DOI:
10.1039/C1SC00225B
(Edge Article)
Chem. Sci., 2011,
2, 21992213
Multistructural variational transition state theory. Kinetics of the 1,4hydrogen shift isomerization of the pentyl radical with torsional anharmonicity†
Received
7th April 2011
, Accepted 25th June 2011
First published on 10th August 2011
We present a new formulation of variational transition state theory (VTST) called multistructural VTST (MSVTST) and the use of this to calculate the rate constant for the 1,4hydrogen shift isomerization reaction of 1pentyl radical and that for the reverse reaction. MSVTST uses a multifaceted dividing surface and provides a convenient way to include the contributions of many structures (typically conformers) of the reactant and the transition state in rate constant calculations. In this particular application, we also account for the torsional anharmonicity. We used the multiconfiguration Shepard interpolation method to efficiently generate a semiglobal portion of the potential energy surface from a small number of highlevel electronic structure calculations using the M06 density functional in order to compute the energies and Hessians of Shepard points along a reaction path. The M062X density functional was used to calculate the multistructural anharmonicity effect, including all of the structures of the reactant, product and transition state. To predict the thermal rate constant, VTST calculations were performed to obtain the canonical variational rate constant over the temperature range 200–2000 K. A transmission coefficient is calculated by the multidimensional smallcurvature tunneling (SCT) approximation. The final MSCVT/SCT thermal rate constant was determined by combining a reaction rate calculation in the singlestructural harmonic oscillator approximation (including tunneling) with the multistructural anharmonicity torsional factor. The calculated forward rate constant agrees very well with experimentallybased evaluations of the highpressure limit for the temperature range 300–1300 K, although it is a factor of 2.5–3.0 lower than the singlestructural harmonic oscillator approximation over this temperature range. We anticipate that MSVTST will be generally useful for calculating the reaction rates of complex molecules with multiple torsions.
Introduction
Computational modeling methods for the kinetics of complex reactions by directdynamics algorithms that include tunneling can be expensive when applied with reliable electronic structure methods because of the need to obtain Hessian information, at least along a large portion of the minimumenergy reaction path (MEP). Furthermore, when kinetics modeling is based on the harmonicoscillator approximation, errors can accumulate from the anharmonic internal rotational motions (i.e., torsions) in complex molecules, especially at high temperature. In the present article, we present and illustrate a new formulation of variational transition state theory (VTST) with multidimensional tunneling that is especially designed for treating complex molecules with many conformers generated by torsions, as is often needed for combustion modeling.
Combustion reactions of hydrocarbon fuels play a significant role as energy sources. However, the mechanisms of most combustion reactions have not been fully elucidated. This hampers the rational design of hydrocarbon fuels and pollution and energy efficiency issues are not optimally decided. The situation has now been complicated by the use of biofuel additives. Therefore, understanding the kinetics of hydrocarbon combustion reactions is becoming more and more important. At the same time, computational chemistry is advancing to the stage where it has quantitative predictive capability for the reaction rates of complex species. This predictive capability is also important for other research areas involving gasphase reactions of complex organic molecules, e.g., atmospheric chemistry. The goal of the work reported here is to develop and apply a more accurate method of calculating the rates of such reactions, in particular, reactions with potential energy barriers and multiple conformations of reactants, transition states, or both.
The 1,4hydrogen shift isomerization reaction of 1pentyl radical is a prototype for an important class of reaction in the combustion of hydrocarbons. Alkyl isomerization reactions have been investigated both experimentally^{1–3} and computationally.^{4} The first direct measurement of the 1,4hydrogen shift isomerization of 1pentyl was reported in 2002 by Miyoshi et al.^{2} Hydrogen shift reactions are expected to be dominated by tunneling contributions at 500 K and below. The evaluation of such contributions requires more global information about the potential energy surface than is required to evaluate the overbarrier contributions. Torsions can lead to multiple structures for the transition state. In the case at hand, the reactant 1pentyl, the product 2pentyl and the saddle point all display multistructural character due to internal rotation in the reactants and products and pseudorotation in the transition state. The harmonic oscillator approximation, using only one structure to represent the transition state or reactants, is inaccurate for rateconstant calculations. Therefore, a multistructural approach is required.
In the present work, we propose an efficient method, based either on conventional transition state theory (TST)^{5} or—by using the reaction path and reactionpath potential^{6} on VTST,^{5,7} to compute the thermal rate constants of molecules with multistructural anharmonicity and multidimensional tunneling and we apply it to the 1,4hydrogen shift reaction of 1pentyl radical. In order to include tunneling for this hydrogen shift reaction, instead of applying expensive straight direct dynamics calculations^{8,9} to obtain the reaction path and reactionpath potential, we used the multiconfiguration Shepard interpolation (MCSI) method^{10,11} to efficiently generate the potential energy surface from a small number of highlevel electronic structure calculations. The M06 density functional^{12} is used to optimize the geometry and compute the energies, gradients and Hessians of stationary Shepard points (reactant well, product well and saddle point), as well as nonstationary Shepard points along the reaction path. The Shepardinterpolated potential surface is used to compute the transmission coefficient by utilizing the multidimensional smallcurvature tunneling (SCT) approximation.^{13,14}
A key aspect of the present calculations is that we analyze all of the different conformers (hereafter called structures) of the critical configurations (1pentyl, 2pentyl and the saddle point) that are generated by internal rotation or pseudorotation. For this purpose, the M062X density functional^{12} was used to obtain all of the structures. Then, the conformational–vibrational–rotational partition function was calculated by the multistructural allstructures^{15} (MSAS) and multistructural referencestructure^{15} (MSRS) methods. The thermal rate constant over the temperature range 200–2000 K was evaluated by the POLYRATE program^{16} employing a new formulation of VTST, called multistructural variational transition state theory.
We do not consider pressure effects^{18} in the present article; without considering pressure effects, transition state theory predicts the equilibrium limit of reaction rates, which is sometimes called the highpressure limit for unimolecular reactions.
Theory
1. Rate constant
VTST has been widely applied to barrier reactions in the gas phase,^{7,17} barrierless reactions in the gas phase^{19–21} and reactions at gassolid interfaces,^{22} in solids,^{23} in liquid solutions^{24,25} and in enzymes,^{26,27} and the most appropriate formalism is different for each case.^{28,29} In the present article, we recognize a distinction between two kinds of gasphase barrier reactions, which we may loosely categorize as smallmolecule reactions and complexmolecule reactions. In the smallmolecule case, both the reactants and transition states have only one conformation or they have more than one conformer, but only one is low enough in energy to merit consideration or the others can be accounted for by a separable torsional anharmonicity approximation. An example with only one conformation for both the reactant and transition state would be Cl·+ CH_{4} → H· + CH_{3}Cl. The previous formulation^{7,14,17,30–33} of VTST for gasphase reactions with a barrier is applicable to this kind of case and we label this formulation of VTST as singlestructural VTST. An example of a complex reaction would be the case considered in this paper, namely the isomerization of 1pentyl radical to 2pentyl radical. Here, the reactant has 15 distinguishable conformations and the transition state has four. We label these conformations as “structures” and, here, we propose a multistructural formulation of VTST theory that includes them. Firstly, we need to review singlestructural VTST.
1.1. Singlestructural VTST.
In singlestructural VTST, as is usually applied, the quasiclassical thermal rate constant of canonical variational theory (CVT: the formulation of VTST in which a best compromise transition state is found for each temperature) is^{7,14,17,30–33} 
 (1) 
where s is the progress variable parameter for a sequence of trial transition states and k^{GT} is the generalized transition state rate constant at temperature T for a transition state at s. In practice, we usually take s as the signed distance along the minimum energy path (MEP), which is the path^{6} of steepest descent in isoinertial coordinates from the saddle point down toward the reactants joined at s = 0 to the path of steepest descent down toward the products. By quasiclassical, we mean that the bound vibrations are quantized, but the motion along the reaction coordinate is classical. To include quantum effects on reaction coordinate motion (e.g., tunneling), we multiply by a transmission coefficient: 
k^{CVT/T} = κ(T)k^{CVT}(T)  (2) 
where /T or /…T in a superscript denotes the inclusion of tunneling and κ is the transmission coefficient.
The transmission coefficient that we usually use is called a groundstate (G) transmission coefficient^{17,32} and it is taken as the ratio of the quantum mechanical thermally averaged flux through the groundstate level^{34,35} of the transition state dividing surface divided by the same quantity computed with the classical reaction coordinate motion. The quantity averaged in the divisor is equal to a Heaviside step function at the threshold energy implied by CVT. This threshold energy is the value of the vibrationally adiabatic groundstate potential energy curve^{6,32}V^{G}_{a} at the temperaturedependent CVT transition state, where

