Article pubs.acs.org/JPCA

Quantum Dynamics of the Abstraction Reaction of H with Cyclopropane Xiao Shan* and David C. Clary* Physical and Theoretical Chemistry Laboratory, Department of Chemistry, University of Oxford, South Parks Road, Oxford OX1 3QZ, U.K. ABSTRACT: The dynamics of the abstraction reaction of H atoms with the cyclopropane molecule is studied using quantum mechanical scattering theory. The quantum scattering calculations are performed in hyperspherical coordinates with a two-dimensional (2D) potential energy surface. The ab initio energy calculations are carried out with CCSD(T)F12a/cc-pVTZ-F12 level of theory with the geometry and frequency calculations at the MP2/cc-pVTZ level. The contribution to the potential energy surface from the spectator modes is included as the projected zero-point energy correction to the ab initio energy. The 2D surface is fitted with a 29-parameter double Morse potential. An R-matrix propagation scheme is carried out to solve the close-coupled equations. The adiabatic energy barrier and reaction enthalpy are compared with high level computational calculations as well as experimental data. The calculated reaction rate constants shows very good agreement when compared with the experimental data, especially at lower temperature highlighting the importance of quantum tunnelling. The reaction probabilities are also presented and discussed. The special features of performing quantum dynamics calculation on the chemical reaction of a cyclic molecule are discussed.

1. INTRODUCTION In a reduced-dimensionality (RD) quantum scattering calculation,1−6 only a subset of the internal degrees of freedom (DoFs) are treated explicitly. Such methods provide opportunities for dynamical studies of the chemical reactions of polyatomic (typically over 6 atoms) systems, and some of the early works can be found in refs 7−12. In the past decade, our group has applied the RD method to various chemical reactions involving H atom abstractions and exchange processes. These reactions include H + CH4 → H2 + CH3,13−15 H + C2H6 → H2 + C2H5,16−18 H + CH3OH → H2 + CH2OH/CH3O,17 H + C3H8 → H2 + n-C3H7/i-C3H7,17,19 H + CH3NH2 → H2 + CH2NH2/CH3NH,20 Cl + CH4/CHD3 ↔ HCl + CH3/ CD3,21−23 CH3 + CH4 → CH4 + CH3,24 H + C4H10 → H2 + n-C4H9/i-C4H9,25 and H + HCF3 ↔ H2 + CF3.26 In our RD calculations, normally two internal DoFs are treated explicitly in the quantum dynamics calculations and potential surface: the chemical bonds that are formed and broken in a reaction. The contribution of the rest of the DoFs, the spectator modes, to the reaction dynamics is accounted as zero-point energies (ZPEs) in the construction of the two-dimensional (2D) potential energy surface (PES). In this case, the 2D PES has the correct barrier height and energetics of reactants and products. In addition, other modes can be included in the quantum dynamics calculations if of particular interest.18 In most of the earlier works, the 2D PESs were fitted with 2D potential functions, in particular a 25-parameter doubleMorse function16 and two 29-parameter double-Morse functions.13−15,17−24 The two 29-parameter functions differ by the function used to define the position of the transition state of a reaction on the surface.24 The latest improvement to our © XXXX American Chemical Society

method focuses on the construction of the 2D PES, in particular reducing the number of necessary ab initio quantum chemistry calculations. The so-called (1 + 1)D methods utilizes the minimum energy path (MEP) of a reaction and approximate the rest of the PES of such reaction with a harmonic function27 or a one-dimensional Morse function.25,26 The approach we use has some similarities to other methods, such as the reaction path Hamiltonian28 and reaction surface Hamiltonian.29 All of the RD methods are largely dependent on the accuracy of the PES for a reaction, which in turn depends on the ab initio quantum chemistry calculations. In principle, the highest level of theory with complete basis set would yield the most accurate results. However, it would be computationally too expensive to use such a method to construct even a full PES for a chemical reaction such as the title reaction H + cyc‐C3H6 → H 2 + cyc‐C3H5

(R1)

Von Horsten et al.27 tested a series of ab initio methods for the single energy calculations of each grid point on the PES for the H abstraction reactions from noncyclic alkane molecules. We follow this work and make comparisons of two commonly used high level methods in the present study to further test their performances in different systems. The H abstraction from cyclopropane (cyc-C3H6) is of particular interest because of the unusually strong C−H bond30−32 and because of its relevance in combustion processes.33 Received: August 28, 2014 Revised: September 30, 2014

