Calculation of Kinetic Isotope Effects for Intramolecular Hydrogen Shift

Article pubs.acs.org/JPCA

Calculation of Kinetic Isotope Effects for Intramolecular Hydrogen Shift Reactions Using Semiclassical Instanton Approach Maksym Kryvohuz* Chemical Sciences and Engineering Division, Argonne National Laboratory, Argonne, Illinois 60439, United States S Supporting Information *

ABSTRACT: Primary H/D kinetic isotope effects for the [1,5] hydrogen shift reaction in 13-atomic 1,3-pentadiene and [1,7] hydrogen shift reaction in 23-atomic 7-methylocta-1,3,5triene are calculated using the semiclassical instanton approach. All 33 and 63 internal degrees of freedom, respectively, are treated quantum mechanically with multidimensional tunneling automatically accounted for by the instanton approach. Reactive potential energy surfaces are calculated on-the-fly using mPW1K/6-31+G(d,p) and mPWB1K/6-31+G(d,p) electronic structure methods. The calculated kinetic isotope effects agree well with the previously reported experimental measurements. The analytical expressions of the semiclassical instanton approach allow one to determine quantitative contributions of various physical mechanisms to the calculated kinetic isotope effects. Multidimensional tunneling is found to play important role in both studied hydrogen shift reactions.

I. INTRODUCTION

shown to provide accurate estimates of quantum reaction rate constants15,16,34 and KIEs15,35 for several reactions. In this paper, we calculate primary KIE for two intramolecular hydrogen-transfer reactions. The first one is [1,5] hydrogen shift in (Z)-1,3-pentadiene, which has been experimentally studied in ref 36. Sufficiently large KIEs have been observed for this reaction in the temperature range 460− 500 K, and a significant contribution of quantum mechanical tunneling has been emphasized in earlier theoretical studies with different methods.37−40 The latter reaction has been also studied in ref 35 using the semiclassical instanton approach with empirical valence bond (EVB) potential energy surface (PES). Though the calculated KIEs were found to be in reasonable agreement with the experimental and previous theoretical studies, the accuracy of EVB PES was questionable and resulted in unusual behavior of ZPEs of transverse degrees of freedom. In the present paper, we improve the latter approach by not using a priori constructed empirical potential surfaces. Instead, we compute PES on-the-fly using mPW1K/631+G(d,p) electronic structure method. The second reaction studied in this paper is [1,7] hydrogen shift in 7-methylocta-1,3(Z),5(Z)-triene. Sufficiently large primary KIEs in the temperature range 333.2−388.2 K have been measured in this system,41 suggesting a significant contribution of tunneling to the hydrogen-transfer step. Previous theoretical studies42,43 indicated the important role of hydrogen tunneling in the mechanism of the latter hydrogen

Kinetic isotope effects (KIEs) are very useful tools for studying mechanisms of chemical reactions.1 KIE is defined as the ratio of rate constants for two isotopologue reactants, kl/kh, where kl is the rate constant for lighter isotopologue, and kh is the rate constant for heavier isotopologue. For hydrogen transfer reactions, KIE is usually the ratio kH/kD between rate constants kh and kD for hydrogen and deuterium transfer, respectively. Kinetic isotope effects associated with hydrogen that is directly involved in bond breakage or formation are classified as primary KIEs and are usually larger than the secondary KIEs associated with nonreacting H atoms. Primary KIEs serve as a sensitive probe of the rate limiting reaction steps, whereas secondary KIEs provide useful information about reaction’s transition state. The magnitude of KIE, thus, contains valuable information on the reaction mechanism, and theoretical interpretation of KIE is highly demanded. In many cases, however, large primary KIEs in hydrogen transfer reactions are experimentally observed and are often associated with quantum mechanical tunneling under the reaction barrier.2−12 Inclusion of nuclear quantum effects are therefore required for accurate theoretical prediction of KIEs.7,13,14 In the present paper, we employ a new theoretical approach for evaluation of KIEs in multiatomic systems integrated with first-principles electronic structure calculations.15 The approach is based on the recently proposed version16 of semiclassical instanton theory,17−33 which can estimate thermal reaction rate constants in multiatomic systems at arbitrary temperatures and incorporates nuclear quantum effects such as tunneling and zero-point energy (ZPE) contributions. The method was © 2014 American Chemical Society