V^{G}_{a} = V_{MEP}(s) + ε^{G}(s)  (3) 
where V_{MEP} denotes the minimum potential energy of the transition state dividing surface at s relative to that of the reactants and ε^{G} denotes the local zeropoint vibrational energy of the modes transverse to the reaction coordinate at s. Note that potential energy is also called the Born–Oppenheimer energy, classical energy and zeropointexclusive energy.
The other factor, k^{CVT}, in eqn (2) is computed from eqn (1), where the generalized transition state theory rate constant of eqn (1) may be written as

 (4) 
where β is (k_{B}T)^{−1}, k_{B} is Boltzmann's constant, h is Planck's constant, Q is a partition function, GT denotes a generalized transition state at location s along the reaction coordinate, R denotes reactant, “el” denotes electronic, “rovib” denotes rotationalvibrational, and 
 (5) 
where Φ^{R}_{rel} is the relative transitional partition function of two reactants (note that Φ^{R}_{rel} may be written as Q^{R}_{rel}/V, where V is the volume). Note that Q^{GT}_{rovib} is missing a degree of freedom, in particular, the reaction coordinate, which is the degree of freedom normal to the transition state dividing surface. In eqn (4), the zero of energy for the reactant partition functions is the minimum potential energy of reactants (which we set to equal to zero by convention), not the reactant zeropoint level; and the zero of energy of Q^{GT}_{el} and Q^{GT}_{rovib} is at V_{MEP}, not at a zeropoint level. All of the symmetry numbers for the overall rotation are included in the rotational partition functions.^{36} The symmetry numbers for the internal rotation are included in Q^{R}_{rovib}. Full details of eqns (1)–(5) are given elsewhere^{14,32,33} and general background may be found in introductory treatments.^{38}
Note that, when using eqn (4), there are two approximations that can be employed to evaluate Q^{GT}_{rovib} and Q^{R}_{rovib}. If neither the generalized TS nor the reactant include any torsional motion, Q^{GT}_{rovib} and Q^{R}_{rovib} are taken as a product of a rotational partition function and a vibrational one that can be calculated using the harmonic approximation. This will be called the SSHO approximation, where SS denotes singlestructural. If the generalized TS or the reactant includes any torsional mode, the corresponding partition functions should be calculated by using the singlestructural torsional method proposed previously^{37} or the one introduced below, called SST. The details of calculations using the SSHO and SST approximations are presented below.
The theory reviewed above has usually just been labeled as VTST, although it will be labeled SSVTST when we are specifically distinguishing it from the multistructural generalization, which is presented next.
1.2. Multistructural VTST.
Note that eqn (1) corresponds to minimizing the generalized free energy of activation.^{7} For a multistructural case, we want a generalization where all quantities are computed along a single reaction path. However, in the limit of conventional transition state theory, we want the hypersurface (through which the transition state reactive flux is calculated) to pass through all of the conformations of the transition state, being normal to each of the imaginary normal modes at each of the saddle points. This requires a new formulation of generalized transition state theory that effectively has a multifaceted dividing surface. We present a practical formulation of the new generalization here.
We label the distinguishable saddle point structures as c = 1, 2, …, C (where C is the number of saddle point structures). Similarly we label the distinguishable structures of the reactants and products by j = 1, 2, …, J (where J is the total number of reactant structures) and l = 1, 2, …, L (where L is the total number of product structures), respectively. Correspondingly, U_{j}^{R} and U_{l}^{P} denote the potential energies of structures j and l with respect to the lowest potential energy structure of the reactants, which is always numbered as j = 1; thus U_{1}^{R} is zero by definition. Also, we use the convention that the l = 1 structure is the lowest potential energy structure of the products.
We choose the reaction path to be the MEP through the saddle point with lowest value of V^{G}_{a} that connects the reactants to products and we label this structure with c = c^{*}. For a unimolecular reaction with a simple barrier and a single product, this MEP terminates at a reactant structure for negative s and at a product structure for positive s; we label these structures with j = j^{*} and l = l^{*}. (For a bimolecular reaction, the MEP for negative s would terminate at a well in the entrance valley of the potential energy surface.) The potential energy relative to structure j = 1 as the reactant moves along the MEP that passes through structure c of the saddle point is called V. Then, we replace eqn (1) and 4 by

 (6) 
where 
 (7) 
where Q^{X}_{con–rovib}, with X = GT or R, denotes a conformational–rotational–vibrational partition function (as in section 1.1, it is missing a degree of freedom when X = GT); Q^{R}_{el} and Q^{R}_{conrovib }have their zero of energy at the lowest potential energy of the j = 1 structure of the reactant and Q^{GT}_{el}and Q^{GT}_{conrovib} have their zero of energy at the sdependent V. By writing eqn (7), we assume that reaction barriers are much larger than the internal rotational barriers of the reactant, so that all the conformational structures of the reactant are in local equilibrium with each other during the reaction. If this is not true, one needs to use a multiwell^{18} formalism.
For the reactants, we have

 (8) 
For the generalized transition state, we approximate Q^{GT}_{con–rovib} as the sdependent rotational–vibrational partition function of the structure c = c^{*} (computed just as Q^{GT}_{rovib} for eqn (4)) times the ratio of the multistructural result to the singlestructural result at the saddle point, which is denoted ‡. This yields

Q^{GT}_{con–rovib} = Q^{GT}_{rovib,c*}(T,s)F^{‡}_{MS}(T)  (9) 
where 
 (10) 
where V^{‡}_{c} denotes the potential energy of saddle point structure c with respect to the potential energy of the lowest energy structure, j = 1, of the reactant and Q^{‡}_{rovib,c} has its zero of energy at V^{‡}_{c}. Note that V^{‡}_{c*} is not necessarily the lowest energy saddle point energy.
For computational purposes, we rewrite the generalized transition state theory rate constant of eqn (6) as

k^{MSCVT} = F^{MS}(T)k^{SSCVT}_{1,c*}(T)  (11) 
where 
 (12) 
where k^{SSGT}_{j,c} and k^{SSCVT}_{j,c} are the singlestructural generalized transition state rate constant and the singlestructural CVT rate constant, respectively, computed using the MEP through structure c of the saddle point and using structure j for the reactants. Note that eqn (11) corresponds to j = 1 and c = c^{*}. Comparing eqn (11) and (12) to eqns (6)–(10) yields 
 (13) 
where 
 (14) 
2. Transmission coefficient
Eqn (2) is used for the transmission coefficient for either the singlestructural case of eqn (1) or the multistructural case of eqn (6). In either case, one uses the MEP through the saddle point with the lowest V^{G}_{a} that connects the reactants to produces, and κ^{CVT}^{/T} is the ratio of a thermally averaged approximate quantal transmission probability divided by a thermally averaged quasiclassical one.^{14,32,33} Although it is not done here, one could extend the theory to average variational effects and tunneling probabilities over more than one reaction path.
3. Vibrational anharmonicity
When using eqns (8), (9), (10), (13) and (14), Q^{GT}_{con–rovib}, Q^{‡}_{con–rovib}, and Q^{R}_{con–rovib} can be calculated by using the recently proposed internalcoordinate multistructural approximation^{15} in either its full version, called the multistructuralallstructures (MSAS) approximation, or its more practical version, called the multistructural referencestructure (MSRS) approximation. When employing MSAS calculations, we denote such partition functions as Q^{MSAS, R}_{con–rovib} and Q^{MSAS, ‡}_{con–rovib}. Similarly, we denote the reactant and transition state partition functions as Q^{MSRS, R}_{con–rovib} and Q^{MSRS, ‡}_{con–rovib} when using the MSRS method. Then eqns (10), (11), (13) and (14) can be rewritten as 
 (15) 

 (16) 

 (17) 

