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Published: November 30, 2011
r 2011 American Chemical Society 319
dx.doi.org/10.1021/jp209360u | J. Phys. Chem. A 2012, 116,
319332
ARTICLE
pubs.acs.org/JPCA
Tunneling in Hydrogen-Transfer Isomerization of n-Alkyl
RadicalsBaptiste Sirjean,*,, Enoch Dames, Hai Wang,*, and Wing
Tsang
Department of Aerospace and Mechanical Engineering, University
of Southern California, Los Angeles, California 90089-1453,United
States
National Institute of Standards and Technologies, Gaithersburg,
Maryland 20899, United States
bS Supporting Information
1. INTRODUCTION
Understanding the thermal decomposition reactions of
alkylradicals is critical to the development of combustion models
ofhydrocarbon fuels. In n-alkane oxidation at high
temperatures,alkyl radicals are initially formed by CC or CH ssion
in thefuel molecule or H-abstraction reaction by a free radical.
Smallalkyl radicals (Ce3) decompose mainly by CC -scission, asthe
12 and 13 hydrogen shifts can occur only through three-or
four-membered ring transition state structures with largeenergy
barriers due to strain energies.1 For large alkyl radicals,the
thermal decomposition is complicated by internal hydro-gen
shifts.27 These isomerization processes occur throughcyclic
transition state structures. Internal isomerization can
occurthrough ve-, six-, or seven-member ring transition state
struc-tures, which may be denoted as 14, 15, and 16 hydrogenshifts,
respectively. Some of these transition state structures
areassociated with ring strain energies, while others are
practicallyunstrained. H-atom shifts are the predominant
unimolecularreactions under lower temperature conditions. As the
tempera-ture is increased, -scissions become increasingly
important,but H-atom shift remains critical to the dissociation of
alkylradicals and especially the product distributions resulting
fromthe -scission processes. The combustion chemistry of olens
and smaller alkyl radicals, formed in the decomposition
process,play a key role in the reactivity of the fuel and formation
ofpollutants and polycyclic aromatic hydrocarbon (PAH)
andsoot.812
The basic problemwith studying alkyl isomerization processesis
that they involve large organic moieties (C5 or above). Exceptat
the lowest temperatures, -scission will always make contribu-tions.
Unless careful attention is paid to setting the reactionconditions
to minimize mechanistic artifacts, it is usually dicultto derive
truly quantitative kinetic parameters. In the case of singlepulse
shock tube experiments, the isomerization rate constantsderived are
all dependent on the initial -scission rate constants,which, in
turn, are derived from literature values. Thus, theuncertainty in
the -scission rate constants inevitably propagatesinto the
isomerization rate constants. For this type of tightlycoupled
unimolecular reaction involving competitive reactionpaths,
theoretical reaction kinetics is often necessary to interpretthe
experimental data.
Received: September 28, 2011Revised: November 22, 2011
ABSTRACT: The role of quantum tunneling in hydrogen shift
inlinear heptyl radicals is explored using multidimensional,
small-curvature tunneling method for the transmission coecients and
apotential energy surface computed at theCBS-QB3 level of
theory.Several one-dimensional approximations (Wigner, Skodje
andTruhlar, and Eckart methods) were compared to the
multidimen-sional results. The Eckart method was found to be
sucientlyaccurate in comparison to the small-curvature tunneling
results fora wide range of temperature, but this agreement is in
fact fortuitousand caused by error cancellations. High-pressure
limit rate con-stants were calculated using the transition state
theory with treatment of hindered rotations and Eckart transmission
coecients for allhydrogen-transfer isomerizations in n-pentyl to
n-octyl radicals. Rate constants are found in good agreementwith
experimental kinetic dataavailable for n-pentyl and n-hexyl
radicals. In the case of n-heptyl and n-octyl, our calculated rates
agree well with limited experimentallyderived data. Several
conclusions made in the experimental studies of Tsang et al.
(Tsang, W.; McGivern, W. S.; Manion, J. A. Proc.Combust. Inst.
2009, 32, 131138) are conrmed theoretically: older low-temperature
experimental data, characterized by small pre-exponential factors
and activation energies, can be reconciled with high-temperature
data by taking into account tunneling; at lowtemperatures,
transmission coecients are substantially larger for H-atom
transfers through a ve-membered ring transition state thanthosewith
six-membered rings; channelswith transition ring structures
involving greater than8 atoms can be neglected because of
entropiceects that inhibit such transitions. The set of
computational kinetic rates were used to derive a general rate rule
that explicitly accounts fortunneling. The rate rule is shown to
reproduce closely the theoretical rate constants.
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320 dx.doi.org/10.1021/jp209360u |J. Phys. Chem. A 2012, 116,
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Reactions involving the transfer of a hydrogen atom are
in-herently subject to quantum tunneling eect.13 The role of
tunnel-ing in n-alkyl radical isomerizations was recently
highlighted byTsang and co-workers.47 They showed that tunneling
must beconsidered to reconcile older data obtained at low
temperatureswith measurements made in their high-temperature
single-pulseshock tube. Low-temperature kinetic data have been
subject tocontroversies because of the unusually small
pre-exponential factors(A) that were dicult to rationalize with the
thermochemicalkinetic theory.14Directmeasurements of the
isomerization reactionrates of alkyl radicals are dicult, and
kinetic data were derivedusually from analyses of complex reaction
mechanisms. Moreover,most experiments were performed in the fallo
region and/orinvolved a chemically activated process. In such
cases, the high-pressure limit rate constant must be extrapolated
from the RiceRamspergerKasselMarcus (RRKM) theory and, for the
lateststudies, from Master Equation (ME) analysis.
Among all n-alkyl radicals, the unimolecular reactions
of1-pentyl and 1-hexyl have been studied most extensively in
thepast. Figure 1 presents selected literature rate constants.