Received: October 6, 2013 Revised: December 26, 2013 Published: January 3, 2014 535

dx.doi.org/10.1021/jp4099073 | J. Phys. Chem. A 2014, 118, 535−544

The Journal of Physical Chemistry A

Article

In ref 16 we provided compact analytical expressions for reaction rate constants based on the semiclassical instanton theory

transfer. We calculate primary KIE for this reaction using the semiclassical instanton approach together with on-the-fly computation of PES via the mPWB1K/6-31+G(d,p) method. The same electronic structure method has been employed in ref 43 for theoretical calculation of KIE in 7-methylocta-1,3(Z),5(Z)-triene using variational transition state theory with multidimensional tunneling. Although the primary objective of this paper is to demonstrate the performance of the proposed semiclassical instanton approach in combination with on-the-fly electronic structure computations, application of the method to H-transfer reactions in the considered organic systems provides interesting insights on the physical mechanisms contributing to KIEs. The paper is organized as follows. In section II, the semiclassical instanton approach is reviewed. In section III, the algorithm for locating instanton trajectories is described. The results of computation of the primary KIEs for the [1,5] sigmatropic rearrangement reaction in (Z)-1,3-pentadiene and [1,7] sigmatropic rearrangement in 7-methylocta-1,3(Z),5(Z)-triene are provided in sections IV and V, respectively. The paper is concluded in section VI with discussion of the mechanisms contributing to the calculated KIEs.

k=

⧧ α1(β) 1 e−βV0− βFvib Q r 2ℏβc sin(πβ /βc)

⎛ α (β ) k = ⎜⎜ 2 ⎝ Qr

⎤ ⎞ −>(β)/ ℏ d −1 d ⎡ ⧧ ⎟ E ( ) ( F ) β − β ⎢ vib ⎥ ⎟e ⎦⎠ dβ 2π ℏ2 dβ ⎣

for β ≥ βc

(II.4)

which are valid in the temperature range β ≤ βc and β ≥ βc, respectively. Parameter βc corresponds approximately to the temperature at which instanton collapses to a point. Other parameters appearing in eqs II.3−II.4 are the reactant’s partition function Qr, classical reaction barrier V0, and the effective “ZPE-corrected” Euclidean action > and energy E >(β) = >inst(β) + ℏβFvib(β)

II. INSTANTON APPROACH The semiclassical instanton approach is based on the imaginary free energy (ImF) formulation of reaction rate theory17−19,21−33 and can be considered as a multidimensional generalization of the semiclassical WKB approach.44 The ImF formulation relates reaction rate constant k with the imaginary part of the partition function Q Im Q ReQ

∮ D[x(τ)]e

d [βFvib(β)] dβ

(II.5)

where >inst(β) and Einst(β) are the Euclidean action and energy of the classical trajectory xinst, and the overall “ZPE” contributions of N − 1 degrees of freedom orthogonal to the reaction coordinate are expressed by Fvib(β) =

1 β

N−1

⎛

∑ ln⎜⎝2sinh n=1

λ n (β ) ⎞ ⎟ 2 ⎠

(II.6)

where λn are N − 1 stability parameters of the instanton. The quantity Fvib(β) is the effective total Helmholtz free energy of N − 1 transverse degrees of freedom in the harmonic approximation. At the saddle point Fvib has the following form: 20

(II.1)

where f is a proportionality constant. The Feynman path integral representation of the partition function Q=

(II.3)

and

E(β) = E inst(β) +

k=f

for β ≤ βc

⧧ Fvib (β ) =

−>[x(τ )]/ ℏ

(II.2)

1 β

N−1

⎛

n=1

⎝

∑ ln⎜⎜2sinh

ℏβωn⧧ ⎞ ⎟⎟ 2 ⎠

(II.7)

ℏβω⧧n

because λn = at the saddle point. Coefficients α1(β) and α2(β) account for the barrier effects such as barrier anharmonicity and restricted quantum fluctuations in the vicinity of the saddle point. These coefficients are explicitly given by

diverges in the barrier region of PES. Here, ℏβ 1 >[x(τ )] ≡ ∫ ⎡⎣ 2 x(̇ τ )2 + V (x(τ ))⎤⎦dτ is the classical action 0 functional along the closed quantum trajectories x(τ) of duration ℏβ on inverted PES and β1/(κBT) is the inverse temperature. To converge the integral of eq II.2, analytic continuation into the complex plane is used,17,30 resulting in the appearance of the imaginary part of Q. The steepest descent approximation is then applied to the integral in eq II.2, which allows one to evaluate the right-hand side of eq II.1 analytically and constitutes the basis of the semiclassical instanton approach. The classical trajectories xinst(τ) satisfying the stationary condition δ >(x(τ )) = 0 of the steepest descent approximation are called instantons and are central objects in the semiclassical instanton theory. We note that in addition to the instanton approach, several other stationary-phase methods for evaluation of quantum rate constants have been proposed in the literature, such as quantum instanton approximation by Miller et al45 and stationary phase evaluation of imaginary-time flux−flux correlation functions by Cao et al.46 Application of stationary phase approximation significantly reduces the complexity of quantum integrals and allows one to address quantum phenomena in large multiatomic systems.