A

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of the transition state (TS), reactants, and products of the R1. The important bond lengths and angles are highlighted in Table 1. It can be seen that our optimized geometry at MP2 level is in a close agreement with Yu et al.43 at CCSD/cc-pVDZ level of theory. In particular when comparing to Yu et al.,43 for the TS structure, our method only underestimates the bond length of H−Ha and Ha−C by ∼3.7% and ∼0.070%, respectively, and the bond angle H−Ha−C by ∼0.95%. Note that in this work we refer to the abstracted H atom as Ha for clarity. In Figure 1d, we illustrate the van der Waals (vdW) complex that is formed by the reacting particles, where the H atom is directly above the center of the C3 ring of cyc-C3H6. On the 2D PES, it can be found in the reactant channel and is a dominant feature. The key geometry coordinates of this complex are also shown in Table 1. The distance between H and the center of the C3 ring is 3.6204 Å. The equivalent H−Ha distance is 3.0731 Å. This structure is a minimum on the 2D PES. When the bond lengths H−Ha and C−Ha are frozen at 3.0731 and 1.0785 Å, respectively, the vdW complex structure would be obtained even if one starts the partial optimization from a collinear geometry for the three atoms. On the 2D PES, the area surrounding this point shows similar bending behavior of the three atoms. At a greater H−Ha distance, the vdW interaction is not as strong, and hence this structure is no longer favored energetically. A near collinear geometry would be obtained from a partial optimization starting from a collinear geometry. These results directly affected our choice of coordinate system to run the scattering calculation with. We shall discuss this point in detail in the next section. Table 2 shows the reaction energetics. Once again we compare our results to ref 43. where the energy is obtained using an extrapolated coupled-cluster/complete basis set (CBS) method with their CCSD/cc-pVDZ geometry, and the extrapolation was based on the cc-pVDZ, cc-pVTZ, and cc-pVQZ basis sets and CCSD and MP2 level of theory. Our energy calculations for the stationary points were done, as mentioned before, at both CCSD(T) and F12a levels of theory. It can be seen that both methods (13.61 and 13.47 kcal/mol for the CCSD(T) and the F12a methods, respectively) show good agreement in estimating the reaction barrier height comparing to the CBS data (13.03 kcal/mol). Both methods show that the reaction is endothermic, which is also calculated by Yu et al.43 Our results for the endothermicity are again in good agreement with the CBS data. However, both our results and the CBS data somewhat overestimate the reaction enthalpy when comparing to the experimental value52−56 of 2.06 kcal/mol. The barrier height for the R1 is much larger than that for the H abstraction reaction from noncyclic C3H8 by a H atom from either the primary or the secondary C atoms (10.56 and 7.76 kcal/mol, respectively).27 Also for C3H8, the H abstractions by H atom are both exothermic reactions. In fact the reaction energetics here is more comparable to the benchmark H + CH4 → H2 + CH3 reaction,13−15,27 for which the barrier and enthalpy are 14.2 and 0.6 kcal/mol. The vdW well depth for R1 is predicted to be 0.21 and 0.26 kcal/mol by the CCSD(T) and F12a methods, respectively. Note that they both overestimate the binding energy when comparing with the CBS result of 0.06 kcal/mol. This is mainly due to the ZPE contribution, which depends on the frequency calculation done at MP2 level of theory. In comparison, the F12a method shows better performance than the CCSD(T) method in predicting the adiabatic reaction barrier and the reaction enthalpy. In addition, it is also approximately 2−3 times faster than the CCSD(T) method.

An earlier study shows the bond dissociation energy, being 445 kJ/mol,34 which is similar to the C−H bond strength in methane. In addition, the product of the abstraction reaction, cyclopropyl (cyc-C3H5), can undergo a ring opening isomerization to form the thermodynamically more stable allyl radical (CH2CHCH2).35 This procedure is often used by experimentalists36−40 to produce the allyl radical. In such experiments, cyc-C3H5 are first generated by the H abstraction reaction from cyc-C3H6 using a radical or atoms. Recently, the H + cyc-C3H6 → H2 + C3H5 reaction has been studied experimentally41,42 and theoretically43 to investigate the mechanism behind the production of rovibrationally hot H2 molecule in the reaction. Two coexisting reaction mechanisms were found:43 R1 and H-addition/ring opening. In addition to the recent theoretical studies on the energetics of R1 in ref 43, the reaction rate constants of R1 have been analyzed using methods that are based on conventional transition state theory (TST).44,45 The present work performs RD quantum calculations on R1 to investigate its kinetics and dynamics in more detail. In particular the cumulative and product-state- and reactant-state-dependent reaction probabilities are reported. In addition, the contribution of the quantum tunnelling effect to the overall reaction rate constant is analyzed and compared with experiment. This is the first time to our knowledge that a quantum scattering calculation has been applied to a chemical reaction involving a cyclic molecule, which also introduces some new features such as the treatment of the center-of-mass (CoM) of the cyc-C3H5 fragment. The remainder of the article is organized as follows. In section 2, we discuss the 2D PES of R1, in particular the ab initio calculations for the stationary points on the surface and the fitting to the ab initio data with a 29-parameter doubleMorse potential function. The theoretical background including key steps of the R-matrix propagation method46 to solve the 2D time-independent nuclear motion Schrödinger equation and the calculations of the reaction rate constants in quantum theory and transition state theory are in section 3. The results of our scattering calculations are presented in section 4 with discussions. Our main conclusions are in section 5.