End-renyi and Le Roy15 proposed the rst experimental rate
constantfor the gas-phase 14 hydrogen migration in n-pentyl
radicals:
1 C5H11 / 2 C5H11 R1They reported:
k1 s1 1:4 107e5440=T 1for temperatures 439 e Te 503 K at low
pressures. Within theframework of transition state theory, they
concluded that theunusual low pre-exponential factor can be
explained only byconsidering a quantum tunneling eect. Watkins16
derived a rateexpression from experiments performed 297 e T e 435 K
andpressures from 1 to 30 Torr:
k1 s1 3:3 108e7600=T 2Watkins also noted the unusually low A
factor of rate constantand proposed that errors resulting from
photochemical activationcould lead to an underestimation of the A
factor. In a follow-uppaper,17 Watkins proposed a rate expression
with a larger A factor
based on a newmechanistic interpretation of the data of
Endrenyiand Le Roy and proposed:
k1 s1 5:0 1011e10600=T 3Similar low A factor values were
observed from low-temperatureexperiments on n-hexyl
radicals.Watkins andOstreko18 proposeda rate expression for the 15
hydrogen migration in n-hexylradicals:
1 C6H13 / 2 C6H13 R2for temperatures from 352 to 405 K and a
pressure of 46 Torr:
k2 s1 2:0 107e4180=T 4The small A factor has been a subject of
debate in theliterature.19,20 In the late 1980s, Dobe et al.21
reported an experi-mental investigation of k2 for 300 e T e 453 K
and pressuresfrom 100 to 200 Torr and proposed a rate expression
that againfeatures a small A factor:
k2 s1 3:16 107e5840=T 5Using the same experimental approach,
these authors also pro-posed rate expressions for n-octyl radicals
isomerization withsimilarA factors.Marshall22 examined the thermal
decompositionof n-pentane in the temperature range of 737923 K and
pres-sures below 200 Torr. The rate expression for the 14
hydrogenshift of reaction R1 was derived from the distribution of
majorproducts as
k1 s1 9:1 1011e11800=T 6By considering earlier low-temperature
measurements, he pro-posed that
k1 s1 1:2 1011e10100=T 7for 438 e T e 923 K. Imbert and
Marshall23 followed a similarapproach to determine the
high-pressure limit rate constant15 hydrogen transfer in n-hexyl
radicals by n-hexane pyrolysis
Figure 1. Arrhenius plots of (a) 14 hydrogen shift in n-pentyl
radical, and (b) 15 hydrogen shift in n-hexyl radical.
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and proposed:
k2 s1 3:16 1010e8560=T 8for 723 e T e 823 K. Miyoshi and
co-workers24 derived theexpressions for k1 and k2 from an analysis
of shock-tube experi-ments for 900eTe 1400 K near the atmospheric
pressure usinga complex reaction mechanism. They proposed
expressions forthe high-pressure limit rate constants k from an
RRKM analysisof their experimental results along with earlier, low
temperaturedata. They included the tunneling eect via a Wigner
correctionwith the imaginary frequencies taken fromHF/6-31G(d)
calcula-tions. They proposed:
k1, s1 4:9 108T 0:846e9830=T 9and
k2, s1 6:7 107T 0:823e6570=T 10for 350 e T e 1300 K. The
somewhat strong and positivetemperature exponents point to an
apparent upward curvature ofthe Arrhenius form largely because of
the tunneling eect beingconsidered in the rate calculation.
Miyoshi and co-workers25 reported the rst direct measure-ment of
the rate coecients of 14 hydrogen shift in 1-pentylradical in the
temperature range of 440520 K and pres-sures from 1 to 7 Torr. From
an RRKM/ME analysis includingEckart tunneling, they evaluated the
high-pressure limit rate con-stant to be
k1, s1 1:9 1010e8870=T 11for 440 e T e 520 K and
k1, s1 2:4 103T2:324e8180=T 9:1
105e5300=T 12300eTe 1300 K. The use of a bi-Arrhenius form is
the result ofa large tunneling eect, leaving a high Arrhenius
curvature.
More recently, Tsang and co-workers carried out a series
ofsystematic studies on the thermal decomposition of n-pentyl,6
n-hexyl,7 n-heptyl,4 and n-octyl5 radicals in single-pulsed
shocktube for temperatures from 850 to 1050 K and pressures
between1.5 and 7 bar. From experimentally determined product
branch-ing ratios, they deduced high-pressure limit rate
expressions for14, 15, and 16 hydrogen transfers using an
RRKM/MEanalysis. In the case of n-octyl, they also explored a
hydrogenmigration channel through the eight-membered ring
transitionstate structure, but found that such a transition is
unimportantcompared to other reaction channels. Earlier, lower
temperaturedata on n-pentyl and n-hexyl were included in their
theoreticalanalysis considering Eckart tunneling. Notably, they
proposedthe high-pressure limit rate constant:
k1, s1 1:0 1012e11300=T 13for 14 hydrogen shift in 1-pentyl from
850 to 1000 K, and
k2, s1 1:8 102T2:55e5550=T 14for 15 hydrogen migration in
1-hexyl for 500 e T e 1900 K.
Theoretically, Viskolcz et al.1 computed the energy barriers
ofseveral isomerization reactions of linear alkyl radicals and
thebarrier of the 14 hydrogen shift in 1-pentyl at the
MP-SAC2//UHF/6-31G* level of theory. They discussed their results
within
the framework of strain energy in the transition state
structure.Using B3LYP/ccpVDZ calculations and transition state
theory(TST) on prototypal reactions, Matheu et al.26 proposed
raterules for 12, 13, 14, 15, and 16 hydrogen shifts in
alkylradicals. Hayes and Burgess27 calculated the energy barriers
ofhydrogen transfer in alkyl, allyilc, and oxoallylic radicals
usingseveral composite methods and showed that an
EvansPolanyicorrelation can be developed in the case of linear
alkyl radicals.Quantum tunneling eects were not considered in these
studies.Truong and co-workers28 studied the 1,4-intramolecular
hydro-gen migration in linear alkyl radicals using the class
transitionstate theory. They calculated the rate constant for the
referencereaction of n-C4H9 using canonical variational transition
statetheory (CVTST) with the small curvature tunneling
(SCT)correction. It is important to note here that SCT
approximationallows for a multi-dimensional treatment of tunneling
throughcomputationally expensive calculation of the Hessian for
numer-ous points along the reaction path. In their study, the
poten-tial energy surface (PES) was computed at the
CCSD(T)/cc-pVDZ//BH&HLYP/cc-pVDZ level of theory. They found
asignicant tunneling contribution at temperatures below 1000
K.Following their study on 14 hydrogen shift, they recentlyproposed
high-pressure limit rate expressions for 13 to 16hydrogenmigrations
in linear alkyl radicals.29 They described theisomerization
reactions of n-propyl to n-hexyl radicals using thesame
methodology. The rate constants for analogous reactionswere
proposed on the basis of these prototype reactions withinthe
framework of reaction class TST. Again, the tunneling eectwas found
to be less prominent above 1000 K. Additionally, theycompared
transmission coecients obtained from multidimen-sional SCT
tunneling with those obtained from the one-dimen-sional Eckart
function and showed that the Eckart tunnelingeect is more
pronounced than SCT tunneling below 400 K.
