states.
Scheme 2. Synthesis of Enyne−Allenes 11a−c
RESULTS AND DISCUSSION
■
Synthesis. In order to address the above mechanistic
scenario, we decided to modulate the energy barrier of the
second TS in the two-step cyclization while keeping the energy
of the initial C2−C6 TS and the surface topology as constant as
possible. Thus, enyne−allenes 11 were designed with either
different groups R or R1 in a way to generate different degrees
of hindrance during rotation of the probing aryl ring. We chose
methoxy and N,N-dimethylamino substituents as group R and
TMS and tBu units as R1. Enyne−allenes 11a−c were prepared
following a reported procedure (Scheme 2).14 The propargyl
acetates16 10a,b were reacted with the corresponding
arylmagnesium bromide in the presence of ZnCl2 and Pd(0)
to afford 11a−c in excellent yields.17
of the allene unit (CCC) located at ∼210 ppm in the 13
C
NMR spectrum.
Thermolysis. To monitor the effect of temperature on the
product ratio, thermolysis of enyne−allenes 11a−c was
performed in dry and degassed toluene at different temper-
atures using a thermostat operating at a temperature constancy
of 0.1 °C (Scheme 3). Differential scanning calorimetry
(DSC) measurements revealed that the onset temperatures of
enyne−allenes 11a−c are in the range of 87−112 °C allowing
the thermolysis to be carried out from 30 to 100 °C while
maintaining partial conversion. The constitutionally isomeric
products 12 and 13 were separated by a long-bed flash column
chromatography and characterized with the help of IR, 1H, 13C,
1
All compounds were fully characterized by IR, H, 13C, and
2D NMR (1H−1H COSY) spectroscopy as well as elemental
analysis. Furthermore, formation of the enyne−allenes from the
corresponding propargyl acetates was confirmed by following
1
1
and H−1H COSY NMR spectroscopy and elemental analysis.
the disappearance of the H NMR signal of CH3CO− at 2.0
ppm and appearance of the highly characteristic carbon signal
In 1H NMR, the main characteristic signal of group R in 12a−c
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Scheme 3. Thermolysis of Enyne−Allenes 11a−c
is diagnostically upfield shifted by 0.90−0.46 ppm in
comparison to that in 13a−c because of the adjacent π-
electronic shielding effect of the phenyl ring.
corrected exchange functional, in conjunction with the Lee−
Yang−Parr nonlocal correlation functional (BLYP)19 using the
moderate 6-31G* basis set20 along with broken spin symmetry
(involving the mixing of the HOMO and LUMO to break the
spin and spatial symmetries), was found to generate rather
reliable results in Schmittel (C2−C6) and Myers−Saito (C2−
C7) cyclizations of enyne−allenes as documented by Schreiner
et al..21 Therefore, we utilized the (BS)-UBLYP method in
conjunction with the 6-31G* basis set to optimize all stationary
points in the gas phase. Calculated minima and first-order
saddle point structures were verified by analyzing their
harmonic vibrational frequencies via their analytical second
derivatives (NIMAG = 0 and 1, respectively). Free energies
including unscaled zero-point vibrational energies after thermal
correction are given in kcal mol−1 relative to the corresponding
starting materials. All TSs were optimized using the qst2 and
qst3 method as implemented in Gaussian 09 and the lowest
energy TSs were taken into consideration in each case. Free
energy values at 25 °C are provided for all stationary points of
the cyclization of 11a−c (Figure 1). A thermochemical analysis
of the optimized structures at different temperatures reveals
that the free energy differences are insignificant within 30−100
°C.
The computed reaction profiles of 11a−c (Figure 1) confirm
a stepwise thermal cyclization via the diradical 11a-cINT as the
minimum energy path (MEP). For all systems, the free energy
of the C2−C6 TSs is higher than that of follow-up TSs; hence,
according to TST the C2−C6 TS is the rate-limiting step and
the second TS the product-determining step. Notably, though,
the computed data on the final ring closure of the diradical, see
11aTS2⧧, 11cTS2⧧ vs 11aTS2′⧧, 11cTS2′⧧, predict a
preferential formation of the less hindered products 13a and
13c, quite in contrast to the experimental outcome. Such
finding is not in line with TST but may hint the involvement of
nonstatistical dynamics in the product formation. In contrast,
the preferential formation of the more hindered product 12b is
predicted by the computational data, but there is a large
discrepancy between experimental and computed ratio.