2. AB INITIO POTENTIAL ENERGY SURFACE This section is divided into two parts. In the first, we discuss the energies of the stationary points on the PES of R1. The fitting of the surface is reported in the second part. All the ab initio calculations were carried out using MOLPRO package.47 For the geometry optimizations on a grid of fixed lengths for the bonds being broken and formed and frequency calculations, we used the second order Møller−Plesset perturbation theory (MP2) with a correlation consistent polarized valence triple-ζ Dunning basis set48 (cc-pVTZ). A single-point energy calculation was carried out, for each optimized stationary structure, at coupled cluster level including single, double, and perturbative triple excitations [CCSD(T)] with the augmented cc-pVTZ basis set (aug-cc-pVTZ). We also employed the explicitly correlated coupled cluster method, CCSD(T)-F12a/cc-pVTZF12 method,49,50 for the single point energy calculations in the current study, where cc-pVTZ-F12 stands for the triple-ζ correlation consistent F12 MOLPRO basis set of Peterson et al.51 For simplicity reasons, we shall for the rest of this article use “CCSD(T)” and “F12a” for the CCSD(T)/aug-cc-pVTZ and CCSD(T)-F12a/cc-pVTZ-F12 methods, respectively. 2.1. Optimized Geometry and Reaction Energetics. We show respectively in Figure 1a−c the molecular geometries B

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Figure 1. Optimized structure of the stationary points on the 2D PES for R1. The H atom being abstracted in the reaction is marked as “Ha” in the TS structure. The remaining H atom on the C atom after the abstraction is marked as “Hb” in the product cyc-C3H5 structure. In the vdW complex structure, one of the H atoms in CH2 on the same side as the incoming H atom to the C3 ring is marked as “Ha”.

Table 2. Reaction Energetics of the H + cyc-C3H6 → H2 + cyc-C3H5 Reactiona

Table 1. Key Geometry Coordinates of the Stationary Points on the 2D PES for the H + cyc-C3H6 → H2 + cyc-C3H5 Reactiona

TS

cyc-C3H6 cyc-C3H5 vdW Complex

bond lengths and angles

MP2 /cc-pVTZ

CCSD /cc-pVDZ43

C−Ha H−Ha H−Ha−C C−H C−Hb H-CoM of C3 ring C−Ha

1.4375 0.85084 174.32 1.0785 1.0758 3.6204 1.0785

1.4385 0.8832 176.0 1.0955 1.0938 4.2734 1.0953

ΔV‡a ΔrH ΔEvdW

MP2 /cc-pVTZ

CCSD(T) /aug-cc-pVTZ

CCSD(T)-F12a /cc-pVTZ-F12

ref 43

19.14 9.54 −0.28

13.61 3.27 −0.21

13.47 4.18 −0.26

13.03 4.03 −0.06

a

All the energies are in kcal/mol. The ZPE corrections were included for all the computational results. The geometry optimization and the frequency calculations that lead to the ZPE for our data were calculated at the MP2/cc-pVTZ level of theory.

before projection, and their values are also reported in Table 3. It can be seen that the projection procedure has significant impact on the frequencies of only two vibrational modes, in particular, from 1165.87 and 871.08 cm−1 to 1171.22 and 859.88 cm−1. Both modes correspond to the bending of H−Ha−C from collinear configuration. 2.2. Potential Energy Surface. In our quantum scattering calculation we utilize the hyperspherical coordinates, ρ and δ. They are defined to this reaction as

a

All bond lengths are in Å, and the bond angles are in degrees. Note that the positions of specified H-atoms, Ha and Hb, can be found in Figure 1.