 (18) 
where X could be either AS or RS.
In addition, in the present article, we consider two versions each of the MSAS and MSRS approximations. The first versions are the MSASHO and MSRSHO, which treat the included structures harmonically. This includes multistructural anharmonicity (which is the difference between the MSASHO or MSRSHO result and the singlestructural HO result) but not the torsional, bend or stretch anharmonicity. The second versions are called MSAST and MSRST, where T denotes torsional anharmonicity, and these are reviewed below.
We note that the MSAST and MSRST methods explicitly only include the multistructural and torsional anharmonicity. However, other forms of anharmonicity can be implicitly included by using scale factors for vibrational frequencies or, for example, using the Morse approximation for stretches^{31} and the quadraticquartic^{38,39} approximation for bends. In the present article, we make the former choice, i. e., scaling.
Calculation methods
1. Electronic structure calculations
The M062X density functional with the 6311+G(2df,2p) basis set^{40} was applied to optimize the geometry and obtain the frequency for all of the conformers of the reactant 1pentyl radical, the product 2pentyl radical and the transition state. These structures are generated by internal rotation around C–C bonds or by pseudorotation of the ring. The 6311+G(2df,2p) basis set for H and C is also called MG3S and we will use this shortened name for brevity.
For the global minimum conformers of 1pentyl, 2pentyl and the transition state, the M06, M062X, and M08SO^{41} density functionals were used to optimize the geometries of the reactants, products and transition states to obtain the reaction energies and barrier heights. In addition, two multilevel methods, BMCCCSD^{42} and MCG3TS,^{43} and the ab initio CCSD(T)F12b^{44,45} method were used to calculate singlepoint energies at the M062X and M08SO stationary points.
In addition to the already mentioned MG3S basis set, several other basis sets were also utilized for the electronic structure calculations, including 631+G(d,p),^{46} maugccpVTZ,^{47} augccpVTZ,^{48} julccpVTZ,^{49} and def2TZVP.^{50} The M06 and M062X density functional calculations were performed by using the Gaussian 09 program,^{51} and the M08SO density functional calculations were carried out via Gaussian 03 locally modified by MNGFM4.1.^{52} CCSD(T)F12a calculations were performed using Molpro^{53} and the BMCCCSD and MCG3TS calculations were carried out using MLGAUSS.^{54} The integration grid employed for the density functional calculations of frequencies had 99 radial shells and 974 angular points per shell. The frequencies used for the partition function and F^{MSX} calculations were obtained by using M062X/MG3S density functional calculations and multiplying the directly calculated values by an empirical frequency scaling factor^{55} of 0.970.
2. The conformational–vibrational–rotational partition function calculations
The most structurally complete conformational–vibrational–rotational partition functions were calculated by the MSAS method^{15} mentioned above. In the MSASHO and MSAST versions of this method we have 
 (19) 
and 
 (20) 
where Q_{rot,j} is the classical rotational partition function of structure j, Q^{HO}_{j} is the usual normalmode harmonic oscillator vibrational partition function calculated at structure j, Z_{j} is a factor designed to ensure that the partition function reaches the correct highT limit (within the parameters of the model) and f_{j,τ} is an internal coordinate torsional anharmonicity function that, in conjunction with Z_{j}, adjusts the harmonic partition function of structure j for the presence of the torsional motion τ. As mentioned above, the frequencies used for Q^{HO}_{j}, Z_{j}, and f_{j,τ} calculations were scaled by an empirical frequency scaling factor.^{55} We use the label MSASHO to denote the partition function calculated without Z_{j} and f_{j,τ}, that is, with all Z_{j} and all f_{j,τ} equal to unity. When using either the MSASHO or the MSAST versions of the MSAS method, it is not necessary to assign each torsional motion to a specific normal mode. The MSAST approximation reduces to the MSASHO approximation in the lowtemperature limit and it approaches the freerotor result for torsions in the hightemperature limit. The Z_{j} and f_{j,τ} factors are designed to interpolate the partition function (between these limits) in the intermediate temperature range. In principle, more accurate interpolations could be carried out^{15} if one calculated the barrier heights for torsional motions that interconvert the reactant structures with one another and the transition state structures with one another, but an advantage of the method employed here is that we forego this expensive step in the present application.
In the notation of ref. 15, when applying the MSAST method, we assume that some of the torsions are strongly coupled, as specified further in the Results and Discussion section. We use eqns (19) and (20) to obtain the conformational–vibrational–rotational partition functions for the reactant, transition state and product, labeled as Q^{MSASX, R}_{con–rovib}, Q^{MSASX, ‡}_{con–rovib}, and Q^{MSASX, P}_{con–rovib}, respectively, where X is either HO or T.
Note that if one includes only a single conformer in eqns (19) and (20), that is, if J equals one, the corresponding equations reduce to

Q^{SSHO}_{rovib} = Q_{rot,1}Q^{HO}_{vib}  (21) 
and

 (22) 
which can be used to calculate the SSHO and SST partition functions for the SSVTST calculations.
We also calculated the conformational–vibrational–rotational partition functions by the MSRSHO and MSRST methods.^{15} These methods only include the structures that can be obtained by a oneatatime torsion from a reference structure. This restriction yields^{15}

 (23) 

 (24) 
where

 (25) 

 (26) 
where we use
j = 1 as the reference structure. The
P_{j,τ} factor in
eqns (25) and
(26) denotes the number of distinguishable minima along torsion
τ in structure
j.
3. Transmission coefficient calculation using the MCSI algorithm
The transmission coefficient κ(T) that incorporates the tunneling for the hydrogen shift reaction was evaluated using the smallcurvature tunneling approximation^{13} (SCT). The MEP and groundstate vibrationally adiabatic potential curve (V^{G}_{a}) were obtained by the MCSI method^{10,11,56–60} using the MCSI^{61} and MCTINKERATE^{62} programs. Although the reactant, product and transition state all have a multistructural character, we only need one structure of each to do MCSI calculations. We always use the transition state structure c^{*} with the lowest V^{G}_{a} as the saddle point to determine the V_{MEP} and V^{G}_{a} curves in the MCSI calculations. We choose the two structures, j^{*} and l^{*}, that are connected to the transition state c^{*} as the reactant and product wells, respectively, in the MCSI interpolation of the potential energy surface and they were determined by following the paths of steepest descent.
To build the potential energy surface by the MCSI method, nine electronic structure Shepard points were used. All of the information about Shepard points, namely energies, gradients and Hessians, was obtained by M06/631+G(d,p) calculations. The first Shepard point was placed at saddle point c^{*}, as optimized by M06/631+G(d,p). The six nonstationary Shepard points were placed close to the minimum energy reaction path (MEP) using a similar strategy to that in a previous paper^{47} and they were at the following locations: 6.91, 12.21, and 17.30 kcal mol^{−1} below the saddle point on the 1pentyl radical side and 6.26, 14.03, 19.96 kcal mol^{−1} below the saddle point on the 2pentyl radical side. The final two Shepard points are the reactant well and product well structures, j^{*} and l^{*}, mentioned at the end of the previous paragraph.
The MCSI method involves interpolating the offdiagonal element of a diabatic potential matrix,^{56,63} where the diagonal elements are given by molecular mechanics. We use the MM3 method^{64–66} for the molecular mechanics. It was found that using a Morse potential for the C–H bond in the MM3 force field for the Shepard interpolation provided smoother V^{G}_{a}curves than using quadratic stretch potentials and so we used Morse stretches in all of the MM3 calculations. The parameters are: bond energy: 98 kcal mol^{−1},^{67} bond length: 1.12 angstrom^{64–66} and force constant: 4.74 N cm^{−1}.^{64–66}
As discussed in the previous work,^{58,60} we used three different internal coordinate sets for the MCSI calculations: set q for molecular mechanics calculations, set r for Shepard interpolation and set s to calculate the Shepard weighting function. In the present work, the set q is the standard set of MM3 internal coordinates, the set r consists of 42 nonredundant internal coordinates (15 bond distances, 20 bond angles and 7 dihedral angles), and the set s consists of three intraatomic distances shown in Fig. 1.

 Fig. 1 Three intraatomic distances r_{1}, r_{2} and r_{3} used to calculate the weight function (set s) for Shepard interpolation.  
4. Thermal rate constant calculation
All of the rate constant calculations were carried out using the MCTINKERATE^{62} and POLYRATE^{16,68} programs. All of the multistructural anharmonicity calculations were carried out using the MSTor program.^{69}
First, the singlestructural CVT^{17} forward rate constant using harmonic approximation was calculated by eqn (12). Here, the transition state and reactant wells are c = c* and j = j* = 1, which are the same structures used in the MCSI calculations, since j^{*} = 1 in the present work. We labeled the singlestructural CVT forward rate constant as k^{SSCVT}.
In the next step, we calculated F^{‡}_{MSX} and F^{R}_{MSX} (X = AST or RST) using eqns (15) and (16). The final thermal rate constant was determined by eqn (27)