To obtain further insight, partial potential energy surfaces
(PES) were computed for enyne−allene 11a (Figures 2 and 3).
The first PES was obtained by varying the C2−C6 distance and
C6−C7−C8−C9 dihedral angle sampling 221 points (13 × 17)
at the (BS)-UBLYP/6-31G* level (Figure 2). It shows a broad
single well for the diradical intermediate with the energy only
raising by 0.35 kcal mol−1 above the lowest energy point when
the dihedral angle C6−C7−C8−C9 changes between 70 and
125° (Figure 4, left). Interestingly, the two initial C2−C6 TSs
on the PES, 11aTS1⧧ and 11aTS1′⧧, are separated by a
significant energy barrier of ca. 5.85 kcal mol−1 (observed from
11aTS1⧧ in the PES scan, Figure 2). The C6−C7−C8−C9
dihedral angle was initially chosen to encompass all of the
stationary points in one surface, but the scan started to produce
an erroneous warping near the second TSs. In order to avoid
problems with the surface topology, we chose the C2−C6 and
Similar to results in another C2−C6/DA reaction,14 in all
thermolyses astonishingly the seemingly more hindered
product 12 is formed preferentially, a finding that suggests
the involvement of nonstatistical dynamics in guiding the
product ratio 12:13 (Table 1). However, in all present cases,
Table 1. Experimental Product Ratios 12:13 in the
Thermolysis of Enyne−Allenes 11a−c
b
c
temp
(°C)
time
(min)
exptl ratio
12:13
nonstatistical dynamics
(%)
a
compd
11a
30
40
50
60
70
80
90
100
30
40
50
60
70
80
90
100
40
50
60
70
80
90
100
1020
780
420
180
75
1.48:1
1.62:1
1.64:1
1.65:1
1.62:1
1.60:1
1.58:1
1.56:1
1.72:1
1.69:1
1.66:1
1.64:1
1.63:1
1.61:1
1.57:1
1.54:1
2.50:1
2.46:1
2.38:1
2.29:1
2.21:1
2.11:1
2.06:1
71
75
76
76
76
76
76
76
71
72
72
72
72
72
73
74
85
86
85
85
84
83
70
50
40
11b
1020
780
420
180
75
70
50
40
11c
720
420
180
75
70
50
30
83
a
b
Tonset (DSC) = 90 °C (11a), 87 °C (11b), and 112 °C (11c). Ratios
were determined from the proton NMR spectrum of the crude
product after thermolysis with experimental accuracy of 0.03.
c
Calculated using eqs 1 and 2.
the product ratio is not temperature independent, at first a
seemingly convincing argument not to consider nonstatistical
dynamics. Indeed, thermolysis of 11b,c shows a continuous
decrease in the product ratio 12/13 when increasing the
temperature from 30 to 100 °C. Differently, the ratio 12a/13a
first augments from 1.48 to 1.65 in the temperature range from
30 to 60 °C but then equally decreases continuously with
increasing temperature.
Computation. To evaluate the positional selectivity and the
origin of the temperature dependence of the product ratio
12:13 in the thermal cyclization, reaction profiles were
computed using DFT as implemented in the Gaussian 09
program package.18 The unrestricted Becke pure gradient-
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Figure 1. Reaction profile for the thermal cyclization of 11a−c in the gas phase at the (BS)-(U)/BLYP/6-31G(d) level. Bond distances (Å) are
shown in italics, and relative free energies at 25 °C are given in kcal mol−1. Numerical values in black, blue, and red correspond to 11a, 11b, and 11c.
TS1 and TS2 refer to the initial C2−C6 and diradical closure transitions states, respectively, and INT to the diradical intermediate. The primed
designations relate to attack at the less hindered aromatic position of the probing ring.
Figure 2. 2D relaxed potential energy surface scan of C2−C6 distance from 2.8 to 1.4 Å with respect to C6−C7−C8−C9 dihedral angle from 16 to
176° (0.1 Å × 10° grid size).
