10988 Phys. Chem. Chem. Phys., 2010, 12, 10988–10995 This journal is c the Owner Societies 2010

Kinetics of thermoneutral intermolecular hydrogen migration in alkyl

radicalsw

Artur Ratkiewicz,a Barbara Bankiewicza and Thanh N. Truong*b

Received 20th April 2010, Accepted 23rd June 2010

DOI: 10.1039/c0cp00293c

High pressure limits of thermal rate constants of intramolecular hydrogen migrations, particularly

1,3 to 1,6 H-shift in propyl, butyl, pentyl and hexyl radicals, respectively, were calculated using

the canonical variational transition state theory (CVT) with a multi-dimensional small-curvature

tunneling (SCT) correction over the temperature range of 300–3000 K. The CCSD(T)/cc-pVDZ//

BH&HLYP/cc-pVDZ method was used to provide necessary potential energy surface

information. Rate constants for these reactions were used to extrapolate rate constants for

reactions of larger alkyls where experimental data are available using the Reaction Class

Transition State Theory (RC-TST). Excellent agreement with experimental data confirms the

validity of the RC-TST methodology and the accuracy of the calculated kinetic data in this study.

1. Introduction

The isomerization via intramolecular hydrogen atom migration,

also known as hydrogen-shift, forms an important class of alkyl

and alkenyl radical reactions. It has long been known that these

reactions are of importance in various complex reaction systems

such as combustion of hydrocarbons.1–5 The formation of some

products in such systems can be explained only in terms of

isomerization. Hydrogen shift reactions were also found to play

a significant role in determining product distribution in the last

stage of paraffin pyrolyses. Furthermore, radical isomerization

appears to compete in hydrocarbon oxidation systems with radical

decomposition and can be quite often encountered in the

postulation of mechanism of the processes involving radical

intermediates. Determination of the thermal rate coefficients of

these isomerization reactions is a prerequisite for the modeling of

combustion systems such as engines and furnaces operating

with hydrocarbon fuels. In spite of the latest experimental

achievements,6,7 relatively little information is currently available

for the kinetics of intramolecular hydrogen transfer reactions.

This is because direct measurements of the rate constants of

isomerization are difficult due to competing reactions in general.

Furthermore, for thermoneutral reactions, product and reactant

are identical, measuring the relative concentrations of reactants

and products is only possible via isotope labeling and is often

difficult. To the best of our knowledge, no direct experimental rate

data are available for these reactions to date.

There are a number of theoretical studies on the activation

energies and geometries of the transition states of H shift reactions

in alkyl radicals. Jitariu et al.8 studied in detail 1,2 to 1,5

isomerizations of 1-pentyl radical. Threshold energies, bottleneck

properties and the canonical variational transition state theory

(CVT) with multi-dimensional small-curvature tunneling (SCT)

rates for selected reactions were reported. Viskolcz et al.9,10

calculated the ab initio activation barriers and the ring strain

energies of the 1,2 to 1,5 H-atom transfer reactions in ethyl,

propyl, butyl, pentyl, and 2-methylhexyl radicals, respectively.

Pressure-dependent thermal rate constants were calculated for

2-methylhexyl radical. The authors showed that the barrier height

decreases as the number of atoms in the ring of the cyclic

transition structure increases. A similar conclusion was derived

by Curran et al.3,4 in their series of proposed hydrocarbon

combustion mechanisms. The authors approximated the activation

energies for isomerization reactions in terms of the number of

atoms in the transition state ring structure (including the transferring

H atom) and the type of carbon atom that the transferring

H atom originally binds to. Matheu et al.11 developed a set of

generic rules to estimate high pressure kinetic parameters of

intramolecular hydrogen shifts (from 1,2 to 1,6) in alkyl

radicals. These rules were derived from the results of DFT

quantum chemistry and TST calculations, for the simplest

reaction in the reaction class. Subsequent rate rules in

the family used the same A and n parameters in the

ATnexp(�Eactivation/RT) rate expression. Such a practice

assumed that the neutral H shift for the primary hydrocarbon is

sufficient to capture the dominant entropic effects, namely the

internal rotor gain and loss in this type of reactions. Since the

thermoneutral hydrogen migrations are the simplest reactions in

their reaction classes, these processes can serve as reference

reactions to investigate all processes within their perspective

classes. For this reason, more accurate kinetic parameters are

needed for the four reactions, from 1,3 to 1,6 H shift, namely:

CCC� 2 C�CC (R1)

CCCC� 2 C�CCC (R2)

CCCCC� 2 C�CCCC (R3)