The single-point energy calculations for the grid points on the PES in this study were therefore done at the F12a level of theory. We report the harmonic frequencies of the TS and cyc-C3H6 in Table 3. This table also includes the vibrational frequencies of the TS after the rectilinear projection.57 We used the rectilinear projection method to project out the contribution of the two active vibrational modes, the remaining vibrational frequencies were then used in the calculation of the ZPEs, which we add to our single-point calculation for the grid points and used in fitting the 2D PES. Note that the ZPEs used to calculate energies in Table 2 are calculated with the frequencies

m1 2 R = [ρ cos(δ)]2 , μ

m2 2 r = [ρ sin(δ)]2 μ

(1)

with m1 = C

2mH(3mC + 5mH) 3mC + 7mH

(2)

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Table 3. Harmonic Frequencies (in cm−1) and ZPEs (in kcal/mol) at MP2/cc-pVTZ Levela frequencies cyc-C3H6

cyc-C3H5

TS

TS (projected)

a

3298.19 1533.89 1169.17 867.94 3287.19 1479.63 1015.57 1342.30 (i,a) 2255.22 (a) 1124.50 886.45 3272.21 1486.41 1074.40 815.53

3279.63 1484.89 1087.00 745.85 3255.54 1283.32 957.30 3272.30 1520.78 1121.60 871.08 3261.80 1269.36 1033.79 760.42

3279.57 1484.88 1056.07 745.80 3249.74 1191.64 882.27 3261.80 1486.41 1074.40 815.53 3255.40 1202.62 1023.56 313.03

ZPE 3197.11 1233.93 1055.94

3188.28 1221.02 909.52

3188.24 1121.01 909.42

51.73

3170.36 1136.66 802.34 3257.22 1267.22 1033.94 760.92 3177.59 1171.22 951.98 272.73

3163.35 1110.77 795.37 3177.59 1202.62 1023.56 313.03 3176.57 1123.81 886.52

1513.09 1077.94 624.21 3176.99 1165.87 951.98 274.70 1521.56 1121.60 859.88

42.91

50.49

47.25

The imaginary frequency at TS is marked with “i” in parentheses. The active modes of TS are marked with “a” in parentheses.

m2 =

mH 2

(3)

m3 =

mH(3mC + 5mH) 3mC + 6mH

(4)

m1m2m3

(5)

and μ=

3

where, in eq 1, R and r are the Jacobi coordinates. Normally (R, r) are defined as the distance between the centers of mass (CoMs) of the H−Ha and cyc-C3H5 fragments and the bond distance of H−Ha, respectively. To construct the 2D PES, whether using a fitting functions or the recent development of the (1 + 1)D method, the starting point is the minimum energy path (MEP). In Figure 2a, we show the MEP of the R1 in hyperspherical coordinates converted from the normal Jacobi coordinates. The black dots, apart from the TS (marked on the graph), are obtained directly from the results of the intrinsic reaction path (IRC) calculation in the MOLPRO program;47 the black curve is the continuation of the ab initio points. The TS is marked as the large red dot on the graph. Points representing the asymptotic region, i.e., large ρ values, are included in the graph as blue dots. It can be seen that this reaction has a late TS, and hence, one would expect that the vibrationally ground state dominates the product. We shall discuss this point in more detail in section 4. A more interesting feature of the MEP can be found in the reactant channel. In particular in the 5.1 ≤ ρ ≤ 5.9 au and 0.42 ≤ δ ≤ 0.61 rad region, the ρ-value of the MEP curve decreases as δ increases. This is in contrast to the normally expected behavior of the curve, which is shown on the graph as the dashed black curve. In the last section, we reported an energetically favored vdW complex structure in the reactant channel of this reaction. Here we show the position of the complex on the MEP as the brown dot in Figure 2a, and its coordinate is (ρ, δ) = (5.119 au, 0.6027 rad). In fact for the MEP points close to the vdW complex, a noncollinear H−Ha−C configuration is observed in the calculation with the bond angle suggesting that the incoming H atom tends to sit above the C3 ring. The presence of this vdW complex in the reactant channel denies the usage of (1 + 1)D reduced dimensionality methods. In the (1 + 1)D calculations,25−27 the geometry of the X−H−Y

Figure 2. Plots of the MEP of R1 projected onto the hyperspherical coordinate (ρ, δ)-plane with ρ and δ values converted from the normal Jacobi coordinate (a) and the alternative Jacobi coordinate (b). The black dots and curves are the ab initio MEP and its continuation, respectively. The blue dots are the ab initio data in the asymptotic regions. The red and brown dots are the position of the TS and the vdW complex, respectively. The dashed black curve (a) is the normally expected route of the MEP in (ρ, δ)-plane.