k^{MSCVT/SCT} = κ^{SCT}(T)F^{MSX}(T)k^{SSCVT}(T)  (27) 
where the transmission coefficient
κ^{SCT} was calculated as discussed above and
F^{MSX} is defined in
eqn (18).
Because we used the SCT approximation^{13,14,68} for the groundstate transmission coefficient, the MSCVT/T rate constant may be labeled k^{MSCVT}^{/SCT}.
The thermal reverse rate constant was calculated by using the partition function of the product instead of that of the reactant. The ratio of the forward to reverse rate constant gives the equilibrium constant.
Results and discussion
1. Structures, energies and conformational–vibrational–rotational partition functions of the reactant, product and saddle point
Table 1 lists the zeropointexclusive barrier heights and energies of reaction for the 1,4hydrogen shift isomerization reaction of 1pentyl radical, as calculated by various theoretical methods. The results obtained by the CCSD(T)F12b/julccpVTZ//M062X/MG3S (our best estimate), BMCCCSD//M062X/MG3S and MCG3TS//M062X/MG3S methods agree very well with each other. However, the M062X and M08SO density functionals, with the 631+G(d,p), MG3S, maugccpVTZ, augccpVTZ and def2TZVP basis sets, predict larger barrier heights. It was found that the M06 functional with the 631+G(d,p) basis gives results that are reasonably close to the CCSD(T)F12b/julccpVTZ//M062X/MG3S calculation. Therefore, we chose M06/631+G(d,p) for the singlestructural components of the kinetics calculation, that is, to calculate k^{SSCVT}and κ^{SCT}. However, we expect M062X/MG3S to be more accurate for conformational energy differences, so we calculated F^{MSX} based on M062X/MG3S results.
Table 1 Calculated forward and reverse zeropointexclusive barrier heights and the energies of reaction for the 1,4hydrogen shift isomerization of 1pentyl radical as calculated by various methods (in kcal mol^{−1})
Method 
V
_{f}
^{‡}

V
_{r}
^{‡}

ΔE 
CCSD(T)F12b/julccpVTZ//M062X/MG3S 
24.65 
27.32 
−2.67 
BMCCCSD//M062X/MG3S 
24.60 
27.11 
−2.51 
MCG3TS//M062X/MG3S 
24.12 
26.91 
−2.79 
M062X/631+G(d,p) 
25.27 
28.27 
−3.00 
M062X/MG3S 
25.83 
28.97 
−3.14 
M062X/augccpVTZ 
25.94 
28.99 
−3.05 
M062X/maugccpVTZ 
25.93 
28.98 
−3.05 
M062X/def2TZVP 
25.83 
28.91 
−3.08 
M08SO/MG3S 
25.99 
29.34 
−3.35 
M08SO/augccpVTZ 
25.87 
29.42 
−3.41 
M08SO/maugccpVTZ 
25.94 
29.30 
−3.36 
M08SO/def2TZVP 
26.04 
29.45 
−3.40 
M06/631+G(d,p) 
24.61 
28.42 
−3.81 
M06/MG3S 
25.18 
28.73 
−3.55 
Based on M062X calculations, 1pentyl radical, 2pentyl radical and the transition state have fifteen, twelve and four distinguishable structures, respectively (Fig. 2). The five carbon atoms in the molecule (1pentyl and 2pentyl) are numbered as: H_{2}C^{(1)}–H_{2}C^{(2)}–H_{2}C^{(3)}–H_{2}C^{(4)}–H_{3}C^{(5)} and H_{3}C^{(1)}–H_{2}C^{(2)}–H_{2}C^{(3)}–HC^{(4)}–H_{3}C^{(5)}. In the 1pentyl and 2pentyl radicals, there exist four torsions, which are around the C^{(1)}–C^{(2)}, C^{(2)}–C^{(3)}, C^{(3)}–C^{(4)} and C^{(4)}–C^{(5)} bonds. Table 2 and Fig. 2 show the naming convention that is used for labeling of the structures. For instance, “1a^{+}g^{−}t” means the conformer of 1pentyl radical with the first, second and third dihedral angles in the ranges of 140 to 163, −85 to −55 and −173 to 173 degrees, respectively, and “2g^{−}a^{+}” means the conformer of 2pentyl radical with the first and second dihedral angles in the ranges of −85 to −55 and 140 to 172 degrees, respectively. The numbering of the structures is specified in Table 3.
The dihedral angles used for torsions are H–C(1)–C^{(2)}–C^{(3)}, C^{(1)}–C^{(2)}–C^{(3)}–C^{(4)} and C^{(2)}–C^{(3)}–C^{(4)}–C^{(5)} for 1pentyl and C^{(1)}–C^{(2)}–C^{(3)}–C^{(4)} and C^{(2)}–C^{(3)}–C^{(4)}–C^{(5)} for 2pentyl, respectively.

1

1pentyl radical 

2

2pentyl radical 

TS

transition state 





Abbreviation

Dihedral angle range (deg)

Antiperiplanar 
a^{+}

[140,172] 
a^{−}

[−172, −140] 
Gauche 
g^{+}

[55, 85] 
g^{−}

[−85, −55] 
Syn 
s

[85, 112] or [−112, −85] 
Synperiplanar 
p

[5, 30] or [−30, −5] 
Trans 
t

[−173, −180] and [180, 173] 
Table 3 Sequence numbers and energies^{a} (kcal mol^{−1}) of the structures of the reactant, product and transition state
Structures 
Number 
Energy 
V

V + ε^{G} 
V is the M062X/MG3S zeropointexclusive energy of the structures and V + ε^{G} is the zeropointinclusive energy that is calculated by adding the zeropoint energy, which is calculated using M062X frequencies multiplied by a scale factor from ref. 55.

1pentyl radical 
1a^{−}g^{+}t, 1a^{+}g^{−}t 
j = 1, 2 
0 
89.01 
1a^{−}g^{+}g^{+}, 1a^{+}g^{−}g^{−} 
j = 3, 4 
0.08 
89.55 
1a^{−}g^{+}g^{−}, 1a^{+}g^{−}g^{+} 
j = 5, 6 
1.00 
92.14 
1a^{+}tg^{+}, 1a^{−}tg^{−} 
j = 7, 8 
0.66 
91.02 
1a^{+}tt, 1a^{−}tt 
j = 9, 10 
0.21 
89.52 
1a^{+}tg^{−}, 1a^{−}tg^{+} 
j = 11, 12 
0.74 
91.20 
1stg^{+}, 1stg^{−} 
j = 13, 14 
0.84 
91.64 
1stt

j = 15 
0.36 
90.11 
2pentyl radical 
2g^{−}a^{+}, 2g^{+}a^{−} 
l = 1, 2 
−3.14 
85.96 
2g^{−}g^{−}, 2g^{+}g^{+} 
l = 3, 4 
−3.12 
86.04 
2tg^{−}, 2tg^{+} 
l = 5, 6 
−2.85 
86.85 
2t^{+}a^{−}, 2t^{−}a^{+} 
l = 7, 8 
−2.98 
86.25 
2sg^{−}, 2sg^{+} 
l = 9, 10 
−2.77 
86.99 
2g^{+}p, 2g^{−}p 
l = 11, 12 
−0.59 
90.93 
Transition sate 
TS1, TS2 
c = 1, 2 
25.83 
112.69 
TS3, TS4 
c = 3, 4 
25.83 
112.80 