2

α1(β) = Δ 2π erf( −Δ)eΔ /2 Δ=

⎞ ββc ⎛⎛ βc ⎞2 ⎤ ⎡ ⧧ ⎜⎜ ⎟ − 1⎟ − d ⎢E(β) − d (βFvib )⎥ ⎜ ⎟ ⎣ ⎦ β β d d 2 ⎝⎝ β ⎠ ⎠

⎛ ⎜ V0 − E(β) + α2(β) = erf⎜ ⎜⎜ d ⎡ − dβ ⎣E(β) − ⎝

β = βc

⎞ ⎟ ⎟ d ⧧ ⎤ ⎟ β ( ) F vib ⎦ ⎟ dβ ⎠

d ⧧ (βFvib ) dβ

(II.8) 15,34,35

We refer the reader to earlier publications for more details on the derivation and physical meaning of eqs II.3−II.4. Evaluation of eqs II.3−II.8 is based on locating ℏβ-periodic classical trajectories xinst(τ), the instantons, on inverted Ndimensional reactive PESs. For a given β, the energy Einst(β) of the instanton trajectory, classical Euclidean action 536



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ℏβ 1 >inst(β) ≡ ∫ ⎡⎣ 2 x ̇ inst(τ )2 + V (x inst(τ ))⎤⎦dτ and N − 1 0 stability parameters λn along the instanton trajectory are then used in eqs II.3−II.8. Instanton trajectories are the only possible periodic trajectories on the inverted reactive PES with a first-order saddle point.20 An N-dimensional reactive PES with a first order saddle point has N − 1 stable degrees of freedom in the saddle point region and a single unstable degree of freedom (the reaction coordinate). The corresponding inverted PES then has N − 1 unstable degrees of freedom and a single stable degree of freedom in the saddle point region. It is periodic oscillations along this stable degree of freedom that are called instantons. However, the latter stable degree of freedom is not necessarily linear in coordinate space and, in most cases, is a curve in N-dimensions, the form of which depends on the geometry of PES. In the following sections, we describe an efficient algorithm for locating ℏβ periodic trajectories on inverted multidimensional reactive PESs (33-dimensional and 63-dimensional PESs in the present paper). The instanton can be assigned with a physical meaning as the optimal tunneling path at a given temperature. At low temperatures (large β’s), the period ℏβ of instanton should be large. Due to the anharmonicities of PES, oscillations with longer periods have larger amplitudes and, thus, spatially longer instantons. The latter corresponds to longer tunneling paths under the multidimensional barrier at lower temperatures. As temperature increases, the magnitude of β as well as the size of instanton decreases, which corresponds to shorter tunneling paths and, thus, to smaller contributions of tunneling. At the crossover temperature β = 2π/(ℏ|ω⧧u |), where ω⧧u is the imaginary frequency of the transition state, instantons shrink down to the saddle point. The latter, however, does not imply that tunneling becomes unimportant at this and higher temperatures. In fact, as it will be shown below, the crossover temperatures for hydrogen shift reactions in (Z)-1,3-pentadiene and 7-methylocta-1,3(Z),5(Z)-triene are lower than the temperatures at which KIEs are reported; however, tunneling still contributes up to the factor of 5 to the rates of these reactions.

the condition δ >(x(τ )) = 0 is satisfied turns out to be a stable and rapidly converging procedure.15,34,57−61 As discussed in the refs 15,34, and 35, it is convenient to represent an ℏβ-periodic trajectory x(τ) in terms of Fourier series over the Matsubara frequencies νm = (2πm)/(ℏβ) ⎛ 2πm ⎞ τ⎟ ⎝ ℏβ ⎠

M

xn(τ ) =

∑ Cnmcos⎜ m=0

(III.1)

The Fourier coefficients Cnm are unknown and are to be found in the optimization procedure δ >(x(τ )) = 0, where we have introduced vector C composed of N × M elements Cnm. Substituting eq III.1 into the expression for Euclidean action ℏβ 1 >[x(τ )] = ∫ ⎡⎣ 2 x(̇ τ )2 + V (x(τ ))⎤⎦dτ , one obtains the 0 expression for >(C) N

>(C) =

M

∑∑ n=1 m=1

π 2m2 2 Cnm + ℏβ

∫0

ℏβ

V (x(C, τ ))dτ (III.2)

An optimization algorithm, such as the Newton−Raphson method, is then used to optimize W(C), that is, to find a solution of the system of N × M algebraic equations

d>(C) =0 dC

(III.3)

The details of the optimization algorithm have been described in refs 15,34. Coefficients C are obtained in the iteration procedure C(i + 1) = C(i) − (D2S(C(i)))−1∇S(C(i))

(III.4)

until the convergence criteria |C − C | < ϵ is reached. Each step requires evaluation of the first and second derivatives of >(C) over C. The latter are explicitly given by (i+1)