CCCCCC� 2 C�CCCCC (R4)

a Chemistry Institute, University of Bialystok, Hurtowa 1, 15-399Bialystok, Poland

bHenry Eyring Center for Theoretical Chemistry, Department ofChemistry, University of Utah, 315 S. 1400 E. Rm. 2020,Salt Lake City, Utah 84112, USA. E-mail: Thanh.Truong@utah.edu

w Electronic supplementary information (ESI) available: Themolecular and frequency data of reagents, products and transitionstates for reactions R1-R8, potential energy data along the MEP forreactions R1-R4, high pressure limits of rate constants for reactionsR1-R4 calculated with POLYRATE, and RC-TST factors and highpressure limits of rate constants for reactions R5-R8. See DOI:10.1039/c0cp00293c

PAPER www.rsc.org/pccp | Physical Chemistry Chemical Physics

Dow

nloa

ded

by B

all S

tate

Uni

vers

ity o

n 17

/04/

2013

12:

14:1

1.

Publ

ishe

d on

27

July

201

0 on

http

://pu

bs.r

sc.o

rg |

doi:1

0.10

39/C

0CP0

0293

CView Article Online / Journal Homepage / Table of Contents for this issue

This journal is c the Owner Societies 2010 Phys. Chem. Chem. Phys., 2010, 12, 10988–10995 10989

In this study, the high pressure limits of thermal rate constants

for the reactions (R1)–(R4) were computed, using the canonical

variational transition state theory augmented with multi-

dimensional semi classical tunneling correction (CVT/SCT), with

potential energy surface calculated at a sufficiently accurate level

of theory. Among these four reaction classes, the 1,4 H-shift class

with the reference reaction (R2) were considered previously.12 In

this study, however, we present a more accurate rate expression

for the reference (R2) reaction.

2. Methodology

All the electronic structure calculations were carried out using

the GAUSSIAN 03 suite of programs.13 Since a hybrid non-

local Density Functional Theory (DFT), particularly Becke’s

half-and-half (BH&H) non-local exchange and

Lee–Yang–Parr (LYP) non-local correlation functionals, has

previously been found to be sufficiently accurate for predicting

the transition state properties,14–22 geometries of reactants,

transition states, and products were optimized at the

BH&HLYP level of theory with Dunning’s correlation-

consistent polarized valence double-zeta basis set [3s2p1d/2s1p],

denoted as cc-pVDZ,23,24 which is sufficient to capture the

physical change along the reaction coordinate for this type of

reaction. In particular, in our previous study,12 for the (R2)

reaction the BH&HLYP/cc-pVDZ method predicts geometries

close to those from the QCISD/cc-pVDZ level of theory for

the reactant and transition state, with the largest difference of

0.014 A. Similarly for frequencies, the average absolute

difference is about 32 cm�1 between those from the

BH&HLYP/cc-pVDZ and QCISD/cc-pVDZ calculations.

This leads to the insignificant (i.e. less than 0.05 kcal mol�1)

differences in the total ZPE’s of the reactant and transition

state for these two levels of theory. For this reason,

all geometry optimizations and frequency calculations were

performed at the BH&HLYP/cc-pVDZ level of theory.

Normal mode analysis was performed at each stationary

point to ensure its characteristics, i.e. stable structure has zero

imaginary vibrational frequency, whereas transition state (TS)

structure has one imaginary vibrational frequency, whose

mode corresponds to the reaction coordinate of the reaction

being considered. The minimum energy path (MEP) is also

obtained at the BH&HLYP/cc-pVDZ level using the

Gonzalez–Schlegel method25 in the mass weighted Cartesian

coordinates, with a step size of 0.01 (amu)12 bohr. In addition,

force constants at 300 points along the MEP were determined

to ensure convergence of the Small Curvature Tunneling

calculations. However, since the MEP is symmetric, thus

actual calculations were only done for 150 points on the

reactant side. The points were chosen based on the curvatures

of the MEP, and the geometrical parameters as functions of

the reaction coordinate according to our auto-focusing

technique.26 Energetic information along the MEP is further

refined by single-point energy CCSD(T)/cc-pVDZ calculations.