 Fig. 2 Fifteen structures of 1pentyl radical (a); twelve structures of 2pentyl radical (b); and four structures of the transition state (c). A vertical dashed line is used to separate the mirror image structures.  
In 1pentyl radical, the torsional motion around the C^{(1)}–C^{(2)}, C^{(2)}–C^{(3)} and C^{(3)}–C^{(4)} bonds can contribute to generate distinguishable conformers and there are 15 structures. However, in the 2pentyl radical, the C^{(1)} and C^{(5)} tails in the molecule are both methyl groups, the internal rotations of which do not generate distinct structures. Therefore, only the torsional motions around the C^{(2)}–C^{(3)} and C^{(3)}–C^{(4)} bonds produce distinguishable conformers and there are twelve structures. The transition state structures (Fig. 2(c)) contain a relatively rigid fivemembered ring and a methyl tail group. The ring structure character reduces the number of conformers to four (two pairs of mirror images). Unlike the structures of 1pentyl and 2pentyl, these transition state structures are not connected to each other through torsion motions around C–C bonds; rather, they are connected by pseudorotation, as in cycloalkanes.
The multistructural character of these critical configurations on the potential surface makes the kinetics of the reaction complicated. The new formulation of VTST presented in this article provides a practical way to compute the rate constants while calculating only a single reaction path based on that the saddle point with the lowest zeropointinclusive energy. We found that this structure is the TS1 structure, which is c^{*} = 1 and the reactant and product wells connected with TS1 are 1a^{+}g^{−}t (j^{*} = 1) and 2g^{−}a^{+} (l^{*} = 1), respectively.
The next step is to calculate the partition functions of 1pentyl, 2pentyl and the transition state. The difficulty comes from treating the multistructural character and torsional anharmonicity correctly. In the literature, the torsional anharmonicity has been treated by a variety of approximations based on hinderedrotor models (several references are cited in our previous papers^{15,37}). Representative recent approximations include methods that assume a onetoone correspondence between the torsions and normal modes^{70,71} or require multidimensional phase space integrals over highdimensional potential energy surfaces.^{70} In the present work, the torsions between C^{(1)}–C^{(2)} and C^{(2)}–C^{(3)} in 1pentyl are strongly coupled and assigning the torsions to specific normal modes is impossible; this kind of problem has been addressed previously by van Speybroeck and coworkers.^{72} Here, we treat this kind of situation with our recently presented^{15} MSAST and MSRST approximations that apply internal coordinate correction factors to multistructural harmonic treatments to treat the nonseparable torsions and the transition to the hightemperature limit of free internal rotation. A brief summary of these methods is given above in connection with eqns (19)–(26). A key feature of the new method is that it does not involve scans of the individual torsion angles, which would be inadequate for 1pentyl radical because, for example, when one rotates the molecule around the CH_{2}–C_{4}H_{9} bond, the other torsions cannot be fixed. In general, scans of the individual torsions are inadequate to find all the structures.
We employed eqns (21) and (22) to calculate the partition functions of the structures TS1, 1a^{+}g^{−}t, and 2g^{−}a^{+} using the SSHO and SST approximations, respectively. The conformational–rovibrational partition functions of 1pentyl, 2pentyl, and their transition state were evaluated by both the MSASX and MSRSX (X = HO or T) methods^{15,69} using eqns (19)–(26). The MSAST and MSRST approximations provide more accurate partition functions than either the singlestructural or MSASHO and MSRSHO approximations. Tables 4–6 list information for each structure of the 1pentyl radical, 2pentyl radical and transition state that is used for the partition function calculations carried out by the MSTor program.^{69}All of the calculated partition functions are given in Tables 7–9.
Table 4 Information used for the 1pentyl radical partition function using the MSAST method^{a}
Torsion 

I

W

M

We used NS : SC = 2:2 and M062X/MG3S for this table. The units are cm^{−1} for torsional barrier heights W and frequencies. The unit is amu Å^{2} for internal moments of inertia, I, and the local periodicity M is unitless. See ref. 15 for details of the method.

Structures 1 and 2 (U = 0) 
C(1)–C(2) 
133 
1.71 
281 
2.53 
C(2)–C(3) 
142 
10.91 
2047 
2.53 
C(3)–C(4) 
99 
15.98 
1040 
3 
C(4)–C(5) 
228 
2.92 
998 
3 
Structures 3 and 4 (U = 0.077 kcal mol^{−1}) 
C(1)–C(2) 
161 
1.71 
426 
2.48 
C(2)–C(3) 
131 
17.09 
2819 
2.48 
C(3)–C(4) 
108 
18.38 
1418 
3 
C(4)–C(5) 
247 
3.05 
1228 
3 
Structures 5 and 6 (U = 1.00 kcal mol^{−1}) 
C(1)–C(2) 
157 
1.71 
382 
2.56 
C(2)–C(3) 
132 
15.88 
2521 
2.56 
C(3)–C(4) 
110 
14.79 
1188 
3 
C(4)–C(5) 
254 
3.06 
1297 
3 
Structure 7 and 8 (U = 0.66 kcal mol^{−1}) 
C(1)–C(2) 
126 
1.67 
81 
4.40 
C(2)–C(3) 
110 
14.45 
533 
4.40 
C(3)–C(4) 
125 
11.48 
1177 
3 
C(4)–C(5) 
229 
3.04 
1049 
3 
Structures 9 and 10 (U = 0.21 kcal mol^{−1}) 
C(1)–C(2) 
118 
1.66 
76 
4.28 
C(2)–C(3) 
119 
11.40 
523 
4.28 
C(3)–C(4) 
117 
11.91 
1079 
3 
C(4)–C(5) 
227 
2.87 
976 
3 
Structure 11 and 12 (U = 0.74 kcal mol^{−1}) 
C(1)–C(2) 
109 
1.67 
64 
4.30 
C(2)–C(3) 
107 
14.52 
536 
4.30 
C(3)–C(4) 
123 
11.34 
1135 
3 
C(4)–C(5) 
228 
3.04 
1041 
3 
Structures 13 and 14 (U = 0.84 kcal mol^{−1}) 
C(1)–C(2) 
103 
1.67 
82 
3.57 
C(2)–C(3) 
109 
15.34 
845 
3.57 
C(3)–C(4) 
124 
11.53 
1161 
3 
C(4)–C(5) 
229 
3.04 
1051 
3 
Structures 15 (U = 0.36 kcal mol^{−1}) 
C(1)–C(2) 
108 
1.65 
90 
3.56 
C(2)–C(3) 
121 
11.88 
816 
3.56 
C(3)–C(4) 
116 
11.96 
1068 
3 
C(4)–C(5) 
232 
2.88 
1024 
3 
Table 5 Information used for the 2pentyl radical partition function using the MSAST method^{a}
Torsion 

I

W

M

We used NS : SC = 2:2 and M062X/MG3S for this table. The units are cm^{−1} for torsional barrier heights W and frequencies. The unit is amu Å^{2} for internal moments of inertia, I, and the local periodicity M is unitless. See ref. 15 for details of the method.

Structures 1 and 2 (U = 0) 
C(1)–C(2) 
216 
3.06 
943 
3 
C(2)–C(3) 
145 
10.93 
1160 
3.44 
C(3)–C(4) 
63 
13.66 
275 
3.44 
C(4)–C(5) 
121 
2.94 
285 
3 
Structures 3 and 4 (U = 0.022 kacl/mol) 
C(1)–C(2) 
223 
3.06 
1001 
3 
C(2)–C(3) 
116 
18.82 
2157 
2.92 
C(3)–C(4) 
77 
15.68 
795 
2.92 
C(4)–C(5) 
136 
3.04 
371 
3 
Structures 5 and 6 (U = 0.29 kcal mol^{−1}) 
C(1)–C(2) 
227 
2.91 
990 
3 
C(2)–C(3) 
105 
16.34 
1240 
2.94 
C(3)–C(4) 
72 
10.27 
362 
2.94 
C(4)–C(5) 
114 
3.02 
260 
3 
Structure 7 and 8 (U = 0.16 kcal mol^{−1}) 
C(1)–C(2) 
228 
2.85 
975 
3 
C(2)–C(3) 
117 
12.13 
551 
4.24 
C(3)–C(4) 
62 
10.78 
138 
4.24 
C(4)–C(5) 
122 
2.82 
278 
3 
Structures 9 and 10 (U = 0.37 kcal mol^{−1}) 
C(1)–C(2) 
238 
3.07 
1147 
3 
C(2)–C(3) 
140 
12.86 
1484 
3.56 
C(3)–C(4) 
61 
15.24 
336 
3.56 
C(4)–C(5) 
98 
3.03 
190 
3 
Structures 11 and 12 (U = 0.37 kcal mol^{−1}) 
C(1)–C(2) 
204 
3.05 
836 
3 
C(2)–C(3) 
175 
18.97 
2660 
3.59 
C(3)–C(4) 
135 
10.42 
879 
3.59 
C(4)–C(5) 
223 
3.05 
1003 
3 
Table 6 Information used for the transition state partition function using the MSAST method^{a}
Torsion 

I

W

M

We used NS : SC = 1:0 and M062X/MG3S for this table. The units are cm^{−1} for torsional barrier heights W and frequencies. The unit is amu Å^{2} for internal moments of inertia, I, and the local periodicity M is unitless. See ref. 15 for details of the method.

Structures 1 and 2 (U = 0) 
C(4)–C(5) 
191 
2.98 
718 
3 
Structures 3 and 4 (U = −0.0028 kcal mol^{−1}) 
C(4)–C(5) 
187 
3.03 
700 
3 
Table 7 Calculated conformational–vibrational–rotational partition function of 1pentyl radical using singlestructural and multistructural methods^{a}
T/K 
SSHO^{b} 
SST^{c} 
MSRSHO 
MSRST 
MSASHO 
MSAST^{d} 
All of the partition functions in this table have their zero of energy at the bottom of the potential well for the j = 1 structure of 1pentyl.
The partition function of the structure j = 1, 1a^{+}g^{−}t calculated using SSHO approximation.
The partition function of the structure j = 1, 1a^{+}g^{−}t calculated using SST approximation.
MSAST partition function is calculated using the MSTor program with NS:SC = 2:2.