2π 2j 2 d> = Cnj + 2 ℏβ dCnj

∫0

ℏβ /2

(i)

dV (x(C, τ )) ⎛ 2πj ⎞ τ ⎟d τ cos⎜ dxn ⎝ ℏβ ⎠ (III.5)

2

22

2π j d> = δnj , mi ℏβ dCnjdCmi

III. LOCATING INSTANTON TRAJECTORIES WITH ON-THE-FLY COMPUTATION OF PES A. General Information. One of the most attractive features of the semiclassical instanton formulation of reaction rate theory is that it only depends on single one-dimensional classical trajectory, the instanton, andunlike other reaction rate theories, such as quantum transition state theory,30,47,48 quantum instanton (QI) rate theory,39,45,49−51 ring polymer molecular dynamics (RPMD) rate theory52−54 or mixed quantum-classical (MQC) rate theory55,56does not require statistical Monte Carlo or molecular dynamics sampling. Because only a single (and spatially short) instanton trajectory is needed, the semiclassical instanton theory can be very efficiently integrated with on-the-fly electronic structure calculations of reactive PES.15,57−61 Locating classical ℏβ-periodic trajectories on inverted PESs by searching for such initial conditions that a classical trajectory will return to its origin after the time ℏβ is a very inefficient procedure.62 The reason for the latter is exponential divergence of any two arbitrarily close classical trajectories at a rate proportional to the largest of the N − 1 stability parameters λn. Instead, variational optimization of the geometry of x(τ) until

+2

∫0

ℏβ /2

d2V (x(C, τ )) ⎛ 2πj ⎞ cos⎜ τ⎟ dxndxm ⎝ ℏβ ⎠

⎛ 2πi ⎞ τ ⎟d τ cos⎜ ⎝ ℏβ ⎠

(III.6)

Numerical calculation of integrals in eqs III.5−III.6 requires knowledge of dV/dx and d2V/(dx dx) along the trajectory x(τ), with τ running from 0 to ℏβ/2. Dividing this trajectory in P intervals, one can calculate the latter derivatives at P + 1 dividing points xp and use accurate interpolation schemes to approximate their values between the dividing points. One therefore needs to know dV(xp)/dx and d2V(xp)/(dx dx) only at P + 1 nuclear configurations, which can be calculated on-thefly (and in parallel15) using accurate electronic structure methods. Most electronic structure packages can calculate single point energy gradients dV(xp)/dx and Hessians d2V(xp)/ (dx dx) at a given nuclear configuration. Thus, one needs to run single-point electronic structure calculations P + 1 times for each iteration step of the optimization algorithm in eq III.4. In 537



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of 1,3-pentadiene transforms into the s-cis form. The s-cis conformer then participates in the hydrogen-transfer step. The overall reaction rate constant is therefore the product of the rate constant of hydrogen-transfer step times the equilibrium constant of the rapid pre-equilibrium between the s-trans and scis conformers. Although the H-transfer step starts from the scis configuration, after being multiplied by the equilibrium constant Keq = Qcis/Qtrans, the rate constant of the total reaction will have the same form as given by eq III.8, but with Qrrot and Qrvip corresponding now to the lowest-energy s-trans conformer. Thus, everywhere below we refer to the lowest-energy conformer, the s-trans conformer, as the reactant and use its rotational and vibrational partition functions in eq III.8. The minimum energy and transition state geometries of 1,3pentadiene optimized using mPW1K/6-31+G(d,p) method are shown in Figures 1a−1c. The energy difference between the transition state and minimum energy s-trans configuration constitutes V0 = 39.34 kcal/mol.

previous works, we have found that P = 16 is sufficient to accurately determine even low-temperature instanton trajectories corresponding to deep tunneling regime. Due to the quadratic convergence of the Newton−Raphson algorithm, the number of iteration steps is usually on the order of 10, making the total number of single point electronic structure calculations for determination of the instanton trajectory to be on the order of 10 × 17. B. Internal Coordinates. In practice, it is convenient to switch into the internal system of coordinates {x} → {Q} consisting of normal coordinates Q at the saddle point. Normal coordinates at the saddle point are very convenient for our purposes because at noncryogenic temperatures instanton trajectories are mostly located in the region of the saddle point. Disregarding the influence of rotations on vibrations,35 the Hamiltonian of internal motion becomes 3N − 6

∑

H vib =

j=1

2 1 ⎛ dQ j ⎞ ⎜⎜ ⎟⎟ + V (Q) 2 ⎝ dt ⎠

(III.7)

3N − 6 vibrational normal coordinates Q are coupled via the anharmonicities of V(Q) and become decoupled at the saddle point. Within the same approximation that neglects rovibrational coupling, the remaining six zero-frequency translational and rotational degrees of freedom can be treated separately from vibrations and result in the following expressions (from II.4−II.3) for reaction rate constants: k=