Jitariu et al.30 carried out direct dynamic calculations
withCVTST/SCT to study the decomposition and isomerizationpathways
of n-pentyl radicals. They reported dual-level calcula-tions at the
PUMP2-SAC/6-311G**///AM1 level of theory.Here, the triple slash
(///) denotes interpolating optimizedcorrections (IOC) in the VTST
calculations.31 In this method,the PES along the reaction path are
corrected using higher-levelvalues at selected points. The
Hessians, the potential energies,and moment of inertia along the
reaction path were computedat the AM1 level of theory.
UMP2/6-311G** geometries, vib-rational frequencies, moments of
inertia, and PUMP-SAC2/6-311G** energies of the stationary points
were then used toscale the low-level AM1 results. This rather
elaborate approachled them to conclude that tunneling is pronounced
at lowtemperatures in the 14 hydrogen shift of 1-pentyl. At 1000K,
however, their proposed rate constant is about a factor 3
largerthan experimental values of Tsang et al.6,24 and Yamauchi et
al.6,24
Zheng and Truhlar studied the 14H shift in the 1-pentyl
radicaland the 14 and 15 hydrogen transfers in the 1-hexyl
radicalusing CVTST/SCT theory.32 They calculated the PES
withseveral levels of density functional theory (DFT) (e.g.,
M06-2X/MG3S andM08-HX/cc-pVTZ+) andmultilevel methods forthe
stationary points. They used the interpolated variationaltransition
state theory by mapping (IVTST-M) in all of theirdynamics
calculations. This method gives PES data (energies,gradients, and
Hessian) along the full reaction path based on alimited number of
points calculated near the saddle point. Theycompared several
one-dimensional approximations (Wigner, zerocurvature tunneling,
and parabolic tunneling approximation) to
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322 dx.doi.org/10.1021/jp209360u |J. Phys. Chem. A 2012, 116,
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their multidimensional SCT results. They noted that at
lowtemperatures the discrepancies among these methods are
ratherlarge. An interesting point is that tunneling is more
important inthe case of 14 hydrogen transfer than for 15 shift, in
agree-ment with the work of Tsang and co-workers. Unfortunately,
thewidely used Eckart method was excluded from their study, andits
ability to reproduce results of higher dimensional
tunnelingtreatments remains unclear.
In the present study, we intend to address several
issuesregarding the role of tunneling in linear alkyl radical
isomeriza-tions.We compute the transmission coecients using
n-heptyl asthe model system employing a multi dimensional
treatment. Theresults allow us to assess accuracy of the various
one-dimensionalmethods for the calculation of transmission
coecients. Beyondthe n-heptyl radical, we use the rigid rotor
harmonic oscillator(RRHO) approximation with corrections for
hindered internalrotation (HR), the TST theory with a selected 1-D
tunnelingmethod, and we systematically determine the rate coecients
for14 to 16 H-shifts in n-alkyl radicals ranging from n-pentyl
ton-octyl. Note that as the size of the linear alkyl chain
increases,most favorable isomerizations (ve-, six-, and
seven-membertransition state structures) can involve hydrogen shift
betweentwo secondary radicals (e.g., 2-heptyl to 3-heptyl). It is
note-worthy to mention that there are almost no kinetic data
availablefor these types of processes. Finally, we discuss the role
oftunneling within the framework of correlations between struc-ture
and reactivity, and its impact on the uncertainty of ratecoecients
derived from theoretical calculations.
2. COMPUTATIONAL DETAILS
Electronic Structure Calculations. Potential energy andmolecular
properties of stationary points were calculated usingGaussian 03
revision B.0333 and QChem version 3.1.34 For allstationary and
critical structures, geometry optimizations wereperformed at the
B3LYP/6-311G(2d,d,p) level of theory.35,36
Frequency calculations were performed at the same level oftheory
for all optimized geometries to determine the nature ofstationary
points. The composite CBS-QB3 method was appliedfor all stationary
geometries and transition states.37 The CBS-QB3 model involves a
five-step calculation starting with ageometry optimization and a
frequency calculation (scaled by afactor 0.99) at the
B3LYP/6-311G(2d,d,p) level of theory,followed by single point
energy calculations at the CCSD(T)/6-31G(d0), MP4SDQ/cbsb4, and
MP2/cbsb3 and a completebasis set extrapolation to correct the
total energy. Dynamic calcu-lations along the reaction path were
performed at the B3LYP/6-311G(2d,d,p) level of theory. We found
that for reactionsexamined here, the energy barrier heights with
zero-point energy
(ZPE) were quite close at the B3LYP/6-311G(2d,d,p) and CBS-QB3
levels of theory (Table 1).Hindered Internal Rotations. The
description of isomeriza-
tion channels of large aliphatic chains requires an accurate
treat-ment of low-frequency internal rotations. Using the
harmonicoscillator (HO) approximation to describe these torsion
modescan lead to large errors in the partition function.
Vansteenkisteet al.38 calculated the thermodynamic properties of
n-alkanesusing a quantum-mechanical treatment and showed that
treatinghindered internal rotation is critical to obtain accurate
entropiesand heat capacities values. As hydrogen transfers occur
throughcyclic transition state structures, most of the internal
rotations ofthe initial reacting radicals are locked in the ring,
and the ratecoefficients will be strongly dependent on the entropy
variation.As an example, Vansteenkiste et al. reported for
n-heptane adifference of 9.3 cal/(mol K) at 1000 K in entropy
betweencalculations using the assumption of harmonic oscillator
(HO)and a more precise quantum mechanical treatment. The
HOapproximation underestimates the entropy change. In the pre-sent
work, hindered internal rotations were treated using thefollowing
procedure. First, the potentials of each internal rotationof
1-pentyl, 2-pentyl, and 3-pentyl radicals were calculated at
theB3LYP/6-311G(2d,d,p) level of theory using a relaxed energyscan.