200 
3.40E − 92 
4.24E − 92 
2.09E − 91 
2.16E − 91 
2.04E − 91 
2.50E − 91 
250 
3.06E − 72 
4.05E − 72 
2.08E − 71 
2.14E − 71 
2.14E − 71 
2.63E − 71 
298.15 
3.24E − 59 
4.52E − 59 
2.37E − 58 
2.45E − 58 
2.54E − 58 
3.12E − 58 
300 
8.52E − 59 
1.19E − 58 
6.25E − 58 
6.44E − 58 
6.69E − 58 
8.22E − 58 
400 
1.13E − 41 
1.72E − 41 
9.26E − 41 
9.66E − 41 
1.05E − 40 
1.28E − 40 
600 
1.01E − 23 
1.71E − 23 
9.51E − 23 
1.01E − 22 
1.14E − 22 
1.34E − 22 
1000 
1.65E − 07 
2.78E − 07 
1.76E − 06 
1.78E − 06 
2.20E − 06 
2.21E − 06 
1500 
1.88E + 03 
2.68E + 03 
2.14E + 04 
1.82E + 04 
2.74E + 04 
2.13E + 04 
2000 
4.05E + 09 
4.63E + 09 
4.79E + 10 
3.29E + 10 
6.18E + 10 
3.71E + 10 
2400 
2.60E + 13 
2.49E + 13 
3.13E + 14 
1.82E + 14 
4.06E + 14 
2.01E + 14 
First, we discuss the MSAST calculations for the 1pentyl radical, 2pentyl radical and the transition state. There are four torsions in 1pentyl and 2pentyl radicals. For 1pentyl, two of the torsions around the C^{(1)}–C^{(2)} and C^{(2)}–C^{(3)} bonds are involved in a strongly coupled^{15} (SC) group. The other two torsions, around C^{(3)}–C^{(4)} and C^{(4)}–C^{(5)} bonds are considered to be nearly separable^{15} (NS). For 2pentyl, the torsions around the C^{(2)}–C^{(3)} and C^{(3)}–C^{(4)} bonds are treated as a strongly coupled (SC) group. The other two torsions, around the C^{(1)}–C^{(2)} and C^{(4)}–C^{(5)} bonds, are considered to be nearly separable (NS). Tables 7 and 8 give the conformational–vibrational–rotational partition functions of the 1pentyl and 2pentyl radicals using NS:SC =2:2. In the transition state, there is only one torsion around the bond between the methyl group and the fivemembered ring. The conformational–vibrational–rotational partition functions are given in Table 9.
Table 8 Calculated conformational–vibrational–rotational partition functions of the 2pentyl radical using singlestructural and multistructural methods^{a}
T/K 
SSHO^{b} 
SST^{c} 
MSRSHO 
MSRST 
MSASHO 
MSAST^{d} 
All of the partition functions in this table have their zero of energy at the bottom of the potential well for the l = 1 structure of 2pentyl.
The partition function of the structure l = 1, 2g^{−}a^{+} calculated using SSHO approximation.
The partition function of the structure l = 1, 2g^{−}a^{+} calculated using SST approximation.
MSAST partition function is calculated using the MSTor program with NS:SC = 2:2.

200 
4.30E − 92 
6.16E − 92 
4.00E − 91 
5.63E − 91 
4.00E − 91 
5.81E − 91 
250 
4.17E − 72 
6.25E − 72 
4.27E − 71 
6.24E − 71 
4.22E − 71 
6.29E − 71 
298.15 
4.27E − 59 
6.59E − 59 
4.71E − 58 
7.00E − 58 
4.98E − 58 
7.50E − 58 
300 
1.22E − 58 
1.88E − 58 
1.34E − 57 
2.00E − 57 
1.31E − 57 
1.98E − 57 
400 
1.69E − 41 
2.67E − 41 
2.08E − 40 
3.11E − 40 
2.00E − 40 
2.99E − 40 
600 
1.57E − 23 
2.38E − 23 
2.17E − 22 
3.09E − 22 
2.06E − 22 
2.84E − 22 
1000 
2.56E − 07 
3.18E − 07 
3.94E − 06 
4.53E − 06 
3.70E − 06 
4.01E − 06 
1500 
2.88E + 03 
2.64E + 03 
4.70E + 04 
3.94E + 04 
4.40E + 04 
3.43E + 04 
2000 
6.18E + 09 
4.19E + 09 
1.04E + 11 
6.39E + 10 
9.68E + 10 
5.57E + 10 
2400 
3.95E + 13 
2.15E + 13 
6.72E + 14 
3.31E + 14 
6.29E + 14 
2.89E + 14 
Table 9 Calculated conformational–vibrational–rotational partition function of the transition state using singlestructural and multistructural methods^{a}
T/K 
SSHO^{b} 
SST^{c} 
MSRSHO 
MSRST 
MSASHO 
MSAST^{d} 
All partition functions in this table have their zero of energy at the bottom of the potential well for the c = 1 structure of the transition state.
The partition function of the TS1 calculated using SSHO approximation.
The partition function of the TS1 calculated using SST approximation.
MSAST partition function is calculated using the MSTor program with NS:SC = 1:0.

200 
2.39E − 90 
2.55E90 
8.05E − 90 
8.60E − 90 
8.05E − 90 
8.60E − 90 
250 
5.63E − 71 
6.13E71 
1.93E − 70 
2.11E − 70 
1.93E − 70 
2.11E − 70 
298.15 
2.40E − 58 
2.66E58 
8.33E − 58 
9.26E − 58 
8.33E − 58 
9.26E − 58 
300 
6.12E − 58 
6.80E58 
2.13E − 57 
2.37E − 57 
2.13E − 57 
2.37E − 57 
400 
2.32E − 41 
2.65E − 41 
8.17E − 41 
9.37E − 41 
8.17E − 41 
9.37E − 41 
600 
5.37E − 24 
6.29E − 24 
1.92E − 23 
2.25E − 23 
1.92E − 23 
2.25E − 23 
1000 
2.64E − 08 
3.02E − 08 
9.55E − 08 
1.09E − 07 
9.55E − 08 
1.09E − 07 
1500 
1.53E + 02 
1.63E + 02 
5.54E + 02 
5.89E + 02 
5.54E + 02 
5.89E + 02 
2000 
2.22E + 08 
2.19E + 08 
8.06E + 08 
7.95E + 08 
8.06E + 08 
7.95E + 08 
2400 
1.14E + 12 
1.06E + 12 
4.13E + 12 
3.85E + 12 
4.13E + 12 
3.85E + 12 
Tables 7–9 show that at low temperature (200–250 K), the differences of the partition functions calculated by the multistructuralallstructure harmonic oscillator method (MSASHO) and MSAST for 1pentyl and the transition sate are within 6–19%. For 2pentyl, the deviations are larger at 31–33%. As the temperature increases, the MSHO approximation changes from being an underestimate to being an overestimate. By 2400 K, the MSASHO results overestimate the MSAST ones by 102% for 1pentyl, 7% for the transition state and 118% for 2pentyl. We will use the more accurate MSAST method for the final kinetics calculations.
Next, we discuss the MSRST calculations for the 1pentyl and 2pentyl radicals. (The transition state conformers are not generated by torsions, thus MSRS is considered to be the same as MSAS for the TS partition function calculations on the reaction considered in this paper, but these methods are not the same for the reactants or products.) When employing the MSRS method, we need to identify all of the structures obtained by independent internal rotations from a reference structure. In this work, we use the global minimum structures 1a^{+}g^{−}t and 2g^{−}a^{+} as the reference structures for 1pentyl and 2pentyl radicals, respectively. For 1pentyl, the internal rotation around the C^{(1)}–C^{(2)} bond in the reference structure 1a^{+}g^{−}t does not generate any distinguishable structures, but the independent rotations around the C^{(2)}–C^{(3)} and the C^{(3)}–C^{(4)} bonds produce the other four structures used in the MSRS calculations. They are 1a^{−}g^{+}g^{+}, 1a^{−}g^{+}g^{−}, 1a^{−}tt and 1a^{−}g^{+}t. For 2pentyl, independent rotations around the C^{(2)}–C^{(3)} and the C^{(3)}–C^{(4)} bonds in the reference structure 2g^{−}a^{+} generate the other four structures: 2g^{+}a^{−}, 2ta^{+}, 2g^{−}g^{−} and 2g^{−}s. The detailed information about each structure of the 1pentyl and 2pentyl radicals used for the MSRS calculation are listed in the electronic supplementary information†and the MSRST partition functions are given in Tables 7 and 8. It is found that the deviations of the MSRST partition functions for the 1pentyl radical from the MSAST ones are 6% at 200 K, increasing to 25% at 400–600 K, then decreasing to 9% at 2400 K. For the 2pentyl radical, the deviations are smaller at 200–1000 K, but they increase to 15% by 2400 K. However, the MSRST method needs much less structural information than the MSAST method and it saves considerable time in the calculations.
2. Transmission coefficient
Fig. 3 displays the V_{MEP} and V^{G}_{a} curves. Both the V_{MEP} and V^{G}_{a} curves are well converged, with only six nonstationary Shepard points, which is consistent with earlier work.^{56} The transmission coefficient calculated using the SCT approximation is also well converged. Therefore, only nine highlevel electronic structure theory Hessians along the reaction path were needed to interpolate the potential curves used for the following rate constant calculations.