⧧ Q rot

⧧ α1(β) e−βV0− βFvib r r Q rot Q vib 2ℏβc sin(πβ /βc)

for β ≤ βc (III.8)

and k=

⧧ Q rot

r r α2(β ) Q rot Q vib

e−>(β)/ ℏ

Qrrot

⎤ d −1 d ⎡ ⧧ E ( ) ( F ) β − β ⎥ ⎢ vib ⎦ dβ 2π ℏ2 dβ ⎣

for β ≥ βc

(III.9)

Q⧧rot

where and stand for rotational partition functions at the minimum energy and transition state configurations, respectively, and Qrvip denotes the vibrational partition function at the minimum energy configuration.

IV. PRIMARY KINETIC H/D ISOTOPE EFFECT FOR THE [1,5] HYDROGEN SHIFT IN (Z)-1,3-PENTADIENE In this section, we apply the semiclassical instanton theory, eqs III.8−III.9, integrated with on-the-fly electronic structure computations of PES to calculate primary H/D KIE for the [1,5] sigmatropic rearrangement reaction of (Z)-1,3-pentadiene. The KIE is calculated as the ratio kH/kD of reaction rate constants for the reactions of H-transfer in D2C(CH)3CH3 and D-transfer in H2C(CH)3CD3. Single-point calculations of electronic energies V(xp), energy gradients dV(xp)/dx and Hessians d2V(xp)/(dx dx) were done at the mPW1K/631+G(d,p) level using the Gaussian09 electronic structure package.63 The same electronic structure method has been previously used in ref 40 for calculation of KIE using variational transition state theory with multidimensional tunneling corrections. The [1,5] sigmatropic rearrangement reaction of (Z)-1,3pentadiene proceeds in two steps.38 The first step involves internal rotation in which the lowest energy s-trans conformer

Figure 1. Optimized geometries of 1,3-pentadiene at the mPW1K/631+G(d,p) level. (a) Global minimum energy s-trans configuration, (b) local minimum energy s-cis configuration, and (c) transition state configuration. 538



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Figure 2. Instanton in D2C(CH)3CH3 at 150 K. (a) Front and (b) top views.

In order to use eqs III.8−III.9, one first needs to find βc. In ref 16, we showed that the value of βc is the root of equation Winst(β) −

ℏ 2

3N − 7

∑ j=1

Euclidean actions >(β) and energies E(β) defined in eqs II.5 as well as 32 stability parameters λn were calculated for several temperatures below Tc⧧ and are provided in the Supporting Information. Interpolation with quadratic polynomial was then used to construct continuous functions >(β), E(β), and λn(β) in the latter temperature range. Parameter βc is determined by numerical solution of eq IV.1. For the isotopologue D2C(CH)3CH3, we find that βc corresponds to the temperature Tc = 347.3 K, and for H2C(CH)3CD3, βc corresponds to the temperature Tc = 278.4 K. Physically, the latter implies that contributions of the transverse degrees of freedom influence the curvature of the effective one-dimensional reaction barrier and make the unstable frequency at the top of the effective 1D barrier to be 357.8/347.3 = 1.03 times smaller than the unstable frequency at the saddle point of the original PES for the D2C(CH)3CH3 isotopologue and 283.4/ 278.4 = 1.02 times smaller for the H2C(CH)3CD3 isotopolgue. Experimental KIEs for the [1,5] sigmatropic rearrangement reaction of (Z)-1,3-pentadiene were reported in the temperature range 460 − 500 K. Since both Tc’s lie below this temperature range, eq II.3 should be used for the calculation of rate constants. The calculated KIE is shown in Figure 3 as a function of temperature and is in rather good agreement with experimental results. While the magnitudes of KIE are slightly lower than the experimentally reported ones, the temperature behavior of KIE is reproduced very well. We also note that the calculated KIE are significantly larger than the KIE of the classical transition state theory (TST) without tunneling, suggesting an important role of tunneling phenomena. In section VI we discuss factors contributing to the computed KIE.