These energies, combined with the BH&HLYP/cc-pVDZ

geometries and frequencies, were then used for rate constant

calculations. Utility software tools were used to compose these

PES data for rate calculations and are available from the

authors upon request.27

High-pressure limits of thermal rate constants were calculated

using both the classical Transition State Theory28 (TST)28 with

the 1D Eckart tunneling, and the Canonical Variational

Transition State Theory (CVT)28 with the Small Curvature

Tunneling (SCT)28 methods, for the temperature range of

300–3000 K. To model vibrations transverse to the reaction

path, we used curvilinear coordinates based on bond stretches,

valence angle bends, and bond torsions from the

POLYRATE29 program. The 1D Eckart transmission coefficients

and partition functions were obtained with the TheRate,

available at the Computational Science and Engineering

Online (CSEO) website.30 The overall rotations were treated

classically and vibrations were treated quantum mechanically

within the harmonic approximation except for the modes

corresponding to the internal rotations of the CH2 and CH3

groups, which were treated as the hindered rotations (HR)

using the method of Ayala and Schlegel.31 In our previous

studies (for example, see ref. 12, 19, 20, and 22), this

methodology was found to be accurate by direct comparison

of our calculated results with experimental data available. The

reaction symmetry number of 3 was used to account for the

number of symmetrically equivalent reaction paths.

3. Results and discussion

3.1 Stationary points

It is known that intramolecular hydrogen migration proceeds

through formation of a cyclic transition structure.3,9,32 The

formation of a strained ring leads to a high reaction barrier.

The optimized transition state structures for all reference

reactions are shown in Fig. 1. The optimized geometries,

frequencies, zero point energies for reagents and transition states

of these reactions are available in the ESIw (Tables S1–S9).

From these geometries, ring strain at the transition state

increases from (R4) to (R1). Since in the more strained ring

more energy is needed for its formation, the barrier for

intramolecular atom transfer is expected to increase when ring

size decreases.

The classical barrier heights of the reference reactions

(R1)–(R4), calculated at various levels of theory with the

inclusion of ZPE corrections, are listed in Table 1. Comparisons

on the performance of different levels of theory enable the

selection of the most cost effective method for generating

potential energy surface information along the MEP for

CVT/SCT rate calculations.

Amongst different correlated methods considered here, the

compound CBS-QB333 model chemistry is expected to be the

most accurate and has been validated for its ability to predict

accurate hydrogen shift barriers.34,35 CBS-QB3 barrier heights

are 42.1, 24.2, 15.4 and 12.8 kcal mol�1 for the reactions

(R1)–(R4), respectively, and are used as reference points for

determining accuracy of other methods. Mean Absolute

Deviation (MAD) of barrier heights calculated with a given

method from the CBS-QB3 value is also given in Table 1.

CCSD(T)/cc-pVDZ//BH&HLYP/cc-pVDZ was found to have

the smallest MAD of 0.7 kcal mol�1. The performance of the

DFT methods is slightly worse, however it is still acceptable.

This fact is consistent with our previous work,17–20 DFT

Dow

nloa

ded

by B

all S

tate

Uni

vers

ity o

n 17

/04/

2013

12:

14:1

1.

Publ

ishe

d on

27

July

201

0 on

http

://pu

bs.r

sc.o

rg |

doi:1

0.10

39/C

0CP0

0293

C

View Article Online

10990 Phys. Chem. Chem. Phys., 2010, 12, 10988–10995 This journal is c the Owner Societies 2010

methods give rather accurate results, comparable to those

obtained with more expensive MO-based correlation methods.

Due to spin contamination, larger errors were found for the

MPn (n = 2,4) methods. Consequently, CCSD(T)/cc-pVDZ//

BH&HLYP/cc-pVDZ method is used to provide energetic

information for rate calculations.

As shown in Table 1 the largest barrier was found for the

1,3 H-shift and the lowest for the 1,6 H-shift. In agreement

with the expectation and the previous findings,9,10 the barrier

heights increase as the size of the ring in the transition state

decreases. Fig. 2 shows the classical, Vc, and vibrationally

adiabatic ground state, VGa , potential energy curves along the

MEP for the four title reactions. Total zero-point energies of

the reactants and transition states and the imaginary frequencies

at the transition states are listed in Table 2. The other potential

energy surface data are available in the ESIw (Tables S19–S22).ZPE corrections lower the classical barrier heights about

2 kcal mol�1, which corresponds to 5–15% of the total barrier

Fig. 1 Optimized, at the BH&HLYP/cc-pVDZ level of theory, geometries (distances in A and angles in degrees) of the transition states of the

reactions: C�CC - CCC�, C�CCC - CCCC�, C�CCCC - CCCCC�, and C�CCCCC - CCCCCC�.