 Fig. 3 Calculated V_{MEP} and groundstate vibrationally adiabatic potential (V^{G}_{a}) vs. the reaction coordinate s (scaled to a reduced mass of one amu) for the 1,4hydrogen shift isomerization reaction of 1pentyl radical. This figure is based on M06/631+G(d,p).  
Because the MCSI method uses just a few Hessian calculations, it saves considerable computation time relative to straight direct dynamics calculations. Also, compared to the IVTSTM algorithm,^{73} the MCSI method provides a more accurate interpolation of the V_{MEP} and V^{G}_{a} curves; IVTSTM needs relatively large number of Hessians for a converged result and those Hessians are only located close to the saddle point, but the Hessians in the MCSI calculations are widely spaced along the reaction path. This makes the interpolation more reliable and robust.
Fig. 4 shows an Arrhenius plot of the SCT transmission coefficient. This shows that the reaction has a very large tunneling contribution at temperatures below 500 K. In particular, the quantum effects on the reaction coordinate motion increase the rate constant by factors of 7.6, 29, 821 and 2.6 × 10^{4} at 500, 400, 300 and 250 K, respectively. However, above 1000 K, the quantum effects on the reaction coordinate motion increase the rate by less than 60%.

 Fig. 4 The calculated common logarithm of the SCT transmission coefficient κ vs. reciprocal temperature (times a thousand).  
3. SSVTST rate constant calculations
The forward and reverse singlestructural rate constants, k^{SSCVT}^{/}^{SCT}, were calculated using both the SSHO and the SST approximations and they are labeled as k^{CVT/SCT}_{SSHO} and k^{CVT/SCT}_{SST}, respectively. The results are given in the first and second rate constant columns of Tables 10 and 11. The SS results of Tables 10 and 11 replace the older singlestructure results of ref. 4. The calculated rate constant in ref. 4 agreed well with experiments, but that resulted from a cancellation of three errors: (1) the reactant structure used was not the lowestenergy structure; (2) the V^{G}_{a} curve was poorly interpolated due to a poor choice of interpolation variable, causing the transmission coefficient to be underestimated; and (3) multistructural and torsional anharmonicity were neglected.
Table 10 Forward SSVTST and MSVTST thermal rate constants (in s^{−1}) for the 1,4hydrogen shift isomerization reaction of 1pentyl radical to produce 2pentyl radical at various temperatures^{a}
T/K 
k
^{CVT/SCT}_{SSHO}

k
^{CVT/SCT}_{SST}

k
^{CVT/SCT}_{MSRSHO}

k
^{CVT/SCT}_{MSRST}

k
^{CVT/SCT}_{MSASHO}

k
^{CVT/SCT}_{MSAST}

Includes variational effects, torsional anharmonicity and tunneling.

200 
5.27E − 06 
4.51E − 06 
2.89E − 06 
2.98E − 06 
2.96E − 06 
2.58E − 06 
250 
9.19E − 04 
7.56E − 04 
4.63E − 04 
4.92E − 04 
4.50E − 04 
4.01E − 04 
298.15 
4.80E − 02 
3.81E − 02 
2.28E − 02 
2.45E − 02 
2.13E − 02 
1.92E − 02 
300 
5.12E − 02 
4.07E − 02 
2.43E − 02 
2.62E − 02 
2.27E − 02 
2.06E − 02 
400 
2.11E + 01 
1.58E + 01 
9.07E + 00 
9.97E + 00 
8.00E + 00 
7.52E + 00 
600 
3.27E + 04 
2.26E + 04 
1.24E + 04 
1.37E + 04 
1.04E + 04 
1.03E + 04 
1000 
2.59E + 07 
1.76E + 07 
8.78E + 06 
9.91E + 06 
7.03E + 06 
7.98E + 06 
1500 
9.34E + 08 
6.98E + 08 
2.96E + 08 
3.71E + 08 
2.32E + 08 
3.17E + 08 
2000 
6.02E + 09 
5.19E + 09 
1.85E + 09 
2.65E + 09 
1.43E + 09 
2.35E + 09 
2400 
1.56E + 10 
1.51E + 10 
4.71E + 09 
7.53E + 09 
3.62E + 09 
6.81E + 09 
Table 11 SSVTST and MSVTST thermal rate constants (in s^{−1}) for the 1,4hydrogen shift isomerization reaction of 2pentyl radical to produce 1pentyl radical at various temperatures^{a}
T/K 
k
^{CVT/SCT}_{SSHO}

k
^{CVT/SCT}_{SST}

k
^{CVT/SCT}_{MSRSHO}

k
^{CVT/SCT}_{MSRST}

k
^{CVT/SCT}_{MSASHO}

k
^{CVT/SCT}_{MSAST}

Includes variational effects, torsional anharmonicity and tunneling. Note that the reaction in this table is the reverse of that in Table 10.

200 
1.94E − 10 
1.44E − 10 
7.02E − 11 
5.33E − 11 
7.03E − 11 
5.18E − 11 
250 
2.14E − 07 
1.56E − 07 
7.17E − 08 
5.37E − 08 
7.26E − 08 
5.32E − 08 
298.15 
3.45E − 05 
2.47E − 05 
1.08E − 05 
8.11E − 06 
1.11E − 05 
8.20E − 06 
300 
4.09E − 05 
2.95E − 05 
1.29E − 05 
9.67E − 06 
1.32E − 05 
9.73E − 06 
400 
7.93E − 02 
5.73E − 02 
2.27E − 02 
1.74E − 02 
2.36E − 02 
1.81E − 02 
600 
5.87E + 02 
4.54E + 02 
1.52E + 02 
1.25E + 02 
1.60E + 02 
1.36E + 02 
1000 
1.67E + 06 
1.67E + 06 
3.91E + 05 
3.89E + 05 
4.16E + 05 
4.38E + 05 
1500 
1.15E + 08 
1.34E + 08 
2.54E + 07 
3.23E + 07 
2.74E + 07 
3.73E + 07 
2000 
1.03E + 09 
1.49E + 09 
2.22E + 08 
3.55E + 08 
2.38E + 08 
4.08E + 08 
2400 
3.12E + 09 
5.33E + 09 
6.66E + 08 
1.26E + 09 
7.15E + 08 
1.45E + 09 
4. Multistructural anharmonicity torsional factors
As mentioned in section 1 of the Results and Discussion, for the transition state, MSRS is the same as MSAS in the present work. Thus, here we denote both F^{‡}_{MSXT} as F^{‡}_{MSAST}. Table 12 gives the F^{‡}_{MSAST} factors at various temperatures and it is found that the F^{‡}_{MSAST} factors range 3.4–4.2. The magnitude can be qualitatively rationalized by the existence of four conformers for the transition state and the similar energies and frequencies of these conformers. F^{R}_{MSXT} and F^{P}_{MSXT} (X = RS or T) are also given in Table 12. The table shows that the deviations between F^{R}_{MSAST} and F^{R}_{MSRST} are larger than those between F^{P}_{MSAST} and F^{P}_{MSRST}. This observation is consistent with the deviations of the conformational–rotational–vibrational partition functions calculated using the two methods mentioned in section 1 of the Results and Discussion. In addition, it shows that the reactant and product wells have larger multistructural anharmonicity factors than the transition state over the whole temperature range. This result is consistent with their larger number of conformers. However, there is no simple relationship between the multistructural anharmonicity factor and the number of conformers. For instance, the 1pentyl radical has 15 conformers, but its F^{R}_{MSAST} and F^{R}_{MSRST} factors are 7.4–9.6 and 6.4–7.6, instead of 15 at 200–300 K.
Table 12 Multistructure torsional anharmonicity factors with respect to the structure TS1 as the transition state, 1a^{+}g^{−}t as the reactant well and 2g^{−}a^{+} as the product well
T/K 
F
^{‡}_{MSASX}

