Thermal Rate Constants for the O(3P) + CH4 â OH + CH3 Reaction

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Thermal Rate Constants for the O(P)+CH#OH+CH Reaction: The Effects of Quantum Tunneling and Potential Energy Barrier Shape Huali Zhao, Wenji Wang, and Yi Zhao J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.6b07029 • Publication Date (Web): 17 Sep 2016 Downloaded from http://pubs.acs.org on September 18, 2016

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The Journal of Physical Chemistry

Thermal Rate Constants for the O(3P)+CH4→OH+CH3 Reaction: The Effects of Quantum Tunneling and Potential Energy Barrier Shape Huali Zhao1, Wenji Wang1*, and Yi Zhao2

1

College of Science, Northwest A&F University, Yangling, 712100, Shaanxi Province, P. R. China.

2

State Key Laboratory for Physical Chemistry of Solid Surfaces and Fujian Provincial Key Lab of

Theoretical and Computational Chemistry, College of Chemistry and Chemical Engineering, Xiamen University, Xiamen, 361005, P. R. China

ABSTRACT: The rate constants and kinetic isotope effects for the O(3P)+CH4 reaction have been investigated with the quantum instanton method in full dimensionality. The calculated rate constants are in good agreement with the experimental values above 400 K, below which the measured values are scattered. Compared to other theoretical approaches, the quantum instanton method predicts the largest quantum tunneling effect, so it gives the largest rate constants at low temperatures. The calculated kinetic isotope effects are always much larger than 1 and increase with the decreasing of temperature, which are due to the zero-point energy and quantum tunneling. Our calculations on different potential energy surfaces demonstrate that the potential energy barrier shape dominates the magnitude of quantum tunneling and gives a great effect on the kinetic isotope effect. 1

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1. INTRODUCTION The reaction of ground state atomic oxygen with methane is substantially important in hydrocarbon combustion, as the simplest saturated hydrocarbon oxidation reaction, it is considered as the benchmark for both experimental techniques and theoretical methods. In particular, this heavy-light-heavy type reaction is very suitable for examining the quantum tunneling effect at room temperature and below. The O(3P)+CH4→OH+CH3 (R1) reaction has been extensively investigated over the past decades1. Experimentally, the crossed beam technique has been used to provide detailed information of products translational, vibrational and rotational energy distributions, products angular distributions, bond and mode selectivities, and rate constants. Zhang et al.2 studied this reaction at a hyperthermal collision energy of 64 kcal/mol, and found that the OH product was scattered in the forward direction, which showed a stripping mechanism. However, with extended collision energies ranging from 7.5 to 13.5 kcal/mol, Zhang and co-workers3 revealed that the products angular distributions were backward scattered for the O(3P) + CHD3 reaction, which indicated a direct rebound mechanism. Zhang and Liu4 investigated the effect of the bending excitation of methane on the title reaction, and found that the bend-excited methane yielded more vibrational excitation of the hydroxyl coproduct. Wang and Liu5 observed that the CH stretching excitation significantly enhanced the reaction rate of O(3P)+CHD3, and the products angular distributions shifted from a backward-dominance to a sideways-peaking. Pan and Liu6 investigated the antisymmetric-stretching mode of CH4, they found that the products angular distributions extended toward sideways from the backward dominance of the ground-state reaction, and the vibrational

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excitations exerted a positive effect on reactivity. Although all of the vibrational excitations are beneficial to the reaction, the translational energy is more efficient in promoting the rate of this central-barrier reaction6. Baulch et al.1 summarized all of the available data, and recommended that the rate constants over the temperature range 400-2500 K could be expressed as follows, k = 7.3 ×10−19 T 2.5exp( − 3310 / T )cm3molecule-1s-1 , while the measured rate constants were found to be scattered below 400 K. Theoretically, electronic structure calculations at different levels show that the transition state structure of the O(3P)+CH4 reaction possesses Cs symmetry7, and the classical potential energy barrier height8-11 is reported to be 14 kcal/mol by dual-level coupled cluster calculation with single, double, and perturbative triple excitations (CCSD(T)). The dynamical investigation of the title reaction usually needs an accurate potential energy surface (PES). Up to now, several kinds of PESs for the O(3P)+CH4 reaction have been constructed12-14. Among these PESs, the most widely used is that constructed by Espinosa-Garcia and coworkers7,14,15. These authors have constructed a series of PESs for the title reaction. In 1998, they created the first full dimensional analytical potential energy surface for the O(3P)+CH4 reaction (labelled as PES1998). Two years later, they obtained PES-2000 by making PES-1998 completely symmetric with respect to any permutation of the four methane hydrogens. Most recently, they further corrected PES-2000 with high level ab initio data, and fitted it (named PES-2014) to reproduce experimental results. Rigorous quantum dynamics calculation involving all degrees of freedom for the O(3P)+CH4 reaction is still not practical, even though a number of reduced dimensional studies, such as 2D16,17,4D18-20, 5D21,

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6D22,23, 7D24,25 and 8D26, have been carried out. These results show that all of the vibrational excitations (the stretching24, bending22 and umbrella19 modes of CH4) can promote the reaction, although none of them is more efficient than the translational energy26. Moreover, these data reveals that the values of the rate constants depend on the numbers of degrees of freedom adopted in the calculations21. Full dimensional quasiclassical trajectory method is also widely used for the title reaction, it is an important complement to the reduced dimensional quantum dynamical calculation. Many important conclusions27-39 have been drawn. Varandas et al.29 showed that the topology of the entrance channel had strong implications on the dynamics of the title reaction. Gonzalez-lavado et al.35 found that the scattering distribution was forward, which predicted a stripping mechanism. Espinosa-Garcia et al.39 reported that the symmetric stretch mode was more reactive than the antisymmetric stretch mode. The quasiclassical trajectory method suffers from the drawbacks that it cannot properly incorporate the effects of zero-point energy and quantum tunneling40, unfortunately, these effects are very important for the present heavylight-heavy type reaction. Another highly efficient dynamical method is the multiconfigurational time dependent hartree41 (MCTDH). Within the J-shifing approximation, Huarte-Larranaga and Uwe Manthe found that the MCTDH rate constants for O(3P)+CH4 reaction were in good agreement with experiment. Recently, several approximate methods have been applied to calculate the rate constant of the title reaction in full dimensionality, in which the tunneling is taken into account in different ways. EspinosaGarcia and Garcia-Bernaldez15 calculated the rate constants using the canonical variational transition state theory (CVT) and canonical unified statistical model (CUS) with semiclassical transmission coefficients.