The energy barriers for these hindered internal rotors werethen
calculated at the CBS-QB3 level of theory. The character-istics of
the rotational potentials and the barriers of rotationobtained were
used to correct the HO partition function usingPitzer andGwinn
tabulations.39 Rotational potential functions andbarriers for each
internal rotor in n-pentyl were used for similarinternal rotors
found in largest alkyl radicals. Reduced moments ofinertia for
internal rotations were calculated for each species
usingB3LYP/6-311G(2d,d,p) geometries with the method of
Pitzerimplemented in ChemRate.40,41 For cyclic transition states,
thevibrational modes of the cyclic part of themolecular structure
weredescribed within the HO approximation, and the lateral
alkylgroup internal rotations were treated as hindered rotors (HR)
viarelaxed scans (with cyclic bond lengths frozen) in the critical
geo-metries for 1-pentyl to 2-pentyl and 1-hexyl to 3-hexyl.
Parametersfor HR corrections for these two critical geometries were
used forall other similar internal rotors found in the saddle point
geome-tries of larger alkyl radicals. Hindrance potentials and
barrierheights are given in the Supporting Information.Transmission
Coefficient.Within a canonical TST or VTST
framework, quantum tunneling is taken into account by
atemperature-dependent transmission coefficient k(T):
kT kTAeE=T 15Within the framework of the RRKM theory and to
account fortunneling, Miller42 proposed one-dimensional tunneling
prob-ability P(E,J) in the sum of states N(E,J) of the transition
state:
NE, J nPE Eqn, J 16
where nq is the nth vibrational energy level and J is the
angular
momentum. The standard microcanonical rate expression
iscalculated using N(E,J):
kE, J NE, JhFE, J 17
where h is the Planck constant and F(E,J) is the density of
energystates of the reactant. Note that k(T) can be calculated from
P(E,J)
Table 1. Critical Energies (kcal/mol) Computed for H-AtomShifts
in n-Heptyl Radicals at 0 K
B3LYP/
6-311G(2d,d,p)a CBS-QB3a G3MP2B327
7ps (1-heptyl/ 2-heptyl) 14.1 14.7 15.86ps (1-heptyl/ 3-heptyl)
14.6 15.0 16.15ps (1-heptyl/ 4-heptyl) 21.3 22.0 23.55ss (2-heptyl/
3-heptyl) 23.2 22.8 23.9
aThis work.
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323 dx.doi.org/10.1021/jp209360u |J. Phys. Chem. A 2012, 116,
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by integrating over the Boltzmann-weighted energy
distributionand that k(T) also includes nonclassical reflection
effects.For a one-dimensional system, the transmission coecient
may be calculated by solving the Schrodinger equation with
anappropriate potential function. In this approach, the
potentialenergy surface is tted with a function for which the
tunnelingprobabilities are known analytically. The simplest method
is thatof Wigner43 who assumed a parabolic function for the
PESleading to the transmission coecient:
kT 1 124
hImqkBT
" #218
where q is the imaginary frequency corresponding to thereaction
coordinate, and kB is the Boltzmann constant. Skodjeand Truhlar44
proposed an improvement over the Wignermethod by treating the
energy barrier as a truncated parabola.The Eckart function is
particularly useful as it provides realistic
potential function that has the correct asymptotic properties
ofthe one-dimensional PES. Moreover, the parameters of theEckart
potential can be tted to reproduce the curvature at thesaddle point
and the exothermicity of the reaction.45 This one-dimensional
method is widely used in the theoretical combustionkinetics because
they are rather inexpensive computationally yetaccurate enough for
the temperatures of interest.More sophisticated methods to
calculate the transmission co-
ecient have been proposed by Truhlar and co-workers.
Parti-cularly, they developed semiclassical methods to calculate
trans-mission coecients that take into account the
multidimensionalnature of tunneling; that is, tunneling paths
deviate drama-tically from the reaction path. Thesemethods are
computationallyexpensive as they required detailed information
(e.g., Hessians)along the reaction path. For reactions with a path
curvature as-sumed to be small, the small curvature tunneling (SCT)
methodwas proposed.46,47 It allows for corner-cutting approximately
onthe concave side of the turning points of the vibrations
transverseto theminimum energy path. For larger curvatures,
complicationsarise as contributions from tunneling into excited
states may haveto be considered. In that case, the large curvature
tunneling (LCT)method can be used to calculate transmission
coecients.48
For the n-heptyl radicals, we carried out calculations
oftransmission coecients with the SCT method. The minimumenergy
paths (MEP) were determined at the B3LYP/6-311G(2d,d,p) level of
theory with direct dynamics calculations. MEPcalculations were
performed in Cartesian coordinates with a stepsize of 0.01 using
the Euler steepest-descents integrator. Thisstep size was found to
be suciently small to converge thereaction path and the
transmission probabilities in the range ofreaction coordinates
ranging from 2.0 to +2.5 for 16 and15 hydrogen shifts, and2.8 to 3
and2.5 to +3 for 14and 25 hydrogen migrations, respectively. All of
the SCTtransmission coecients reported here were calculated
usingthe POLYRATE 9.749 and GAUSSRATE 9.750 codes. LCTcalculations
were also performed to determine if a large curvaturemechanism
could be important for the range of reaction co-ordinate studied
here. We do not expect to obtain accuratetransmission coecients
from the LCT method as the reactioncoordinate ranges may be too
small for a proper description of alarge curvature path, and it may
be necessary to include tunnelingin excited states to obtain
accurate results.