F
^{R}_{MSRSX}

F
^{R}_{MSASX}

F
^{P}_{MSRSX}

F
^{P}_{MSASX}

HO 
T 
HO 
T 
HO 
T 
HO 
T 
HO 
T 
200 
3.37 
3.60 
6.15 
6.36 
6.00 
7.36 
9.30 
13.26 
9.31 
13.50 
250 
3.43 
3.75 
6.80 
6.98 
6.99 
8.58 
10.25 
15.00 
10.13 
15.10 
298.15 
3.47 
3.86 
7.31 
7.54 
7.84 
9.62 
11.02 
16.27 
10.78 
16.23 
300 
3.48 
3.87 
7.34 
7.57 
7.85 
9.65 
11.02 
16.32 
10.74 
16.27 
400 
3.52 
4.04 
8.19 
8.56 
9.29 
11.34 
12.29 
17.93 
11.82 
17.65 
600 
3.58 
4.19 
9.42 
9.98 
11.29 
13.20 
13.83 
18.69 
13.13 
18.14 
1000 
3.62 
4.12 
10.67 
10.71 
13.33 
13.34 
15.40 
16.34 
14.46 
15.67 
1500 
3.62 
3.86 
11.38 
9.64 
14.57 
11.29 
16.33 
12.41 
15.25 
11.90 
2000 
3.63 
3.59 
11.83 
8.11 
15.26 
9.14 
16.76 
9.36 
15.68 
9.01 
2400 
3.62 
3.39 
12.04 
6.97 
15.62 
7.71 
17.01 
7.56 
15.90 
7.31 
In the k^{CVT/SCT}_{MSAST} calculation, F^{‡}_{MSAST}/F^{R}_{MSAST} varies from 0.49 at 200 K to 0.31 at 1000 K, then back up to 0.44 at 2400 K. A corresponding analysis for the reverse reaction yields F^{‡}_{MSAST}/F^{P}_{MSAST} equal to 0.27, 0.26, and 0.46 at these three temperatures.
All of the multistructural torsional anharmonicity factors F^{‡}_{MSAST}, F^{R}_{MSAST} and F^{P}_{MSAST} depend on the frequencies used for partition function calculations by the MSTor program.^{69} As mentioned above, the frequencies used in the present work are the ones obtained from the M062X/MG3S density functional calculations multiplied by the scale factor of 0.970.^{46} This scale factor is designed to yield an accurate zeropoint energy and, thus, applying it makes the V^{G}_{a} curve more reliable for the transmission coefficient calculations. However, when calculating the conformational–rotational–vibrational partition functions using the multistructural methods, the contributions from the lowfrequency modes dominate and the current scaling factor is not needed to improve the results for those low frequencies. Therefore, we tested the effect of calculating the multistructural torsional anharmonicity factors without using the scale factor. The ratio of the multistructural torsional anharmonicity factors calculated with the scaling factors over that without the scaling factors, that is (F^{X}_{MSAST})_{scale}/(F^{X}_{MSAST})_{noscale} (where X = ‡, R or P), are listed in the electronic supplementary information†. One finds that the effect is 6% or less for 1pentyl, 2% or less for the transition state and 7% or less for 2pentyl. Thus, neither the forward or reverse rate constants are sensitive to scaling the frequencies of the low frequency modes.
5. Final thermal rate constants
We employed the k^{CVT/SCT}_{SSHO} and κ^{SCT}(T) results discussed in sections 2, 3 and 4 of the Results and Discussion to calculate the final thermal rate constant in the highpressure limit using eqn (27). The results are given in the final four columns of Tables 10 and 11, and Fig. 5 shows how the final forward and reverse thermal rate constants vary with the temperature. Fig. 5a compares our final result to the experimentallybased recommendations of the highpressure limit for the forward reaction from refs 1 and 2; the figure demonstrates that the calculated forward rate constants are in excellent agreement with the experimental results of Yamauchi and Miyoshi et al.^{1,2} over the whole temperature range of 300–1300 K for which they make an experimental recommendation. The experimental evaluations of the highpressure limit are less certain above 800 K due to falloff effects.^{2}Fig. 5b shows the calculated results for the reverse reaction.

 Fig. 5 Arrhenius plots of calculated (a) forward and (b) reverse rate constant k^{CVT/SCT}_{MSAST} calculated by MSVTST (black curve) and previous experimental data for the 1,4hydrogen shift isomerization reaction of the 1pentyl radical to produce the 2pentyl radical.  
The next issue to consider is a comparison of the MSCVT/SCT rate constants calculated with the less expensive MSRST approximation to those calculated with the full MSAST approximation. Table 10 shows deviations of only 16% at 200 K, increasing to 33% at 400–600 K, then decreasing again to 11% at 2400 K. The performance is even better for the reverse reaction, where Table 11 shows errors of 8% or less at 200–600 K, increasing to 13% at 1500–2400 K. However, one's sense of encouragement in the accuracy of the less expensive reference structure method must be tempered by the realization that it results in part from a cancellation of errors. Further testing is warranted.
Based on the calculated thermal rate constant, we calculated the Arrhenius activation energies and fitted the rate constant to obtain a formula to predict the rate at any temperature in the highpressure limit. The Arrhenius activation energy is calculated using 
 (28) 
and the results are given in Table 13. Table 13 shows that the E_{a} increases with temperature, but much more slowly than a linear increase in T, whereas linear behavior is predicted by the very popular fitting expression 
 (29) 
Table 13 Forward and reverse activation energies (in kcal mol^{−1}) calculated by MSVTST for the 1,4hydrogen shift isomerization reaction of the 1pentyl radical to produce the 2pentyl radical at various temperatures
T/K 
300 
400 
600 
1000 
1500 
2000 
Forward (MSVTST) 
12.6 
15.6 
18.7 
21.7 
23.4 
26.1 
Reverse (MSVTST) 
16.4 
19.0 
22.9 
25.4 
27.9 
29.5 
Therefore, as recommended previously,^{74} we abandon this popular form and instead fit the calculated rate constants to the more physical functional form given by^{74}

 (30) 
We obtain A = 1.06 × 10^{8} s^{−1}, n = 3.2897, E = 11.436 kcal mol^{−1} and T_{0} = 185.34 K for the forward reaction, and we obtain A = 8.81 × 10^{6} s^{−1}, n = 4.01577, E = 13.865 kcal mol^{−1} and T_{0} = 165.90 K for the reverse reaction. A plot is shown in the electronic supplementary information†.
Conclusions
We have addressed the challenge of developing an efficient and accurate way to predict the thermal rate constant of a reaction with multiple conformational states, and we illustrated the new method here by calculating the rate constant of the 1,4hydrogen shift reaction of the 1pentyl radical and the reverse reaction. The calculation involved several steps. First, we applied the multiconfiguration Shepard interpolation (MCSI) method to obtain the potential energy surface of the reaction by using molecular mechanics to interpolate density functional calculations in the reaction valley that passes through the transition state with the lowest zeropointinclusive energy. Then a singlestructural variational transition state calculation was carried out and the multidimensional tunneling contribution was calculated by the SCT approximation. Finally, the multistructural effect, including the torsional anharmonicity, was added based on the optimizations and frequency calculations for all of the structures of the reactants, products and transition state. These steps together constitute a new formulation of variational transition state theory labelled MSVTST.
We found that multistructural anharmonicity in the harmonic approximation lowers the forwardreaction rate by factors of 0.56, 0.32 and 0.23 at 200, 600 and 2400 K, respectively, and torsional anharmonicity lowers it (again with respect to the singlestructural harmonic approximation) by factors of 0.49, 0.32 and 0.44 at these three temperatures. A large transmission coefficient is found for this reaction at lower temperatures, in particular, 1.2 × 10^{7} and 3.90 at 200 and 600 K, respectively. The calculated final forward thermal rate constant agrees very well with experimentallybased data^{1,2} over the entire temperature range (300–1300 K). The MSVTST method used here can be applied to calculate thermal rate constants for other complex unimolecular or bimolecular reactions involving molecules of even larger size.
Acknowledgements
The authors are grateful to Dr Osanna Tishchenko for help in using the MCSI method. Also, we appreciate helpful discussions with Dr John Alecu. This work was supported by the U. S. Department of Energy, Office of Basic Energy Sciences, under grant no. DEFG0286ER13579 and by the Combustion Energy Frontier Research Center under award no. DESC0001198.
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Footnote 
† Electronic supplementary information (ESI) available: Information used for calculating the conformational–rotational–vibrational partition functions of 1pentyl, 2pentyl and the transition state by MSRSHO and MSRST methods, the transmission coefficient of the reaction using SCT approximation, the optimized geometries of the 1pentyl radical, the 2pentyl radical and the transition state structures, the effect of scaling factors on the multistructural torsional anharmonicity factors and a plot of the calculated MSVTST rate constants with the fitting curves. See DOI: 10.1039/c1sc00225b 

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