Table 1 Calculated barrier heights and reaction energies for the C�CC2 CCC�, C�CCC2 CCCC�, C�CCCC2 CCCCC� and C�CCCCC2CCCCCC� reactions (numbers are in kcal mol�1). Zero-point energy corrections were included

Level of theory

DVGa for reaction

Mean absolutedeviation fromCBS-QB3 resultsaC�CC - CCC� C�CCC - CCCC� C�CCCC- CCCCC�

C�CCCCC-CCCCCC�

B3LYP/cc-pVDZ 38.711 23.111 15.311 10.5 2.2B3LYP/cc-pVTZ 39.6 23.1 15.1 12.2 1.2BH&HLYP/cc-pVDZ 43.3 26.9 17.8 14.4 2.2BH&HLYP/cc-pVTZ 44.2 27.9 19.1 15.8 3.1CCSD(T)/cc-pVDZ//BH&HLYP/cc-pVDZ 41.1 24.8 16.2 13.2 0.7QCISD/cc-pVDZ//BH&HLYP/cc-pVDZ 42.5 29.7 17.8 14.6 2.5MP2/cc-pVDZ//BH&HLYP/cc-pVDZ 42.0 8.1 16.7 13.5 4.8MP2/cc-pVTZ//BH&HLYP/cc-pVDZ 41.3 9.5 16.8 14.1 4.6MP4/cc-pVDZ//BH&HLYP/cc-pVDZ 42.6 7.4 16.5 13.4 4.8MP4/cc-pVTZ//BH&HLYP/cc-pVDZ 42.2 8.7 — — 7.8CBS-QB3 42.1 24.2 15.4 12.8 —MP2/6-311G**//HF/6-31G*9 43.1 26.4 19.1 — 2.4

a MAD ¼PjbarrierCBS�QB3�barrier for specific methodj

number of barriers calculated

Dow

nloa

ded

by B

all S

tate

Uni

vers

ity o

n 17

/04/

2013

12:

14:1

1.

Publ

ishe

d on

27

July

201

0 on

http

://pu

bs.r

sc.o

rg |

doi:1

0.10

39/C

0CP0

0293

C

View Article Online

This journal is c the Owner Societies 2010 Phys. Chem. Chem. Phys., 2010, 12, 10988–10995 10991

height. In consistence with the trend in the barrier height, the

imaginary frequencies decrease as the size of the transition

state ring increases. In addition, the positions of the maxima of

the ground-state adiabatic potential energy curves VGa are very

close to the corresponding transition states for all reactions.

This suggests the re-crossing effect would be small for these

reactions.

3.2 Rate constants

The high pressure limits of rate constants of the four titled

H-shift reactions studied were calculated using the Canonical

Variational Transition State Theory (CVT) with the Small

Curvature Tunneling (SCT) and Ayala–Schlegel31 hindered

rotations treatment (HR), over a wide range of temperature

from 300 to 3000 K. The final CVT/SCT/HR rate constants

for reactions (R1)–(R4) are plotted in Fig. 3. For practical

applications, the predicted rates were fitted by least-squares

analysis as given below:

kC�CC$CCC�

¼4:71�10�71T25:836 expð�3897:78=TÞ 300K�To700K

5:91�106T1:938 expð�18716:81=TÞ 700K�To3000K½s�1�

(

ð1Þ

kC�CCC$CCCC� ¼1:187�102�T3:042�exp �8826:73T

� �½s�1�;

ð2Þ

Fig. 2 Potential energy curves along the reaction coordinates of the reactions: C�CC - CCC�, C�CCC - CCCC�, C�CCCC - CCCCC� and

C�CCCCC - CCCCCC�. VGa is the vibrationally adiabatic ground state potential curve and VC is the classical adiabatic ground state potential curve.

Table 2 Calculated total zero point energies for reactants and transition states and imaginary frequencies of the C�CC 2 CCC�, C�CCC 2CCCC�, C�CCCC 2 CCCCC� and C�CCCCC 2 CCCCCC� reactions

Reaction Reactant ZPE/kcal mol�1 Transition state ZPE/kcal mol�1 Imaginary frequency/i cm�1

C�CC 2 CCC� 56.8 54.0 2285.2C�CCC 2 CCCC� 75.3 73.3 2073.5C�CCCC 2 CCCCC� 94.0 91.8 1925.0C�CCCCC 2 CCCCCC� 112.3 110.3 1881.8

Dow

nloa

ded

by B

all S

tate

Uni

vers

ity o

n 17

/04/

2013

12:

14:1

1.