One-dimensional transmission coecients were also com-puted using
Wigner and Skodje and Truhlar approximationsdirectly from CBS-QB3
results on stationary points of the PES.The one-dimensional Eckart
transmission coecients were calcu-lated using ChemRate software
where the characteristic lengthof the Eckart function is obtained
from forward and reversebarrier heights (E1 and E1) at 0 K along
with the imaginaryfrequency (i) of the transition state using the
equations reportedby Johnston and Heicklen.51 Note that in
Chemrate, canonicaltransmission coecients are calculated by
integrating microca-nonical transmission probabilities over the
Boltzmann-weightedenergy distribution. Lennard-Jones parameters
were taken fromthe JetSurF version 1.0 transport database.52,53
Calculated char-acteristic lengths, as well as parametersE1, E1,
and i, are given inthe Supporting Information.
3. RESULTS AND DISCUSSION
Potential Energy Surfaces. The potential energy for hydro-gen
shift in n-heptyl is presented in Figure 2. As the heptylradicals
allow for secondary to secondary H shift, we will followthe
notation proposed byHardwidge et al.19 that explicitly retainsthis
information. Within this framework, a particular hydrogentransfer
will be described as an iab process, where i is the ring sizeof the
cyclic transition state structure, a refers to a primary
orsecondary radical (noted p or s) for a reactant, and b is p or s
for aproduct. Within this nomenclature, the 14 hydrogen transfer
of1-pentyl radical would be referred to as a 5ps isomerization.It
is known that transmission coecients are sensitive to the
barrier heights. Therefore, to be able to analyze the SCT
trans-mission coecients obtained with molecular parameters
ob-tained at the B3LYP/6-311G(2d,d,p) level of theory and theEckart
k from CBS-QB3//B3LYP/6-311G(2d,d,p), the criticalenergies computed
at these two level of energies were compared.From Table 1, it can
be seen that the CBS-QB3 and DFT energybarriers computed for heptyl
are very close to each other with thelargest deviation being 0.7
kcal/mol. Consequently, comparisonscan be made directly between the
one-dimensional and SCTtransmission coecients.Comparisons are also
made with the G3MP2B3 results of
Hayes and Burgess27 (Table 1). The CBS-QB3 critical energiesare
found to be systematically lower than the G3MP2B3 values.The
discrepancies range from 1.1 kcal/mol (5ss) to 1.5 kcal/mol(5 ps),
with a mean absolute deviation of 1.2 kcal/mol. Thesedierences are
perhaps within the limits of the uncertainty of the
Figure 2. Potential energy of H-atom shift in the n-heptyl
radicalscomputed using the CBS-QB3 method at 0 K.
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324 dx.doi.org/10.1021/jp209360u |J. Phys. Chem. A 2012, 116,
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The Journal of Physical Chemistry A ARTICLE
two methods, but we expect the present CBS-QB3 results to bemore
reliable. It has also been reported that CBS-QB3 is able
toaccurately predict thermochemical and kinetic data for
hydro-carbon combustion.5458 In addition, Wang et al.59 studied
thethermal decomposition and isomerization of 1-hexyl radical
andshowed that the CBS-QB3 calculations were able to
reproduceexperimental kinetic data.A comparison of the critical
energies computed for H-shift in
dierent n-alkyl radicals is presented in Table 2. It can be
seenthat for a given type of reaction, all critical energies lie
within0.3 kcal/mol of each other, indicating that the reaction
energeticsis primarily a function of the ring strain energy in the
criticalgeometry, as will be discussed later.Transmission
Coefficients. We shall start here using the
heptyl radical as the focal point of discussion. As shown
inTable 3, transmission coefficients are calculated for 1-heptyl
/4-heptyl hydrogen transfer considering four approximations. ForT g
800 K, the transmission coefficients are close to each otherfor all
approximations considered. Below 800 K, however,discrepancies
become progressively larger. For 300eTe 500 K,transmission
coefficients calculated with the SkodjeTruhlar(S&T)
approximation give the highest values, while Wignersmethod produces
the lowest values; both are substantially dif-ferent from the
higher-order SCT results (Figure 3). In com-parison, the Eckart
method provides a reasonably good approx-imation of the SCT results
above 300 K. The largest discrepancyis for the 7ps isomerization
(1- to 2-heptyl) with the Eckartk value 2.8 times that of the SCT
value at 300 K. This difference isgenerally smaller than that
resulting from the various uncertain-ties in the electronic
structure calculation itself. At 400 K, thedeviation becomes
substantially smaller and is only about 20%.Above 500 K, SCT and
Eckart transmission coefficients areessentially identical. It is
noteworthy to mention that Ratkiewiczet al.29 compared Eckart and
SCT transmission coefficient for ippisomerizations, with i = 4, 5,
6, and 7. They found somewhatlarger differences between the Eckart
and SCT k(T) values at
300 and 400 K than the present results. In particular, their
Eckarttransmission coefficients are larger than our values. This
facthighlights the strong sensitivity of Eckart k(T) to the
barrierheight and the imaginary frequency, an issue to be discussed
later.Although the discrepancy between the Eckart and SCT
methods is small, the fundamental cause for the apparent
agree-ment is clear. The Eckart method inherently underestimates
thetunneling probability by neglecting corner-cutting. In most
cases,this underestimation is balanced by the fact that the
Eckartfunction yields a potential energy curve narrower than the
actualPES, as illustrated in Figure 4, thus leading to an
overestimationof the tunneling probability. Hence, the good
agreement betweenEckart and SCT transmission coecients is in many
waysfortuitous because of error cancellation in the Eckart
approxima-tion. For the same reason, the ability of the Eckart
method inreproducing the SCT results should not be generalized to
otherreaction systems. For hydrogen shift reactions studied
here,however, the Eckart method is accurate.Zhang and Dibble60
studied the impact of tunneling on
hydrogen-transfer isomerizations in n-propylperoxy radicalusing
multidimensional SCT calculations as well as the one-dimensional,
Eckart, and Wigner transmission coecients. Theyreached a similar
conclusion for their system. The Eckart methodworks well as
compared to the SCT result, but they cautionedthat the agreement
may not be generalized to other systems.Multidimensional tunneling
calculations allow for the calcula-
tion of representative tunneling energies. This energy
representsthe path with the greatest tunneling probability at a
giventemperature. SCT representative tunneling energies for
hydro-gen transfers in n-heptyl radicals system are presented in