triatomic fragment in the reacting complex is assumed to be collinear or near collinear along the MEP. For the MEP around the vdW complex in R1, assuming a collinear configuration of D

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the surface, but we do not face the complication that it is lying outside the normally expected route of the MEP as shown in the last coordinate system. Our double-Morse potential fitting function can handle this with ease. The potential yields an RSS value of 2.8 × 10−4, and a total of 163 ab initio grid points were used in the fit. The values of the parameters are reported in Table 4, and a contour plot of the fitted potential function is shown in Figure 3, with the positions of the TS and the vdW complex indicated on the plot.

the X−H−Y fragment or freezing the bond angle at 180° in the ab initio calculations would clearly introduce large errors in the PES. Therefore, in order to study the dynamics of the reaction with RD methods, we have to use a fitting function. Previously, a 29-parameter double-Morse potential function13−15,17−24 has been successfully applied to several H-abstraction reactions by our group, and more detailed discussions of this function can be found in ref 24. It can be written as V (ρ , δ) = f1 (ρ)({1 − exp[g1(ρ)δ + h1(ρ)]}2 − c14)

Table 4. Values of the Parameters Defining the CCSD(T)F12a/cc-pVTZ-F12 2D PES

+ f2 (ρ)({1 − exp[−g2(ρ)δ − h2(ρ)]}2 − c 28) + c 29

(6)

c1 c2 c3 c4 c5 c6 c7 c8

with f1 (ρ) = c1 + c 2ρc3 exp( −c4ρ)

g1(ρ) = c5 + c6ρ + c 7ρ2 h1(ρ) = c8 + c 9ρ + c10ρ − log(c11 + c12ρ + c13ρ2 )

0.1792 −27.3273 1.4881 1.9875 5.5220 −0.3538 0.1360 1.0460

c9 c10 c11 c12 c13 c14 c15 c16

0.5050 −0.1084 −44.6845 26.2979 0.1459 2.4558 0.1718 −0.5894

c17 c18 c19 c20 c21 c22 c23 c24

3.6143 1.8439 3.6370 2.5737 −0.0863 −11.9089 0.0579 −0.0012

c25 c26 c27 c28 c29

8644.2894 1991.9187 −58.1164 1.1323 −117.6630

f2 (ρ) = c15 + c16ρc17 exp( −c18ρ)

g2(ρ) = c19 + c 20ρ + c 21ρ2 h2(ρ) = c 22 + c 23ρ + c 24ρ2 + log(c 25 + c 26ρ + c 27ρ2 )

where ci (i = 1, 2, 3, ..., 29) are parameters. In some of the earlier works, this fitting potential function has been applied to reactions with vdW wells in both reactant and product channels.13,22 However, it should be noted that in those cases, the vdW structures all had a near collinear or collinear configuration for the X−H−Y fragment of a reacting complex. The function is fitted to the ab initio data via a least-squares procedure.58 We measure the convergence of the fitting procedure with the sum of the squared residue (RSS).14,21,24 Typically, a RSS value on the order of 10−4 would mean an adequate potential fit.24 However, even with our best fitted set of parameter values, the RSS obtained is as large as 0.029. Therefore, it is clear that the 29-parameter double-Morse potential function is not appropriate to fit the ab initio PES data in the hyperspherical coordinates. In the previous works of our group,16−20 we used an alternative definition of the Jacobi coordinates, especially for reactions involving alkanes with more than one C atom.16−20 In this alternative system, r is the same as the normal Jacobi coordinates, while R is defined as m R = rCHa + H rHHa (7) 2

Figure 3. Contour plot of the fitted PES, produced with eq 6, and the parameter values are given in Table 4. The position of the TS and the vdW complex are marked on the plot with arrows.

3. SCATTERING THEORY With the PES constructed as described in Section 2, we proceed to solve the nuclear motion Schrödinger equation. In the hyperspherical coordinate system; the Hamiltonian can be written as46 2 2 ⎪ ∂ ⎪ Ĵ ⎫ 1 ⎧ 1 ∂2 3 ̂ + 2 2 − − 2⎬ + V (ρ , δ ) H=− ⎨ 2 2 ⎪ 2μ ⎩ ∂ρ ρ ∂δ ρ ⎪ 4ρ ⎭