Publ

ishe

d on

27

July

201

0 on

http

://pu

bs.r

sc.o

rg |

doi:1

0.10

39/C

0CP0

0293

C

View Article Online

10992 Phys. Chem. Chem. Phys., 2010, 12, 10988–10995 This journal is c the Owner Societies 2010

kC�CCCC$CCCCC� ¼2:51�105�T1:886�exp �5312:82T

� �½s�1�;

ð3Þ

kC�CCCCC$CCCCCC� ¼2:49�106�T1:6037�exp �4527:82T

� �½s�1�;

ð4Þ

The CVT/HR, TST/HR, and TST/Eckart/HR rate constants

are also plotted for the sake of comparison. Numeric values of

all the rates, used to prepare Fig. 3, are available in the ESIw(Tables S23–S26). The Eckart and SCT transmission coefficients

are listed in Table 3. It is well known that tunneling is significant

for hydrogen transfer reactions at lower temperatures. For all the

reactions investigated, the magnitude of SCT transmission

coefficients confirms this expectation as illustrated by the

differences between CVT/SCT/HR and CVT/HR curves shown

in Fig. 3. At T = 300 K, the SCT transmission coefficients are

105, 364, 39, and 18 for (R1)–(R4), respectively. Eckart

coefficients are much larger than SCT values particularly at low

temperatures. It is known34,36,37 that 1D Eckart function often

yields the potential width too narrow compared to the real ones

and thus leads to an overestimation of the tunneling coefficients.

Tunneling effects become less important when T > 1000 K. On

the other hand, not much difference between the TST and CVT

rates was found. This indicates the re-crossing effects are small as

mentioned earlier.

Fig. 3 Arrhenius plots of the calculated rate constants for the C�CC - CCC�, C�CCC - CCCC�, C�CCCC - CCCCC�, and C�CCCCC -

CCCCCC reactions along with those available in the literature. Since, for all four reactions, TST/HR and CVT/HR rates are almost the same, the

corresponding curves coincide and are difficult to resolve.

Table 3 Calculated transmission coefficients for the C�CC2 CCC� , C�CCC2 CCCC�, C�CCCC2 CCCCC� and C�CCCCC2 CCCCCC�

reactions. The third column for each reaction shows the ratio of the Eckart and SCT tunneling coefficients

Temperature/K

C�CC 2 CCC� C�CCC 2 CCCC� C�CCCC 2 CCCCC� C�CCCCC 2 CCCCCC�

Eckart SCT Eckart/SCT Eckart SCT Eckart/SCT Eckart SCT Eckart/SCT Eckart SCT Eckart/SCT

300 5 053 000 136 417 37.1 4686 364.0 12.9 223.1 38.5 5.8 98.9 17.7 5.6400 360.7 128.3 2.8 34.6 21.7 1.6 12.9 7.0 1.8 10.0 5.1 2.0500 15.6 11.6 1.4 7.0 6.5 1.1 4.7 3.3 1.4 4.2 2.8 1.51000 1.9 1.6 1.2 1.7 1.5 1.1 1.6 1.2 1.3 1.6 1.2 1.32000 1.3 1.1 1.2 1.2 1.1 1.1 1.2 1.0 1.2 1.2 1.0 1.23000 1.2 1.0 1.1 1.1 1.0 1.1 1.1 1.0 1.2 1.1 1.0 1.2

Dow

nloa

ded

by B

all S

tate

Uni

vers

ity o

n 17

/04/

2013

12:

14:1

1.

Publ

ishe

d on

27

July

201

0 on

http

://pu

bs.r

sc.o

rg |

doi:1

0.10

39/C

0CP0

0293

C

View Article Online

This journal is c the Owner Societies 2010 Phys. Chem. Chem. Phys., 2010, 12, 10988–10995 10993

4. Comparisons with literature data

The only kinetic data available for reactions (R1)–(R3) in the

NIST database38 are based on the B3LYP/cc-pVDZ calculations

and Transition State Theory performed by Matheu et al.11 In

these calculations, barrier heights were adjusted on a basis of

the value of the reference reaction and the relative heats of

reaction. Tunneling corrections were not included. Except for

the 1,5 H-shift (R3) reaction, where the rate data of Matheu

et al. agree well with our results, for the other two reactions,

1,3 and 1,4 H-shift reactions, their predicted data are too small

compared to our most accurate CVT/SCT/HR data. For

lower temperatures, the differences are noticeable. Since in

the present study, multidimensional tunneling and hindered

rotation corrections were included and thus rate constants are

expected to be more accurate.

Experimental data are available for the following H-shift

reactions:

C�CCCCC 2 CC�CCCC (1,5 H-shift) (R5)

C�CCCCC 2 CCC�CCC (1,4 H-shift) (R6)

C�CCCCCCC 2 CCC�CCCCC (1,6 H-shift) (R7)

C�CCCCCCC 2 CCCC�CCCC (1,5 H-shift) (R8)

among other H migrations in hexyl and octyl radicals done by

Tsang and co-workers.6,7 The experiments were performed in

a single pulse shock tube at temperatures in the 850–1000 K

range. High-pressure rate constants have been derived over

700–1900 K range with an uncertainty factor of less than 2.

The value of a given rate constant, k, could lie between

km/f and km � f, where km is the reported value and f is an

uncertainty factor.