C1−C10 distances for a second surface scan (Figure 3). The
C2−C6 distance was decreased from 2.5 to 1.2 Å and the C1−
C10 distance from 3.7 to 1.3 Å with the step size of 0.1 Å in
both cases. This partial PES reveals the topology about the
Diels−Alder product 11aP′ encompassing additionally the
stationary points 11aTS1′⧧, 11aTS2′⧧, and 11aINT.
mol−1). The two follow-up TSs, 11aTS2⧧ and 11aTS2′⧧,
connecting the product to the diradical intermediate were also
located. Here, the situation is inverted with transition state
11aTS2′⧧ (ΔG = 15.42 kcal mol−1) being 1.06 kcal mol−1
lower in energy than 11aTS2⧧ (ΔG = 16.48 kcal mol−1).
Accordingly, diradical 11aINT being located at 13.30 kcal
mol−1 in a shallow well, is separated from products 11aP and
11aP′ by small barriers ΔG⧧ = 3.18 and 2.12 kcal mol−1,
Transition state 11aTS1′⧧ (ΔG = 20.19 kcal mol−1) is 0.77
kcal mol−1 higher in energy than 11aTS1⧧ (ΔG = 19.42 kcal
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Figure 3. 2D relaxed potential energy surface and corresponding contour plot generated by the scan of C2−C6 distance from 2.5 to 1.2 Å with
respect to C1−C10 distance from 1.3 to 3.7 Å (0.1 Å × 0.1 Å grid size).
Figure 4. 1D scan of C6−C7−C8−C9 dihedral angle keeping the C2−C6 distance fixed at 1.52 Å which is the intermediate distance (left side). 1D
scan of C2−C6 distance from 1.94 to 1.70 Å with 0.02 Å interval (right side). Red dots are the projection of the black points on the plane counting
for two distance coordinates.
respectively, suggesting preferential formation of 13a via 11aP′.
Clearly, the experimental finding with 12a formed preferentially
over 13a is not in accord with a TST-controlled product
formation from the singlet intermediate 11aINT. According to
the computations, a concerted pathway can also be excluded
because no concerted TS for 11aP was located; each attempt
afforded the optimized initial C2−C6 TS. Likewise, the PES
(Figure 3) does not show any separate saddle on the surface
corresponding to direct conversion to 11aP′. Summing up the
experimental and computational results on 11a there is an
apparent difficulty to explain the results in a consistent manner
using TST.
A more detailed analysis of the surface (Figure 3, black
arrow) reveals that the MEP starting from 11aTS1′⧧ leads
initially to a decreasing but later to an increasing C1−C10
distance on the way to diradical intermediate 11aINT.
Therefore, the MEP toward the intermediate involves a
brusque bending on the surface. In general, after crossing a
TS, molecules transform their potential energy into kinetic
energy while following the reaction coordinate along the
steepest descent path. Here, however, the bend of the MEP
requires a sudden reversal of momentum of all atoms to reach
the diradical minimum. As any bent path is very difficult to
follow by molecules that have enough kinetic energy, they will
rather continue their initial momentum and even cross follow-
up barrier(s) if their kinetic energy is sufficient.
Although the PES toward formation of 11aP was not
determined with respect to C2−C6 and C1−C9 distance
coordinates, the PES with respect to the C2−C6 distance and
C6−C7−C8−C9 dihedral angle (Figure 2) equally showed a
bending in the MEP toward formation of 11aINT after crossing
11aTS1⧧. A better visualization of the bending is provided by a
1D scan of the C2−C6 distance from 1.94 to 1.70 Å at 0.02 Å
intervals (Figure 4, right) revealing the C1−C9 distance to
decrease initially from 3.40 Å (at 11aTS1⧧, Figure 5) to 3.13 Å
when the C2−C6 bond distance approaches 1.76 Å. At that
point, the C1−C9 distance starts to lengthen, finally affording
the intermediate with a C1−C9 distance at 3.41 Å. If dynamics
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simplified model systems, but clearly in the present case each
substituent at the enyne−allenes matters.