Effectively, this alternative definition is assuming the CoM of cyc-C3H5 fragment on the C atom, from which the H atom is abstracted in the reaction. This system has been applied to H-abstraction reactions from noncyclic alkanes up to propane.16−19 We plot in Figure 2b the MEP of the R1 in hyperspherical coordinates converted from the alternative Jacobi coordinates. The black dots are the ab initio data, while the black curve is the continuation of the data. The vdW coordinate is once again included as the brown dot, with the coordinates being (ρ, δ) = (5.646 au, 0.5397 rad). It can be seen that the MEP has a normal behavior and that the vdW is presented along the MEP. In this case, the structure of the potential energy well in the reactant channel is maintained on

(8)

where J ̂ is the total angular momentum operator, and V(ρ,δ) is the potential energy, which is calculated using eq 6 with the parameter values given in Table 4. For J = 0, we have the Hamiltonian 1 ⎧ ∂2 1 ∂2 3 ⎫ ⎬ + V (ρ , δ ) Ĥ 0 = − ⎨ 2 + 2 2 − 2μ ⎩ ∂ρ ρ ∂δ 4ρ2 ⎭

(9)

We then carry out the time-independent scattering simulation using the R-matrix propagation scheme, which was developed by E

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respectively. The subscripts “i” and “f ” denote, respectively, the initial and final states of the reactants and products. It should be noted that in the ideal case, PJkk′= 0(E) should be evaluated in the final ρ-sector only. However, since we apply the approximate boundary conditions directly in the hyperspherical coordinates, oscillatory behavior can be observed as a function of sector ρi.63 To remove the problem, we calculate the S-matrix over a number (Snum) of sectors in the asymptotic region, and average over the probability matrix elements to produce PJkk′= 0.64,65 The reaction rate constant, kscatt(T), is then calculated using the cumulative reaction probability with the J-shifting model.3,66 kscatt(T) is given by

Stechel et al.46 We only outline the key steps of the scheme and the application of the approximate boundary conditions. More detailed description of the R-matrix method and its application to the J-shifting model can be found in earlier studies.14,15,24,59 The hyperspherical coordinate space is first divided into even width sectors in ρ; for each sector the δ-dependent Hamiltonian is given by 1 ∂2 Hδ̂ = − + V (ρi , δ) 2μρi 2 ∂δ 2

(10)

We then expand the sector-dependent wave function for the quantum state k as a function of ρ as N

Ψk(ρ , δ ; ρi ) =

kscatt(T ) =

∑ fk′ k (ρ; ρi )φk′(δ ; ρi )

(11)

k′

k TST(T ) =

where W is a diagonal matrix. It is given by

(13)

(14)

(15)

where Ikk =

exp( −iλk ρi ) and Okk =

exp(iλk ρi )

Table 5. Values of the Parameters Used in the Scattering Calculationsa

(16)

with λk (ρi ) =

Wkk(ρi )

(17)

Once the S-matrix is calculated, one can then obtain other physical properties. In the present study, we report the state-tostate and cumulative reaction probability with respect to the total energy, E. When J = 0, they are given by PiJ→=f0(E) = |Si → f (E)|2

final value

parameter

final value

ρmin ρmax Nρ Snum

3.2 au 12 au 750 110

Nδ Emax Einc N

200 1.5 eV 0.001 eV 10

experience of previous works, fix Snum to the final 15% of Nρ. In the energy range (0 to 1.5 eV) that we are interested in, there are only seven open channels. In our calculations, the number of contracted basis function required is 10. We present in Figure 4 the plots of cumulative reaction probability for R1 versus total energy as the black curves.

(18)

∑ |Si→ f (E)|2 i,f

parameter

a Distances are in atomic units. See Ref 24. for a definition of the parameters.

and J=0 Pcum (E ) =

⎛ −ΔV ‡ ⎞ a ⎜⎜ ⎟⎟ k T exp B H cyc ‐ C3H6 k T 2πQ tot (T )Q tot (T ) ⎝ B ⎠ ‡ (m′) Q tot (T )

4. SCATTERING CALCULATION RESULTS AND DISCUSSION In order to test the convergence of the reaction probabilities for R1, we have conducted a series of test calculations varying the scattering parameters. The resulting values of the parameters are reported in Table 5. Note that in this study we follow the

S(ρi ) = (R(ρi )O′(ρi ) − O(ρi ))−1(R(ρi )I′(ρi ) − I(ρi ))

λk−1/2

⎛ E ⎞ J=0 (E) exp⎜ − Pcum ⎟d E ⎝ kBT ⎠ (20)

(21)

To solve eq 12, the R-matrix is now propagated through all sectors from the classical forbidden region at a small ρ to asymptotic ρ.46,61 In the asymptotic region, we apply the approximate boundary conditions in hyperspherical coordinates.21,62 The scattering matrix (S-matrix) is extracted from the R-matrix, for sector ρi,

λk−1/2

∞

where ΔV‡a is the adiabatic vibrational barrier height; the values were reported in Table 2. For the TS partition function, all the vibrational modes except the one that corresponds to the transition state vector are included in the vibrational partition function calculation, and m′ = 3N − 6 − 1. Note that the R1 involves multiple H abstraction sites, it is accounted for in both eqs 20 and 21 by the symmetry number in the rotational partition function.