To make direct comparisons with these experimental data,

rate constants for reactions (R5)–(R8) were extrapolated from

those of (R2)–(R4) using as the reference reactions within the

Reaction Class Transition State Theory (RC-TST) methodology.17

Within the RC-TST framework, the high pressure limit of the

rate constant of an arbitrary reaction (denoted as ka) in a

given reaction class is proportional to the rate constant of a

reference reaction of that class, kr:

ka(T) = f(T) � kr(T) (5)

The key idea of the RC-TST method is to factor f(T) into

different components under the TST framework:

f(T) = fs � fk(T) � fQ(T) � fV(T) � fHR(T) (6)

where fs, fk, fQ, fV and fHR are the symmetry number,

tunneling, partition function, potential energy and hindered

rotations factors, respectively. These factors are simply the

ratios of the corresponding components in the TST expression

for the two reactions:

fs ¼sasr

ð7Þ

fkðTÞ ¼kaðTÞkrðTÞ

ð8Þ

fQðTÞ ¼Qaa ðTÞ

FRa ðTÞ

� �Qar ðTÞ

FRr ðTÞ

� � ¼Qaa ðTÞ

Qar ðTÞ

� �FRa ðTÞ

FRr ðTÞ

� � ð9Þ

fVðTÞ ¼ exp �ðDVaa � DVa

r ÞkBT

� �¼ exp �DDVa

kBT

� �ð10Þ

fHRðTÞ ¼cHR;aðTÞcHR;rðTÞ

ð11Þ

where k(T) is the transmission coefficient accounting for the

quantummechanical tunneling effects; s is the reaction symmetry

number; Qa and FR are the total partition functions (per unit

volume) of the transition state and reactants, respectively;

DVa is the classical reaction barrier height; cHR symbolizes

the correction to the total partition function due to the

hindered rotation treatment, T is the temperature in Kelvin;

kB and h are the Boltzmann and Planck constants, respectively.

Among these, only the symmetry factor can be easily calculated

from the molecular topology of the reactant. Obtaining the

exact value of four other factors requires structures, energies,

and vibrational frequencies of the reactant and transition state

for the reaction investigated. In the present study, we have

employed this formalism for reactions (R5)–(R8). In other

words, we used eqn (6)–(11) to extrapolate rate constants

calculated for small (R1)–(R4) reactions to those of

(R5)–(R8) reactions. Resulting high pressure limits of rate

constants are shown in Fig. 4 (RC-TST). The geometries,

frequencies, ZPE values for reactants and transition states,

explicit values of all the factors for (R5)–(R8) and resulting rates

are available in the ESIw, Tables S8, S10 and 11 and S27–S31.

The potential energy factor can be calculated using the

reaction barrier heights of the arbitrary reaction and the

reference reaction. The classical reaction barrier height DVa

for the arbitrary reaction can be obtained using the Linear

Energy Relationship (LER), similar to the well-known

Evans–Polanyi linear free energy relationship, between

classical barrier heights and reaction energies of reactions in

a given reaction class. Combining averaged values for expressions

(7)–(11) with Linear Energy Relationship (LER),17 one can

extrapolate rate constants for any reaction by knowing only

the reaction energy as has been done for the 1,4 hydrogen

migration12 and others.17,18,20,22 Since application of the

RC-TST/LER methodology for the 1,4 H-shift in alkyl radicals

reaction class was done,12 its prediction for the (R6) reaction,

labeled as RC-TST/LER, is also shown in Fig. 4. Within the

experimental temperature range from 850–1000 K, our

predicted rate constants agree very well with those from

experimental data even with the reasonably large uncertainty

factor of 2. The agreement is still satisfactory within the

temperature range of 700–1900 K, where the high pressure

limits of rate constants were derived in ref. 6 and 7.

Performance of the RC-TST/LER method is slightly worse

than RC-TST. The convenience of rate expressions without TS

knowledge would off-set the lower accuracy of the LER

approach, however.

Dow

nloa

ded

by B

all S

tate

Uni

vers

ity o

n 17

/04/

2013

12:

14:1

1.

Publ

ishe

d on

27

July

201

0 on

http

://pu

bs.r

sc.o

rg |

doi:1

0.10

39/C

0CP0

0293

C

View Article Online

10994 Phys. Chem. Chem. Phys., 2010, 12, 10988–10995 This journal is c the Owner Societies 2010

5. Conclusion

In this work, rate constants of the H-shift reactions in linear

alkyls, namely propyl, butyl, pentyl, and hexyl radicals have

been investigated theoretically. We found that:

� A CCSD(T)/cc-pVDZ//BH&HLYP/cc-pVDZ method is a

cost effective method for exploring the potential energy surface

of the intramolecular hydrogen migration in alkyl radicals.