Apart from costly dynamic trajectory computations, other
protocols, such as the canonical competitive nonstatistical
model (CCNM) by Truhlar22 and the RRKM-Master equation
as elaborated by Glowacki,8d,23 equally allow rationalization of
the branching in nonstatistical reactions, although their validity
is still under scrutiny.3d Our protocol,24 in contrast, offers a very
simple way to approximate the fraction of molecules that follow
nonstatistical dynamics by comparing the experimental ratios
with the statistical partitioning as follows:
XnsQ1 + XsQ2 = Q exp
(1)
Xns + Xs = 1
(2)
Here, Xns = mole fraction of molecules following the
nonstatistical (ns) dynamically controlled pathway; Xs = mole
fraction of molecules following the statistically (s) controlled
pathway; Q1 = partitioning created at the initial (C2−C6) TSs;
Q2 = partitioning created at the second TSs; Qexp
=
experimental product ratio.
In the thermolysis of enyne−allene 11a, for instance, the
experimentally observed product ratio 12a:13a is 1.48:1 at 30
°C. If all molecules continued with conservation of momentum
through the diradical well after crossing the two initial C2−C6
TSs, 11aTS1⧧ and 11aTS1′⧧, then according to computational
results the observed product ratio would be 3.63:1 (ΔΔG⧧ =
0.77 kcal mol−1). Using eqs 1 and 2, 29% of the reacting
molecules do not follow dynamic behavior in the intermediate,
but instead partition in a ratio of 1:6.25 to afford 11aP (4%)
and 11aP′ (25%). With higher temperature, the percentage of
molecules following nonstatistical dynamics increases very
slightly from 71% (30 °C) to 76% (50 °C) but then remains
almost constant up to 100 °C.
In order to test our theoretical model in another system, we
chose a methoxy group as R in the probing aryl ring of 11b.
Surprisingly, while the two initial TSs, 11bTS1⧧ and 11bTS1′⧧,
are located at similar free energies of 20.93 and 21.05 kcal
mol−1 (at 25 °C), respectively, the relative free energy of the
follow-up TSs favors formation of the more hindered product
via 11bTS2⧧ at 16.13 kcal mol−1, which is 0.99 kcal mol−1
lower than that of 11bTS2′⧧. The structural origin favoring the
more hindered product (Figure 6) is hidden in the C2−C1−
C11−C12 dihedral angle of 11bTS2⧧ and 11bTS2′⧧. Since the
phenyl ring at the alkyne terminus provides stabilization to the
reacting σ-radical site, any deviation of the C2−C1−C11−C12
dihedral angle from 90° will destabilize the system. In
11bTS2′⧧, the dihedral angle deviates by 18.5°, which is
more than in 11bTS2⧧ with a deviation of only 7.1°. As one
would expect, the longer C1−C9 bond distance (2.57 Å) in
11bTS2⧧ allows the C2−C1−C11−C12 angle to stay closer to
90°. In contrast, despite the C1−C9 and C1−C10 bond
distances in 11aTS2⧧ and 11aTS2′⧧, respectively, being rather
long at 2.58 and 2.56 Å, the C2−C1−C11−C12 dihedral angle
deviates more in 11aTS2⧧ (by 25.1°) than in 11aTS2′⧧ (by
15.1°) due to excessive steric repulsion between the N,N-
dimethylamino and the adjacent phenyl group in the former.
In accordance with TST, one would expect a product ratio
12b:13b of 5.18:1 (calculated ΔΔG⧧ = 0.99 kcal mol−1) at 30
°C, but we observe only a ratio of 1.72:1, which is rather close
to the partitioning at the initial TSs (1.22:1 as derived from
ΔΔG⧧ = 0.12 kcal mol−1). The large deviation from the TST-
predicted ratio provides some support for the occurrence of
Figure 5. Optimized TSs, 11aTS1⧧, 11aTS1′⧧, 11cTS1⧧, and
11cTS1′⧧ with distances in angstroms.
plays a role after crossing the initial TS, 11aTS1⧧, then one
would expect the product to form in a direct continuation of
the primary momenta of atoms C1 and C9. In such case, the
decrease of the C1−C9 distance would continue even after
reaching a C2−C6 bond distance of 1.76 Å leading to a
deviation from the MEP. On the other hand, increasing the
C1−C9 distance along the calculated MEP as shown in Figure
4 (right) will lead to a loss of the momenta of atoms C1 and
C9, thus enforcing a TST-controlled product formation.