⎛ β ℏω⧧ ⎞⎛ ⎞ dλ j ⎟⎜λj ,inst − β j ,inst ⎟ = 0 coth⎜⎜ ⎟ dβ ⎠ ⎝ 2 ⎠⎝ (IV.1)

and lies in the vicinity of 2π/(ℏ|ω⧧u |), where ω⧧u is the unstable frequency at the saddle point. λj,inst are 3N − 7 nonzero stability parameters of the instanton trajectory. The imaginary frequencies of the optimized TS geometries of D2C(CH)3CH3 and H2C(CH) 3CD3 are 1563i cm−1 and 1238i cm −1, respectively, and correspond to the respective crossover temperatures T⧧c = 357.8 K and T⧧c = 283.4 K. The numerical search for βc should be therefore conducted in the vicinity of these temperatures (i.e., by locating several instanton trajectories at temperatures below T⧧c and using their stability parameters in eq IV.1). After several instantonseach corresponding to some temperature below T⧧c are found, their Einst(β), >inst(β) and Fvib(β) substituted to eqs III.8−III.9 allow one to calculate reaction rate constants k. Instanton trajectories were searched for in the space of 33 vibrational normal coordinates Q of the transition state configuration using the Fourier representation approach described in the previous section Mk

Qk =

⎛ 2πm ⎞ τ⎟ ⎝ ℏβ ⎠

∑ Ckmcos⎜ j=1

(IV.2)

We used Mk = 8 lowest-order Fourier coefficients for the unstable mode and Mk = 6 lowest-order Fourier coefficients for each of the 32 stable modes. Instanton trajectories were optimized using the procedure described in the previous section. The path of the instanton xinst(Q(τ)) between its turning points was divided in 16 equal intervals. The ground state electronic energy, energy gradient, and Hessian were calculated in parallel at each of the 17 dividing point nuclear configurations using the DFT mPW1K/6-31+G(d,p) method and were then used in the optimization algorithm as described in the previous section. A typical instanton trajectory (i.e., tunneling paths of atoms) in D2C(CH)3CH3 is shown in Figure 2a and b.

V. PRIMARY KINETIC H/D ISOTOPE EFFECT FOR THE [1,7] HYDROGEN SHIFT IN 7-METHYLOCTA-1,3,5-TRIENE The semiclassical instanton approach described in previous sections can be directly extended to larger systems, such as 23atomic 7-methylocta-1,3,5-triene compound. Here, we report the results of calculations of the primary H/D KIE for the [1,7] sigmatropic rearrangement reaction in the latter organic molecule. The KIE is calculated as the ratio kH/kD of rate constants for the reactions of H-transfer in 7-methylocta-1,3,5triene and D-transfer in the monodeuterated 7-methylocta1,3,5-triene.41 On-the-fly electronic structure calculations were 539



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Figure 3. Temperature dependence of the primary kinetic H/D isotope effect for the [1,5] hydrogen shift in (Z)-1,3-pentadiene. Experimental results36 (solid circles); canonical variational transition state theory with the multidimensional small curvature tunneling correction on mPW1K/6-31+G(d,p) PES38 (open squares); classical transition state theory (dashed line); semiclassical instanton results of the present paper with mPW1K/6-31+G(d,p) PES (solid line). The experimental data point at 400 K was obtained from the Arrhenius law reported in ref 36.

done at the mPWB1K/6-31+G(d,p) level using the Gaussian09 electronic structure package.63 The same electronic structure method has been previously used in ref 43 for calculation of KIE using variational transition state theory with multidimensional tunneling corrections. The optimized minimum energy and transition state geometries of 7-methylocta-1,3,5-triene are shown in Figures 4a−4c. The most stable conformation (R1) of 7-methylocta1,3,5-triene was found in ref 43 to be of s-trans, s-trans geometry and is pictured in Figure 4a. Similarly to the discussion in section IV, we consider this conformer to be the reactant in our instanton analyses, the vibrational and rotational partition functions of which are used in eqs III.8−III.9. The transition state configuration (TS) is shown in Figure 4c and is V0 = 26.67 kcal/mol higher in energy than the lowest-energy conformation R1. Conformation (R2) which has the most suitable geometry for the [1,7] hydrogen transfer step is also shown in Figure 4b and has energy 7.7 kcal higher than the lowest-energy conformation R1. The frequencies of unstable modes at the saddle point were found to be 1318i cm−1 and 1082i cm−1 for 7-methylocta-1,3,5triene and its monodeuterated isotopologue, respectively, which correspond to the crossover temperatures 301.8 and 247.8 K. The latter temperatures indicate the temperatures below which one should search for the instanton trajectories to determine βc from eq IV.1 and use physical properties of these trajectories in eqs III.8−III.9 to calculate reaction rate constants. Instanton trajectories were searched for in the space of 63 vibrational normal coordinates of the transition state configuration using the procedure described in sections III and IV. A typical instanton trajectory (i.e., tunneling paths of atoms) during the process of hydrogen transfer in 7methylocta-1,3,5-triene is shown in Figure 5. It is interesting to note that many atoms and degrees of freedom are actually involved in the process of H-transfer via multidimensional tunneling at temperatures lower than the crossover temperature, as can be seen from Figure 5. The inverse temperature βc appearing in EqsIII.8−III.9 can be found from eq IV.1 using stability parameters of instantons