� Tunneling is significant in all reactions considered here at

low temperatures even at the room temperature.

� By using the RC-TST formalism, rate constants of H-shift

in larger alkyls can be accurately extrapolated from those of

smaller alkyls when compared to available experimental data.

This further confirms the validity of the RC-TST

methodology.

Although the above observations are based on only four

selected reactions, they are expected to be valid for all

members of the intramolecular hydrogen migrations in alkyls.

Thus, approximate expressions within the RC-TST/LER

framework for these 1,3 to 1,6 H-shift classes for alkyls would

be useful and will be reported in a future study.

References

1 S. W. Benson, Thermochemical Kinetics, Wiley, New York, 1976.2 J. A. Miller, R. Knee and C. K. Westbtook, Annu. Rev. Phys.Chem., 1990, 41, 317.

3 H. J. Curran, P. Gaffuri, W. J. Pitz and C. K. Westbrook,Combust. Flame, 1998, 114, 149–177.

4 H. J. Curran, P. Gaffuri, W. J. Pitz and C. K. Westbrook,Combust. Flame, 2002, 129, 253–280.

5 S. Dobe, T. Berces, F. Reti and P. Marta, Int. J. Chem. Kinet.,1987, 19, 895–921.

6 W. Tsang, J. A. Walker and J. A. Manion, Proc. Combust. Inst.,2007, 31, 141.

7 W. Tsang, W. S. McGivern and J. A. Manion, Proc. Combust.Inst., 2009, 32, 131–138.

8 L. C. Jitariu, L. D. Jones, S. H. Robertson, M. J. Pilling and I. H. J.Hillier, J. Phys. Chem. A, 2003, 107, 8607–8617.

9 B. Viskolcz, G. Lendvay, T. Kortvelyesi and L. Seres, J. Am.Chem. Soc., 1996, 118, 3006–3009.

10 B. Viskolcz, G. Lendvay and L. J. Seres, J. Phys. Chem. A, 1997,101, 7119–7127.

11 D. M. Matheu, W. H. Green and J. M. Grenda, Int. J. Chem.Kinet., 2003, 35, 95–119.

12 B. Bankiewicz, L. K. Huynh, A. Ratkiewicz and T. N. Truong,J. Phys. Chem. A, 2009, 113, 1564–1573.

13 M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria,M. A. Robb, J. R. Cheeseman, J. A. Montgomery, Jr.,T. Vreven, K. N. Kudin, J. C. Burant, J. M. Millam,S. S. Iyengar, J. Tomasi, V. Barone, B. Mennucci, M. Cossi,G. Scalmani, N. Rega, G. A. Petersson, H. Nakatsuji, M. Hada,M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida,T. Nakajima, Y. Honda, O. Kitao, H. Nakai, M. Klene, X. Li,J. E. Knox, H. P. Hratchian, J. B. Cross, C. Adamo, J. Jaramillo,R. Gomperts, R. E. Stratmann, O. Yazyev, A. J. Austin,R. Cammi, C. Pomelli, J. W. Ochterski, P. Y. Ayala,K. Morokuma, G. A. Voth, P. Salvador, J. J. Dannenberg,V. G. Zakrzewski, S. Dapprich, A. D. Daniels, M. C. Strain,O. Farkas, D. K. Malick, A. D. Rabuck, K. Raghavachari,J. B. Foresman, J. V. Ortiz, Q. Cui, A. G. Baboul, S. Clifford,J. Cioslowski, B. B. Stefanov, G. Liu, A. Liashenko, P. Piskorz,I. Komaromi, R. L. Martin, D. J. Fox, T. Keith, M. A. Al-Laham,C. Y. Peng, A. Nanayakkara, M. Challacombe, P. M. W. Gill,B. Johnson, W. Chen, M. W. Wong, C. Gonzalez and J. A. Pople,Gaussian 03, Revision A.1, Gaussian, Inc., Pittsburgh, PA, 2003.

14 T. N. Truong, W. T. Duncan and M. Tirtowidjojo, Phys. Chem.Chem. Phys., 1999, 1, 1061–1065.

15 T. N. Truong, J. Chem. Phys., 2000, 113, 4957–4964.

Fig. 4 Arrhenius plots of the calculated and experimental rate constants for the reactions: (a) (R5): C�CCCCC 2 CC�CCCC, (b) (R6):

C�CCCCC2 CCC�CCC, (c) (R7): C�CCCCCCC2 CCC�CCCCC and (d) (R8): C�CCCCCCC2 CCCC�CCCC. Experimental data are taken

from: ref. 6 for reactions (R5) and (R6) and ref. 7 for reactions (R7) and (R8).