Molecules that originally cross 11aTS1⧧ have little
probability to produce 11aP′ if they are not relaxed in the
intermediate well because the required large change of the C6−
C7−C8−C9 dihedral angle has to be a sluggish process due to
the resultant large movement of a considerable number of
atoms. Such a process needs vibrational energy redistribution in
the intermediate during the overall process 11aTS1⧧ → 11aP′.
To sumarize the results from inspecting the PES, a certain
fraction of the molecules crossing the transition state 11aTS1⧧
will go to product 11aP directly under dynamic control, while
the remaining fraction of molecules will relax vibrationally in
the intermediate well. The relaxed molecules will statistically
partition to produce both 11aP and 11aP′ according to the two
follow-up TSs 11aTS2⧧ and 11aTS2′⧧. Similarly, those
molecules passing initially through 11aTS1′⧧ will produce
11aP′ preferably in a nonstatistical dynamic scenario. Assuming
a nonstatistical setting on the basis of PES considerations may
seem daring, but in related C2−C6 cyclizations, various pieces
of evidence, such as temperature-independent product ratios,14
intra- and intermolecular kinetic isotope effects,5a,13 and
trajectory computations,13 have substantiated the occurrence
of nonstatistical dynamics.
Considering a dynamic model, a huge number of dynamic
trajectory calculations would be necessary to describe the
motion, momenta, and fraction of molecules following
nonstatistical dynamics. Thus, one would like to reduce the
costs of the molecular dynamics computations by turning to
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Figure 7. 1D scan of C1−C2−C3−C4 dihedral angle of the
(cyclopenta-1,3-dien-1-yl)trimethylsilylbenzyl radical and (cyclopen-
ta-1,3-dien-1-yl)-tert-butylbenzyl radical using the UBLYP/6-31G*
method.
structure than the TSs of 11a (Figure 5). As a result, the two
C2−C6 TSs, 11cTS1⧧ and 11cTS1′⧧, are 2.06 and 2.13 kcal
mol−1 higher in energy than those of 11aTS1⧧ and 11aTS1′⧧,
respectively, and the free energy differences ΔΔG⧧ for
11aTS1⧧→11aTS2⧧, 11aTS1′⧧→11aTS2′⧧, 11cTS1⧧→
11cTS2⧧, and 11cTS1′⧧→11cTS2′⧧ are −2.94, −4.77, −3.93,
and −5.46 kcal mol−1, respectively. These values indicate that,
in the case of 11c, the molecules receive much more kinetic
energy after passage through the initial TSs so that overcoming
the second TSs is easier, generating more dynamically reacting
molecules in comparison to enyne−allene 11a.
In systems 11a,c, we thus see an increasing contribution of
nonstatistically reacting molecules with increasing free energy
gap between the initial TS and second TS. The higher fraction
of dynamically reacting molecules in 11c, despite a much
deeper intermediate well, suggests that the well depth in
systems with PES such as in Figure 2 is not the decisive factor
determining the amount of nonstatistical dynamics.
Nonstatistical Dynamic vs Asynchronous Concerted
Reaction Path. Finally, one needs to ask whether the above
suggested nonstatistical dynamic reaction course is any different
from a largely asynchronous concerted one. According to
IUPAC, a concerted reaction is a single-step reaction through
which reactants are directly transformed into products without
involvement of any intermediates. But would this definition still
be of utility when the intermediate zone is passed only at the
rim with conservation of momentum?
An asynchronous concerted reaction is realized as a two-stage
reaction in which the changes in bonding occur in two distinct
stages, some changes mainly taking place between the reactant
and transition state, the others mainly between the transition
state and product.25 For example, 11aTS′⧧ could indeed serve
for a highly asynchronous concerted reaction path if the
trajectories would exclusively develop along the ridge that is
depicted as a blue line in Figure 8. Despite the principal
possibility, the highly asynchronous concerted reaction path
seems to be very unlikely as the initial TS exhibits no
displacement amplitude toward C1−C10 bond formation in
the imaginary frequency. Rather, it describes almost exclusively
the C2−C6 displacement. Hence, the more like scenario
Figure 6. Optimized TSs, 11aTS2⧧, 11aTS2′⧧, 11bTS2⧧, and
11bTS2′⧧ with distances given in angstroms.
nonstatistical dynamic behavior equally in this case. In
comparison with enyne-allene 11a, the depth of the
intermediate well in the reaction of 11b is more pronounced
thus opposing dynamics although due to the initial, higher
energy C2−C6 TSs more kinetic energy should be available for
molecules on the downhill side. These two opposing effects in
summary provide a little less dynamic system (71−74%
nonstatistical dynamics) using eqs 1 and 2.