Table 4.Below 800 K, the representative tunneling energies are
wellbelow the barrier top, and the tunneling paths can
deviatedramatically from the reaction path. As an example, Figure
5presents the PES for 1-heptyl/ 4-heptyl and the
representativetunneling energy at 300 K. Above 800 K, tunneling and
reactionpaths are converging, and a one-dimensional approximation
isgenerally adequate. For large critical ring structures, for
example,6ps and 7ps, and at low temperatures, the representative
tunnel-ing energy is closer to the energy of the saddle point than
for the5ps or 5ss hydrogen shift. Hence, corner-cutting is enhanced
forthe more strained transition state structures. It is for this
reasonthat SCT calculations show that the transmission
coecientincreases with a decrease in the critical ring size. For
example,k(T) values are predicted to be 268 and 270 for 5ps and
5ssH-atom shifts at 300 K, respectively. For 6ps and 7ps
transitions,k(T) values are notably smaller and equal to 30.8 and
14.3,respectively. The dierence remains signicant until 800 K,
thetemperature above which the transmission coecients are closeto
unity.We compare our SCT results to those of Zheng and
Truhlar32
for 1-pentyl 5ps and 1-hexyl 5ps and 6ps H-atom shifts and
thosefor 1-butyl 5pp and 1-pentyl 6pp of Ratkiewitcz et al.29 As
shownin Table 5, k(300 K) values calculated for H-shift in the
ve-membered ring transition structure is in reasonably good
agree-ment with literature values. For six-membered ring
structures,the k(300 K) value of Zheng and Truhlar is larger than
ours byabout a factor of 3 and that of Ratkiewitcz et al. by a
factor of 4.Note that the Hessians were computed for all of the
points alongthe MEP in both Ratkiewitcz et al. and our SCT/VTST
calcula-tions. In particular, Ratkiewitcz et al. performed a
considerablenumber of force constant calculations along the MEP to
ensureconvergence of SCT calculations (150 points on each side
of
Table 2. CBS-QB3 Critical Energies (kcal/mol, 0 K) Com-puted for
ips and iss Hydrogen Shift
species 7ps 6ps 5ps 6ss 5ss
n-pentyl 22.3
n-hexyl 15.3 22.1
n-heptyl 14.7 15.0 22.0 22.8
n-octyl 14.5 15.0 22.0 15.7 22.6
Table 3. Transmission Coecients Computed for 5psH-Atom Shift in
1-Heptyl Radical Using Wigner, Skodje andTruhlar (S&T), Eckart,
and SCT Approximations
transmission coecient, k(T)
T (K) Wigner S&T Eckart SCT
300 4.2 1.1 105 454 268400 2.8 55.5 13.0 15.6
500 2.2 5.4 4.2 4.9
600 1.8 2.7 2.6 2.8
800 1.5 1.7 1.7 1.7
1000 1.3 1.4 1.4 1.4
1500 1.1 1.1 1.2 1.2
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The Journal of Physical Chemistry A ARTICLE
MEP, leading to a total of 300 points due to symmetry of
thepotential function for ipp reactions, with i = 5, 6, and
7).Regardless, ve-membered ring critical geometries are shownto
have a greater tunneling tendency than the six-membered
ringtransition in all of the studies.
As was already mentioned above, large curvature calculationswere
also performed in a somewhat qualitativemanner to examinewhether
the small curvature approximation is adequate forH-shiftin n-heptyl
radicals. The LCT approximation describes tunnelingpaths that are
near and far from the minimum energy path, andallows for a large
degree of corner-cutting. Calculations show
thatanLCTtunnelingmechanismcanbe important over a small,
selectedrange of energies along the reaction path, but for
reactionsexamined here, the SCT approximation alone is
suciently
Figure 3. Comparison of the transmission coecient k(T) for 1-
and 2-heptyl radicals calculated using the Wigner, Skodje and
Truhlar (S&T), Eckart,and SCT approximations. Symbols are
computed values. Lines are drawn to guide the eye.
Figure 4. Comparison of the potential energy () of 14 H shift
in1-heptyl and an Eckart function t (- - -).
Table 4. Representative Tunneling Energies (Ert) as a Func-tion
of Temperature Relative to the Vibrational Ground-StateEnergy of
the Saddle Point (E0K
q) for Hydrogen Shift inn-Heptyl
E0Kq Ert (kcal/mol)
T (K) 5ps 6ps 7ps 5ss
300 8.1 2.3 2.3 4.0
400 5.1 1.3 1.7 2.3
500 1.8 0.7 1.5 2.0
600 1.0 0.4 1.0 1.7
800 0.3
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326 dx.doi.org/10.1021/jp209360u |J. Phys. Chem. A 2012, 116,
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The Journal of Physical Chemistry A ARTICLE
accurate. LCT tunneling, even for a few energy levels, does
leadto larger transmission coecients. For example, for 7ps
H-atomshift in 1-heptyl, the transmission coecient is increased by
17%at 300 K and 41% at 250 K.Eckart transmission coecients are
known to be sensitive to
the energy barrier height and the imaginary frequency at the
saddle point. Figure 6a presents the results of sensitivity
analyseswith respect to the imaginary frequency and barrier height
using1-heptyl/ 4-heptyl as an example. It is seen thatwithin(2
kcal/molvariation, k exhibits a small sensitivity with respect to
the barrierheight. At 300 K, an increase or decrease of 2 kcal/mol
in thebarrier height leads to 30% change in k. What impacts
thetransmission coecient the most is the uncertainty in
theimaginary frequency (i) of the critical geometry, as shown
inFigure 6b. At 600 K, k calculated with 1.1 i is 20% larger
thanthe reference value. At 300 K, however, deviations are as large
as afactor of 4 due to a 10% change in i.High-Pressure Limit Rate
Constants. Figure 7 presents high-
pressure limit rate constants computed for heptyl
isomerizationusing the classical TST-HR approximation with Eckart
tunneling.Molecular parameters were taken from electronic
structurecalculations at the CBS-QB3 level of theory. As observed
byZheng and Truhlar32 and Ratkiewicz et al.,29 almost no
differencewas found between TST and CVT calculations,
indicatingrecrossing to be infrequent (a maximum deviation of
0.3%).Figure 1 presents the computed k for reactions R1 and R2
alongwith literature data. The agreement is generally good,
especiallyfor reaction R1 considering the fairly significant
scatter in thedata. The high-pressure limit rate constants of
reaction R2reported by Imbert and Marshall23 and Watkins and
Ostreko18
are probably outside of the uncertainty bound of the
currentcalculation and those of other studies.The high-pressure
limit rate constants compare well with the
experimentally derived data of Tsang et al.,4,6,7 as illustrated
inboth Figures 1 and 7. In general, the dierence is within a
factorof 2 of each other. A very interesting point is that our
calculationsconrm the conclusion of Tsang et al.4 regarding the
dierence in6ps and 7ps hydrogen transfers. They proposed that the
rate ofthe 6ps transition is about a factor of 2 greater than the
7pstransition and concluded that larger-ring transition structures
willpossibly make a lesser contribution to isomerization because
ofentropic constraints. The present results support this view.