In J = 0 case, it can be reduced to ⎛ 3 ⎞ ⎟⎟ Wkk(ρi ) = 2μ⎜⎜E − εk(ρi ) − 8μρi 2 ⎠ ⎝

∫0

where QHtot(T) and Qtot 3 6(T) are the total partition functions of H and cyc-C3H6, while Q‡(m) tot (T) is the partition function of the TS. Note that for the vibrational partition function of the TS only the frequencies of vibrational spectator modes after the projection procedure are included in the function, and hence here we have m = 3N − 6 − 2. In this study, in order to analyze the contribution from any quantum effects, the classical transition state theory (TST) rate constant is also calculated. It is given by

(12)

⎛ J(J + 1) ⎞ 3 ⎟⎟ Wkk(ρi ) = 2μ⎜⎜E − εk(ρi ) − − 8μρi 2 2μρi 2 ⎠ ⎝

H cyc ‐ C3H6 2πQ tot (T )Q tot (T )

cyc‑C H

where ϕk′(δ;ρi) is the δ-dependent wave function obtained by diagonalizing eq 11 using a discrete variable representation with a particle-in-a-box basis.60 The size of the contracted basis, N in eq 11, is set to be larger than the number of open channels in the asymptotic region.46 The expression of the wave function in eq 11 is applied to eq 8, the problem is reduced to solve the so-called close-coupled equations d2 f(ρ , ρi ) + W(ρi )f(ρ , ρi ) = 0 dρ 2

‡ (m) (T ) Q tot

(19) F

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quantum tunnelling effect, their reaction probabilities both rise before the adiabatic barrier. It should be noted that although from the figure it seems the (v = 0) and (v = 1) reactant-statedependent probabilities have the same threshold, our calculation suggests that the two channels open at a total energy of ∼0.373 and ∼0.435 eV, respectively. At ∼0.849 eV the reactant channel associated with the second excited vibrational state opens. We show our calculation of the rate constants for reaction R1 in Figure 5a,b in the temperature (T) ranges of 200 to 2000 K

Figure 4. Plots of the cumulative reaction probability versus total energy and the product-state-dependent reaction probabilities (a) and the reactant-state-dependent reaction probabilities (b).

The product-state- and reactant-state-dependent contributions are shown in Figure 4a,b, respectively. The adiabatic energy barrier, ΔV‡a , is ∼0.584 eV. It is marked on both graphs as the black dashed line. It can be seen that the reaction probability curve rises before this value, and from our calculation, the reaction starts at total energy of ∼0.443 eV. This result indicates a quantum tunnelling effect is contributing to the reaction mechanism. Another picture of the tunnelling contribution in terms of magnitudes at low temperature can be found when comparing the rate constants of the quantum calculation with the classical TST results, which will be discussed in more detail later in this section. In Figure 4a, the red, blue, and brown curves are the probabilities for the reaction producing the H2 product in the ground, first, and second excited vibrational states. It can be seen that the reaction is dominated by the ground state H2, which was predicted from our discussion before on the reaction having a late TS. The v = 1 product is not found until the total energy is typically above ∼0.85 eV. The v = 2 H2 channel opens at very high energy; its contribution to the overall reaction rate constant is very small. The red, blue, and brown curves in Figure 4b show, respectively, the reactant-state-dependent reaction probabilities from (v = 0), (v = 1), and (v = 2) states. One interesting feature is that the reactants in the first vibrational excited state has a comparable or even higher (at lower energy) contribution to the ground state reactants. Both states are involved in the

Figure 5. Comparison of the reaction rate constants of R1 from the present study to previous experimental and theoretical studies in the temperature ranges of 200 to 2000 K (a) and 333 to 1000 K (b).