In order to generate an even deeper well at the intermediate,
we replaced the trimethylsilyl with a tert-butyl group at the
allene terminus. This suggestion emerged from the computed
energy profile about the dihedral angle in a model benzyl
radical (Figure 7) revealing that the rotation of an adjacent aryl
ring is sterically less hindered with a trimethylsilyl than a tert-
butyl group due to the longer C−Si bond distance. Thus,
enyne−allene 11c was prepared and studied. Figure 7 shows a
double-minimum profile for the TMS-substituted molecule,
while only a single minimum for the intermediate well
(11aINT and 11bINT) of enyne-allenes 11a,b is observed.
In case of enyne−allene 11c, the singlet intermediate
11cINT is placed at 11.42 kcal mol−1 and the two follow-up
TSs, 11cTS2⧧ and 11cTS2′⧧, are located at 17.55 and 16.86
kcal mol−1. The follow-up TSs, 11cTS2⧧ and 11cTS2′⧧, are
thus 1.07 and 1.44 kcal mol−1 higher in energy than those of
11aTS2⧧ and 11aTS2′⧧, respectively, whereas 11cINT is lower
in energy than 11aINT (13.30 kcal mol−1). Clearly, the
cyclization of enyne−allene 11c proceeds through a much
deeper energy well for the diradical intermediate. Surprisingly,
though, the percentage of dynamically reacting molecules is
much higher (83−86%) for 11c than for 11a,b. A closer
analysis of the stationary points reveals significantly longer
bond distances in 11cTS1⧧ (dC1−C9 = 3.48 Å) and 11cTS1′⧧
(dC1−C10 = 3.48 Å), respectively, than in 11aTS1⧧ (dC1−C9
=
3.40 Å) and 11aTS1′⧧ (dC1−C10 = 3.35 Å) indicating that the
C2−C6 TSs of 11c are more displaced from the product
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Figure 8. 2D potential energy surface generated by scanning the C2−C6 distance from 2.0 to 1.4 Å and the C1−C10 distance from 2.1 to 3.7 Å (0.1
Å × 0.1 Å grid size) (truncated Figure 3).
demands that after crossing the initial TS, a C2−C6 bond
quickly forms subsequently supporting C1−C10 bond
formation by maintaining the initial momentum. Finally, a
nonstatistical dynamic course (red line in Figure 8) evolves for
the reaction.
ACKNOWLEDGMENTS
■
We are indebted to the Deutsche Forschungsgemeinschaft for
financial support and to Prof. Jaquet (Siegen) for helpful
support concerning the computational infrastructure.
In summary, we present three closely related enyne−allenes
that follow a stepwise mechanism for their thermal cyclization.
According to an inspection of the PES, two noninteracting
reaction paths share a common intermediate well without
losing memory of their previous transition-state passage
resulting preferentially in a nonstatistical dynamic behavior.
An equation connecting ΔΔG⧧ of the first and second
transition states with the experimentally determined product
ratio allows us to calculate the amount of nonstatistical
dynamics in the overall process, a method that could also be
of use in other systems. The thus-determined percentage of
dynamically reacting molecules only slightly depends on the
depth of the intermediate well but rather more on ΔΔG⧧
between the initial and the follow-up transition states.26 Such
findings should raise the awareness toward the importance of
conservation of momentum15a and of nonstatistical dynamics in
reaction mechanisms.
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ASSOCIATED CONTENT
■
S
* Supporting Information
Synthetic and experimental procedures are described in
addition to spectroscopic and computational data. This material
is available free of charge via the Internet at http://pubs.acs.org.
AUTHOR INFORMATION
■
Corresponding Author
*E-mail: schmittel@chemie.uni-siegen.de.
Notes
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