Fromthe PES displayed in Figure 2, it can be seen that the
criticalenergies for 6ps and 7ps isomerizations are similar, yet
thecomputed k(6ps)/k(7ps) is 5 over the temperature rangeof 5002000
K.
Figure 5. Vibrationally adiabatic ground-state potential energy
as afunction of reaction coordinate for the 14 hydrogen shift in
1-heptyl.The dotted line denotes the representative tunneling
energy at 300 K.
Table 5. Comparison of SCT Transmission Coecients at300 K
reaction k reference
5ps (1-pentyl/ 2-pentyl) 231 Zheng and Truhlar32
5ps (1-hexyl/ 3-hexyl) 245 Zheng and Truhlar32
5pp (1-butyl/ 1-butyl) 364 Ratkiewitcz et al.29
5ps (1-heptyl/ 4-heptyl) 268 this work6ps (1-hexyl/ 2-hexyl) 114
Zheng and Truhlar32
6pp (1-pentyl/ 1-pentyl) 38.5 Ratkiewitcz et al.29
6ps (1-heptyl/ 3-heptyl) 30.1 this work
Figure 6. Sensitivity of Eckart transmission coecient computed
with respect to (a) critical energy (E0Kq) and (b) the imaginary
frequency (i) for 14
H-shift in 1-heptyl with the base case E0Kq and vi computed at
the CBS-QB3 level of theory. Symbols are computed values. Lines are
drawn to guide
the eye.
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327 dx.doi.org/10.1021/jp209360u |J. Phys. Chem. A 2012, 116,
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The Journal of Physical Chemistry A ARTICLE
Branching ratios of the four unimolecular channels of
1-heptylradical (5ps, 6ps, and 7ps isomerizations
andCCbond-scission)are presented in Figure 8, based on the rate
constants computedat the high-pressure limit. For CC -scission, the
rate con-stant is calculated using the critical geometry of Sirjean
et al.determined at the CBS-QB3 level of theory.55 For T
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328 dx.doi.org/10.1021/jp209360u |J. Phys. Chem. A 2012, 116,
319332
The Journal of Physical Chemistry A ARTICLE
is estimated by adding the energy required to abstract an
hydrogenatom of a given type (primary, secondary, or tertiary) to
the ringstrain energy in the cyclic transition structure. The
pre-exponential(A) factor can be calculated by estimating the
entropy of activationby considering the loss or gain of internal
rotor(s) and of opticalisomers. This semiempirical method relies on
the group additivityprinciple, with values of ring strain energies
taken from thosetabulated for cycloalkanes and entropies of
activation estimatedfrom analogous cyclic and acyclic alkanes. For
example, in an earlystudy,63 a4 cal/(mol K) change in the entropy
was proposed foreach rotor locked into the cyclic transition
structure. They derivedthis value on the basis of entropy
difference between n-butaneand cyclobutane (12 cal/(mol K)) and the
three internal rotorsin n-butane.63 Unfortunately, the corrections
were derived fromexperimental data of hydrogen-transfer reactions
that are inherently
subject to the tunneling effect. Additionally, the
pre-exponentialfactors and activation energies derived from these
methods alsoinclude the effects of tunneling. Here, we re-examine
these raterules by decoupling the tunneling effect from the
apparentactivation energies and A factors.We assume that for a
given reaction class, transmission coe-
cients follow the same temperature dependency. This assump-tion
is supported by the current theoretical results shownFigure 10. As
seen, the transmission coecient is primarily afunction of the ring
size in the transition state and not a functionof the reactant
size. Within each reaction class, the maximumdeviation is smaller
than 2% for T > 500 K. Below thistemperature, the maximum
deviation is 4% for all reactionsexcept for ve-member ring
structures where a 14% maximumdeviation is observed. Tunneling can
therefore be taken into
Figure 9. Arrhenius plots for the high-pressure limit rate
constants of hydrogen shifts in n-octyl radicals. The bottom right
plot represents branchingratios computed for the thermal
decomposition of 1-octyl radical.
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329 dx.doi.org/10.1021/jp209360u |J. Phys. Chem. A 2012, 116,
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The Journal of Physical Chemistry A ARTICLE
account within the framework of a semiempirical correlation
byusing a modied Arrhenius equation leading to the
parameterpresented in Table 7. Here, small curvature transmission
coe-cients determined for the n-heptyl system were used as
referencedata. The temperature range for the t was chosen to
minimizethe errors induced by the tting procedure. In all cases,
the ttingerror is
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330 dx.doi.org/10.1021/jp209360u |J. Phys. Chem. A 2012, 116,
319332
The Journal of Physical Chemistry A ARTICLE
values are consistently 2.5 kcal mol1 above those of
cyclicalkanes. Hence, using the ring strain energy of
cycloheptanewould lead to an overestimation of the activation
energy.Contributions of the loss of internal rotation to the
activation
entropy for each hydrogen-transfer reaction studied in this
workare presented in Table 9. Only values at 298 K are reported
here.Temperature was found to have a negligible eect on the
entropyof activation. For ips hydrogen transfers, the loss of one
internalrotor causes the entropy of activation to decrease by 3.61(
0.23cal/(mol K), while a mean value of 4.56 ( 0.23 cal/(mol K)per
rotor was calculated for iss hydrogen shift. The dierence isnot
surprising considering that ips involves CH2CH2 andCH2CH2 3 rotors,
but iss involves CH2CH2 internalrotations only.The recommended
hydrogen-transfer rate rule is summarized
in Table 7. Comparisons between the theoretical k values andthe
rate-rule estimations are shown in Figure 11 for hydrogenshift in
n-heptyl radicals. For the temperature range considered(5001500 K),
the maximum deviation is about a factor of 2.5.