Figure 4. Optimized geometries of 7-methylocta-1,3,5-triene at the mPWB1K/6-31+G(d,p) level. (a) Global minimum energy R1 configuration, (b) local minimum energy R2 configuration, and (c) transition state configuration.

at several temperatures below T⧧c and vibrational frequencies ω⧧j of the transition state configuration. The obtained values of βc correspond to temperatures Tc = 275 K and Tc = 233 K for the nondeuterated and monodeuterated 7-methylocta-1,3,5triene compounds, respectively, and effectively stand for the ZPE-corrected crossover temperatures for transition from under-the-barrier to over-the-barrier reaction mechanism. By comparing T⧧c and Tc, one can see that the imaginary frequency of the effective reaction barrier is 301.8/275 = 1.10 times lower than the corresponding imaginary frequency at the saddle point of classical PES for the reaction of hydrogen transfer and is 247.8/233 = 1.06 times lower for the reaction of deuterium transfer. Experimental KIEs for the [1,7] sigmatropic rearrangement reaction in 7-methylocta-1,3(Z),5(Z)-triene were reported in the temperature range 333.2−388.2 K, which is higher than either of the crossover temperatures Tc. The latter implies that 540



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In addition, we emphasize that disappearance of instantons at temperatures above the crossover temperature does not imply that tunneling becomes unimportant in this temperature range. The latter can be illustrated with Figure.7, where we provide absolute reaction rate constants for H (Figure 7a) and D transfer (Figure 7b) reactions in 7-methylocta-1,3(Z),5(Z)triene. One can see that in the reported temperature range from 333.2 to 388.2 K, which lies far above the crossover temperatures (275 K for H-transfer and 233 K for D-transfer), tunneling increases the rate of hydrogen transfer up to five times.

VI. DISCUSSION In the present paper the primary H/D KIEs of [1,5] hydrogen shift in (Z)-1,3-pentadiene and [1,7] hydrogen shift in 7methylocta-1,3(Z),5(Z)-triene were computed using the semiclassical instanton theory integrated with on-the-fly computations of electronic PES. A good agreement of the calculated KIEs with experimental results is observed for both systems, which is encouraging because no ad hoc correction coefficients or empirical potential energy surfaces were used in the calculations. Additionally, the instanton approach provides a deeper insight on the role of nuclear quantum effects and on the physical mechanisms responsible for the observed KIEs. Indeed, the analytical form of eq III.8 used in the present calculations of reaction rate constant k allows one to divide it into three constituting components k = κanhκtunnkTST, each having clear physical meaning. Here, kTST = (1/2πℏβ)(Q⧧rot/ ⧧ Qrvibrrot)e−βV0−βFvib is the classical reaction rate constant, κtunn = πβ/[βc sin (πβ/βc)] is the tunneling transmission coefficient of a parabolic barrier, and the coefficient κanh = α1(β) accounts for barrier anharmonicity. We analyze each of these components below. A. Classical TST and ZPE Contributions. Zero-point vibrational energies influence the effective height of the reaction barrier. Because the frequency of C−H vibrations is higher than the frequency of C−D vibrations, the effective energy barrier for H atom should be lower than that for deuterium isotope D. The latter difference results in KIE values greater than 1 when calculated using classical transition state theory (TST). The contribution of classical TST part (kTST) to the overall (D) KIEs constitutes k(H) TST/kTST = 3.2 at 400 K and 2.6 at 500 K for the H-shift reaction in (Z)-1,3-pentadiene, 4.07 at 333.3 K, and 3.34 at 388.2 K for the H-shift reaction in 7-methylocta1,3(Z),5(Z)-triene. B. Quantum Tunneling. The tunneling contribution at high temperatures is given by the parabolic barrier transmission (D) coefficient κtunn and constitutes from κ(H) tunn/κtunn = 2.5 at 400 K to 1.5 at 500 K to the overall calculated KIE in (Z)-1,3pentadiene, and from 1.8 at 333.2 K to 1.4 at 388.2 K in 7methylocta-1,3(Z),5(Z)-triene. In the present instanton analyses, tunneling occurs under the effective one-dimensional barrier formed by the potential energy along the reaction coordinate (instanton) plus ZPE-like contributions of transverse degrees of freedom expressed in terms of stability parameters along the instanton. As discussed in sections IV and V, inclusion of the effect of transverse degrees of freedom reduces the effective imaginary frequency of the reaction barrier and thus influences tunneling transmittance of such parabolic barrier.