and 333 to 1000 K, respectively. On both figures, the solid curves are our quantum scattering calculation results. The dashed curves are the classical TST rate constants. It can be seen that the two curves converge at high T, over 1000 K. We can also see that our quantum rate constant curve bends away from the classical TST curve at low T. At around 500 K, the rate constant of the quantum calculation being 8.342 × 10−16 cm3 molecule−1 s−1 is ∼5 times greater than the classical TST result of 1.604 × 10−16 cm3 molecule−1 s−1; at around 300 K, it becomes ∼20 times greater. This clearly shows the strong contribution of the quantum tunnelling effect to the reaction, especially at lower temperature. This tunnelling effect was also observed in the reaction probability calculation of Figure 4. To confirm such G

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some special aspects for the calculation and resulting fitting procedures, which are described in detail. In the quantum scattering calculation, the R-matrix propagation scheme was applied to the fitted 2D PES. We examined the cumulative reaction probability as well as the product-state- and reactant-state-dependent reaction probabilities. The state-dependent probability results confirmed our finding in the ab initio calculations that the reaction has a late transition state. We also found that a quantum tunnelling effect has a significant contribution to the reaction at lower total energies. The reaction rate constants computed from our quantum scattering calculation and classical TST were compared with experimental data at two different temperature ranges. Our quantum results show very good agreement to the experiments in both cases, while the classical TST underestimated the rate constants at lower T due to the lack of quantum tunnelling. This is the first time that quantum dynamical methods have been applied to chemical reaction involving cyclic alkane molecules. The success of the study suggests the approach can be extended to larger cyclic systems in future studies.

effect exists in the reaction, we compare our calculated reaction rate constants to the experimental data. Two experimental studies of R1 were conducted by Marshall and co-workers at different T ranges: 358 to 550 K30 and 628 to 779 K.31 The fitted Arrhenius functions to the experimental data were reported, respectively, as log(k /cm 3 mol−1 s−1) = 14.21 ± 0.13 − (11.7 ± 0.26 kcal mol−1/2.3RT )

and log(k /cm 3 mol−1 s−1) = 13.6 ± 1.0 − (11.6 ± 3.11 kcal mol−1/2.3RT )

We show them in Figure 5a,b as the red and blue curves, respectively. The experimental error estimates are also presented. Comparing to the results in the higher T range (blue curves), both the quantum and classical TST curves show good agreement to the experiment. For the lower T range data, the classical TST method is clearly underestimating the rate constant. Our quantum calculation, however, shows again very good agreement with the experiment. This provides clear evidence that quantum tunnelling is contributing to the overall reaction. The rate constant of a previous theoretical study44 is also included in Figure 5a,b as the brown curves. This study was based on the TST framework, and the values of the parameters in the Arrhenius equation were chosen to fit the lower T experimental data. As a result of the fitting nature, on the graphs it has better agreement to the red curves than our quantum calculation results, but worse agreement to the blue curves. To have clearer comparison with the experiments, the values of the reaction rate constants from our quantum and classical TST calculations as well as the experimental data are presented in Table 6. We can see that in both low and high T cases, the quantum results are in very good agreement to the experiments.

■

*E-mail: (X.S.) [email protected]. *E-mail: (D.C.C.) [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This work was supported by the EPSRC Programme Grant No. EP/G00224X/1 and Levenhulme Trust Project Grant No. RPG-2013-321.

■

a

kTST

kscatt

exptl

360 390 420 450 480 510 540 630 660 690 720 750

6.517(−19) 2.912(−18) 1.059(−17) 3.267(−17) 8.812(−17) 2.128(−16) 4.684(−16) 3.267(−15) 5.593(−15) 9.169(−15) 1.447(−14) 2.209(−14)

8.083(−18) 2.830(−17) 8.379(−17) 2.167(−16) 5.019(−16) 1.061(−15) 2.076(−15) 1.091(−14) 1.730(−14) 2.644(−14) 3.914(−14) 5.632(−14)

2.054(−17)a 7.247(−17)a 2.135(−16)a 5.447(−16)a 1.236(−15)a 2.547(−15)a 4.844(−15)a 6.231(−15)b 9.495(−15)b 1.395(−14)b 1.985(−14)b 2.746(−14)b

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Table 6. Comparison of Rate Constants for the H + cyc-C3H6 → H2 + cyc-C3H5 Reaction in cm3 molecule−1 s−1; Powers of 10 Are in Parentheses temperature (K)

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Corresponding Author

From ref 30. bFrom ref 31.

5. CONCLUSIONS We applied a reduced dimensionality quantum scattering calculation on the chemical reaction of H abstraction by an H atom from the cyclopropane molecule. The latest available ab initio quantum chemistry methods were used to construct the potential energy surface. The cyclic hydrocarbon produced H

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