For most cases, the deviation is well within the
estimateduncertainty of the theoretical calculation (around a
factor of 2).
4. CONCLUSION
The role of quantum tunneling in hydrogen-transfer
isomer-izations of linear alkyl radicals was studied in detail.
Transmissioncoecients are shown to require a multidimensional
treatmentbelow 800 K. Only above800 K is a one-dimensional
treatmentappropriate. The Eckart method is shown to reproduce
themultidimensional transmission coecients over the entire
tem-perature range, but the agreement is due to a favorable
errorcancellation. The inability of the Eckart approach to account
forhigher-dimensional tunneling eect is compensated by therigidity
of the Eckart function, leading to a potential energycurve narrower
than the actual PES. Calculations show thatbelow 600 K the Eckart
transmission coecient is highlysensitive to the value of the
imaginary frequency, and hence issubject to the uncertainty in the
electronic structure calculation.Regardless, high-pressure limit
rate constants calculated usingthe classical transition state
theory with treatment of internalrotation and the use of Eckart
transmission coecients and a PESdetermined at the CBS-QB3 level of
theory are in good agree-ment with literature data.
The present results are consistent with Tsang et al.,
whoconcluded that a large body of literature data at low
temperatures(with small A factor and activation energy) can be
reconciledwith high-temperature data by taking into account
quantumtunneling. Our calculations also conrm their observation
thattunneling is more pronounced with ve-membered ring transi-tion
structures than with six-membered ring structures. Inaddition,
hydrogen transfer through an eight-membered ringtransition
structure is not competitive due to prohibitive entropyeects.
The systematic study of isomerization reactions from n-pentylto
n-octyl led to a large set of theoretical kinetic data that can
berationalized within a reaction class approach. It is shown that
thetransmission coecient can be treated as an explicit
structure/reactivity correlation parameter. This approach allows
for a fastand suciently accurate estimation of the impact of
tunneling on
Table 9. Contributions of Internal Rotor Losses to the Entropy
of Activation in iab Hydrogen Shift
reaction channel
number of internal
rotors lost (n) S298Kq (cal/(mol K))
per rotor entropy loss ((Sq) = Sq/n,
cal/(mol K))
1-pentyl/ 2-pentyl (5ps) 3 10.00 3.331-hexyl/ 2-hexyl (6ps) 4
14.02 3.511-hexyl/ 3-hexyl (5ps) 3 10.23 3.411-heptyl/ 2-heptyl
(7ps) 5 17.79 3.561-heptyl/ 3-heptyl (6ps) 4 14.57 3.641-heptyl/
4-heptyl (5ps) 3 11.27 3.762-heptyl/ 3-heptyl (5ss) 3 13.08
4.361-octyl/ 2-octyl (8ps) 6 19.83 3.311-octyl/ 3-octyl (7ps) 5
18.80 3.761-octyl/ 4-octyl (6ps) 4 16.08 4.021-octyl/ 4-octyl (5ps)
3 11.57 3.862-octyl/ 3-octyl (6ss) 4 18.07 4.522-octyl/ 4-octyl
(5ss) 3 14.41 4.80mean for ips 3.61 ( 0.23mean for iss 4.56 (
0.22
Figure 11. Comparison of the high-pressure limit rate constants
()and rate-rule estimates ( 3 3 3 ) for hydrogen shift in n-heptyl
radials.
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331 dx.doi.org/10.1021/jp209360u |J. Phys. Chem. A 2012, 116,
319332
The Journal of Physical Chemistry A ARTICLE
the rate constant. It can be used also to extrapolate kinetic
dataobtained in a limited temperature range. A rate rule was
proposedfor hydrogen shift in linear alkyl radicals. In this rule,
transmis-sion coecients are calculated using multidimensional
tunnelingfor a representative reaction class. The rate constant
withouttunneling may be estimated by considering the loss or gainof
internal rotations in the transition state structure for
pre-exponential factors. The apparent activation energy may be
esti-mated from that of typical H-abstraction by an alkyl
radicalcorrected for ring strain. The rate rule proposed here is
shown toreproduce the theoretical high-pressure limit rate constant
towithina factor of 2 for almost the entire range of temperatures
considered.
ASSOCIATED CONTENT
bS Supporting Information. Optimized geometries at
theB3LYP/6-311G(2d,d,p) level of theory for all species.
Hindrancepotentials and barrier heights at the CBS-QB3 level of
theory.Forward and reverse reaction barrier heights (E1 and E1) at
0 Kalong with imaginary frequencies (i) of the transition states
andcalculated characteristic lengths. SCT transmission
coecientsfrom 200 to 2000 K for all n-heptyl H-atom shifts. This
material isavailable free of charge via the Internet at
http://pubs.acs.org.
AUTHOR INFORMATION
Corresponding Author*Phone: (33) 383175202 (B.S.), (213)
740-0499 (H.W.). E-mail:[email protected]
(B.S.), [email protected] (H.W.).
Present AddressesLaboratoire Reactions et Genie des Procedes,
Nancy Universite,CNRS, BP 20451, 1 rue Grandville, 54001 Nancy,
France.
ACKNOWLEDGMENT
This work was supported by the U.S. Air Force Oce ofScientic
Research (AFOSR Grant Nos. FA9550-07-1-0168 andFA9550-08-1-0040)
and by the Combustion Energy FrontierResearch Center (CEFRC), an
Energy Frontier Research Centerfunded by the U.S. Department of
Energy, Oce of Science, Oceof Basic Energy Sciences under Award No.
DE-SC0001198. Partof this work was performed using HPC resources
from GENCI-CINES (Grant 2011086686).
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