Figure 5. Instanton (optimal tunneling path of atoms) for H-transfer in 7-methylocta-1,3(Z),5(Z)-triene at 180 K. Red lines represent tunneling paths of carbon atoms; blue lines represent tunneling paths of hydrogen atoms.

instantons at these temperatures are of zero length and one should use the high-temperature eq III.8 for evaluation of reaction rate constants. The calculated KIE as a function of temperature is shown in Figure 6 and is compared there with

Figure 6. Temperature dependence of the primary kinetic H/D isotope effect for the [1,7] hydrogen shift in 7-methylocta-1,3(Z),5(Z)-triene. Experimental results with error bars41 (solid circles); canonical variational transition state theory with microcanonically optimized multidimensional tunneling transmission coefficient and mPWB1K/6-31+G(d,p) PES43 (open squares); classical transition state theory (dashed line); semiclassical instanton results of the present paper with mPWB1K/6-31+G(d,p) PES (solid line).

experimental results as well as theoretical results obtained from canonical variational transition state theory (CVT) with multidimensional tunneling correction and from classical transition state theory. A good agreement between the experimental results and the results of the present calculations is observed. In fact, for the hydrogen shift reaction in 7methylocta-1,3(Z),5(Z)-triene the results of the semiclassical instanton theory agrees with experiment better than the results of CVT, which is encouraging because neither tunneling corrections nor zero-point energy corrections are used in the instanton approach (these effects are automatically incorporated in the theory). 541



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Figure 7. Temperature dependence of reaction rate constants for the reaction of H-transfer (a) and D-transfer (b) in 7-methylocta-1,3(Z),5(Z)triene. Experimental results (solid circles); canonical variational transition state theory with microcanonically optimized multidimensional tunneling transmission coefficient on mPWB1K/6-31+G(d,p) PES (long-dashed line); classical transition state theory (dashed line); semiclassical instanton results of the present paper with mPWB1K/6-31+G(d,p) PES (solid line).

which single-point electronic structure calculations need to be performed is given by the product of the number of discretization points of instanton trajectories (which is independent of the number of atoms) and the number of iterations required for convergence of the optimization algorithm (which is also almost independent of system size). These factors make the semiclassical instanton approach an attractive method for automated evaluation of quantum reaction rate constants and, in particular, for KIEs in large multiatomic systems.

We note that the effect of transverse degrees of freedom on the effective barrier shape originates automatically in the present semiclassical instanton approach and does not require ad hoc ZPE-corrections of the barrier shape such as those required for evaluations of multidimensional tunneling transmission coefficients in variational transition state theory. C. Anharmonicity. The coefficient α1(β) in eq III.8 accounts for the anharmonicity of reaction barrier along the reaction coordinate and regularizes the behavior of the parabolic barrier transmission coefficient, which diverges at temperatures close to the crossover temperature. The contribution of anharmonicity effects to KIEs, α1(H)(β)/ α(D) 1 (β), in the considered temperature ranges constitutes from 0.75 at 400 K to 0.95 at 500 K in (Z)-1,3-pentadiene and from 0.81 at 333.2 K to 0.93 at 388.2 K in 7-methylocta1,3(Z),5(Z)-triene. D. Vibrationally Assisted Tunneling. The role of internal vibrational motions in promoting tunneling can be clearly seen from Figures 2 and 5. At low temperatures, several vibrational modes can participate in rearrangement of atoms in the process of tunneling, whereas some of these modes tunnel themselves. However, in the considered high temperature ranges, instanton trajectories are already shrunk to the saddle point and have zero length. This implies that in the reported temperature range, the dominant role of multidimensional vibrations come from their contributions on the effective barrier shape rather than from promoting tunneling dynamically. To summarize, the employed semiclassical instanton approach for calculation of quantum reaction rate constants and KIEs has several advantages. First, it provides explicit analytical expressions for reaction rate constants, which allow one to have clearer understanding of the mechanism of chemical reaction. Second, it automatically incorporates nuclear quantum effects such as multidimensional tunneling and ZPEeffects without resorting to ad hoc corrections. Third, it requires knowledge of PES at sufficiently small number of points and efficiently scales with system’s size, which allows onthe-fly electronic structure computations of reactive PES.15,57,64,65 In fact, the number of nuclear configurations at

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ASSOCIATED CONTENT

S Supporting Information *

Optimized geometries of 1,3-pentadiene and 7-methylocta1,3,5-triene in transition state and minimum energy configurations, vibrational frequencies of their H/D-isotopologues, physical properties of instantons, and stability parameters at different temperatures. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: (630) 252-2893. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS I would like to acknowledge the support of this research by a Director’s Postdoctoral Fellowship, Office of the Director, Argonne National Laboratory, and Dr. Julius Jellinek for helpful discussions and support. I acknowledge Dr. John D. Thoburn for providing the numerical CVT data on KIE in pentadiene. I also acknowledge Dr. Juan M. Lopez-Encarnacion for fruitful discussions on electronic structure methods. This research used resources of the National Energy Research Scientific Computing Center, which is supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC0205CH11231. 542



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Calculation of Kinetic Isotope Effects for Intramolecular Hydrogen Shift

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