Dow

nloa

ded

by B

all S

tate

Uni

vers

ity o

n 17

/04/

2013

12:

14:1

1.

Publ

ishe

d on

27

July

201

0 on

http

://pu

bs.r

sc.o

rg |

doi:1

0.10

39/C

0CP0

0293

C

View Article Online

This journal is c the Owner Societies 2010 Phys. Chem. Chem. Phys., 2010, 12, 10988–10995 10995

16 T. N. Truong, D. K. Maity and T.-T. T. Truong, J. Chem. Phys.,2000, 112, 24–30.

17 S. Zhang and T. N. Truong, J. Phys. Chem. A, 2003, 107, 1138–1147.18 N. Kungwan and T. N. Truong, J. Phys. Chem. A, 2005, 109,

7742–7750.19 L. K. Huynh, A. Ratkiewicz and T. N. Truong, J. Phys. Chem. A,

2006, 110, 473–484.20 L. K. Huynh, S. Panasewicz, A. Ratkiewicz and T. N. Truong,

J. Phys. Chem. A, 2007, 111, 2035–2252.21 L. K. Huynh, S. Zhang and T. N. Truong, Combust. Flame, 2008,

152, 177–185.22 M. Muszynska, A. Ratkiewicz, L. K. Huynh and T. N. Truong,

J. Phys. Chem. A, 2009, 113, 8327–8336.23 T. H. Dunning, J. Chem. Phys., 1989, 90, 1007.24 D. E. Woon and T. H. Dunning, J. Chem. Phys., 1994, 100, 2975.25 C. Gonzalez and H. B. Schlegel, J. Phys. Chem., 1990, 94, 5523–5527.26 W. T. Duncan, R. L. Bell and T. N. Truong, J. Comput. Chem.,

1998, 19, 1039–1052.27 A. Ratkiewicz, unpublished results.28 A. Fernandez-Ramos, B. A. Ellingson, B. C. Garrett and

D. G. Truhlar, in Reviews in Computational Chemistry, ed.K. B. Lipkowitz, T. R. Cundari, Wiley-VCH, Hoboken;New York, 2007, vol. 23, pp. 125–232.

29 J. C. Corchado, Y.-Y. Chuang, P. L. Fast, W.-P. Hu, Y.-P. Liu,G. C. Lynch, K. A. Nguyen, C. F. Jackels, A. F. Ramos,

B. A. Ellingson, B. J. Lynch, J. Zheng, V. S. Melissas, J. Villa,I. Rossi, E. L. Coitino, J. Pu, T. V. Albu, R. Steckler,B. C. Garrett, A. D. Isaacson and D. G. Truhlar, Polyrate-version9.7, University of Minnesota, Minneapolis, USA, 2007.

30 T. N. Truong, www.cseo.net.31 P. Y. Ayala and H. B. Schlegel, J. Chem. Phys., 1998, 108, 2314.32 J. H. Knox, Combust. Flame, 1965, 9, 197.33 J. W. Ochterski, G. A. Petersson and J. A. Montgomery Jr,

J. Chem. Phys., 1996, 104, 2598.34 K. T. Kuwata, S. T. Dibble, E. Sliz and E. B. Petersen, J. Phys.

Chem. A, 2007, 111, 5032–5042.35 B. J. Lynch and D. G. Truhlar, J. Phys. Chem. A, 2003, 107, 8996.36 J. G. Zhang, Q. S. Li and S. W. Zhang, J. Mol. Model., 2006, 12,

190.37 M. M. Kreevoy, D. Ostovic, D. G. Truhlar and B. C. Garrett,

J. Phys. Chem., 1986, 90, 3766.38 J. A. Manion, R. E. Huie, R. D. Levin, D. R. Burgess Jr.,

V. L. Orkin, W. Tsang, W. S. McGivern, J. W. Hudgens,V. D. Knyazev, D. B. Atkinson, E. Chai, A. M. Tereza,C. Y. Lin, T. C. Allison, W. G. Mallard, F. Westley,J. T. Herron, R. F. Hampson, D. H. Frizzell, NIST ChemicalKinetics Database, NIST Standard Reference Database 17,Version 7.0 (Web Version), Release 1.4.3, Data version 2008.12,National Institute of Standards and Technology, Gaithersburg,Maryland, 20899–8320. Web address: http://kinetics.nist.gov/.

Dow

nloa

ded

by B

all S

tate

Uni

vers

ity o

n 17

/04/

2013

12:

14:1

1.

Publ

ishe

d on

27

July

201

0 on

http

://pu

bs.r

sc.o

rg |

doi:1

0.10

39/C

0CP0

0293

C

View Article Online