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However, the rate constants of the transition state theory are very sensitive to the method used for the tunneling correction, indeed, the transmission coefficient with the microcanonical optimized multidimensional tunneling (µOMT) is nearly 6 times larger than that with the small curvature tunneling (SCT) at 300 K. Most recently, Suleimanov and co-workers42,43 calculated the rate constants with the ring polymer molecular dynamics (RPMD) method, and showed that the quantum tunneling could be included effectively. Richardson and Althorpe44 also gave a connection between the RPMD and semiclassical instanton theory. In the present paper, we will use the quantum instanton (QI) method to calculate the rate constants of O(3P)+CH4 reaction, so as to further investigate the quantum tunneling effect. The quantum instanton method, proposed by Miller et al.45,46, can be used to calculate rate constants in the full Cartesian coordinate space, so it includes all anharmonic and rotational effects47,48. Compared to the semiclassical instanton49-51, the quantum instanton is expressed wholly in terms of the quantum Boltzmann operator, in this case, it gives a better description of the quantum tunneling effect52,53. Moreover, the QI method can be applied to complex systems because the Boltzmann operator can be evaluated with path integral Monte Carlo method. The QI method has been applied to investigate the gas phase54,55 and surface reactions56-58, and it provides accurate estimates of the rate constants and kinetic isotope effects59-61. Theoretical calculations also demonstrate that the kinetic isotope effect is very remarkable for the title reaction. With the canonical unified statistical model (CUS), Espinosa-Garcia and co-workers7,15 reported that the H/D kinetic isotope effects for O(3P)+CH4 reaction were 29.1 and 18.9 on PES-1998 and PES-

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2000, respectively, while Gonzalez-Lavado et al.43 gave a kinetic isotope effect of 18.6 on PES-2014 at 300 K. Gonzalez-Lavado et al. also calculated the kinetic isotope effect with the RPMD on PES-2014, and found that the value was 30.8 at 300 K. These discrepancies reveal that the kinetic isotope effect is sensitive to both the potential energy surface and the dynamical method involving the quantum tunneling. In order to have a deeper insight into the kinetic isotope effect, we will calculate it with the quantum instanton method. The rest of this paper is organized as follows: In Section 2, we summarize the quantum instanton theory and describe the details of the present calculations. Section 3 presents the quantum instanton rate constants and kinetic isotope effects. Section 4 is the conclusions. 2. QUANTUM INSTANTON RATE CONSTANT The quantum mechanical thermal rate constant62 is defined as k (T ) =

1 Qr

∫

∞

0

C ff (t )dt ,

(1)

in which the rate constant is related to the integral of the flux-flux correlation function Cff(t). With the quantum instanton (QI) approximation45,46, the thermal rate constant can be expressed as follows

kQI (T ) =

1 π h C ff (0) . Qr 2 ∆H

(2)

Here, Qr is the reactant partition function per unit volume, and Cff(0) is the zero time value of the flux– flux correlation function, ˆ ˆ ˆ ˆ C ff ( t ) = tr[e − β H 2 Fˆa e − β H 2 ei H t h Fˆb e − i H t h ],

6


(3)

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where β is the inverse temperature 1 ( k B T ) , Hˆ is the Hamiltonian operator, and

Fˆa

and Fˆb are the flux

operators given by i Fˆr =  Hˆ , h (ξγ ( rˆ ) )  (γ = a, b), h

(4)

with r represents a collective of the coordinates, h is the Heaviside function, and

ξa ( r )

and ξb ( r) define

two separate dividing surfaces. These two dividing surfaces correspond qualitatively to the turning point surfaces of the periodic orbit that runs on an upside-down potential energy surface in imaginary time (i.e., the “instanton”). ∆ H is a specific type of energy variance, 1/ 2  C&&dd (0)  ∆H = h  −  ,  2Cdd (0) 

(5)

where Cdd ( 0) and C&&dd (0) are the zero time value and its second derivative of the “delta–delta” correlation function, ˆ ˆ ˆ ˆ Cdd ( t ) = tr e − β H 2 ∆ˆ a e − β H 2eiHt h ∆ˆ b e − iHt h  ,  

with

∆ˆ a

and

∆ˆ b

(6)

being a modified version of the Dirac delta function

∆ˆ r = ∆ (ξ r ( rˆ ) ) = δ (ξ r ( rˆ ) )

N

1 ∑ m ( ∇ ξ ( rˆ ) ) i r

i =1

2

(γ

= a, b ) .

(7)

i

For the present reaction, O(3P)+CH4→OH+CH3, in order to locate the two dividing surfaces, a generalized reaction coordinate s (r ; ξ ) is defined, where ξ is an adjustable parameter that shifts the location of the dividing surface. s ( r ; ξ ) is defined by linearly interpolating between two constituent reaction coordinates s0(r) and s1(r) through the parameter ξ ,

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s ( r; ξ ) = ξ s1 ( r ) + (1 − ξ ) s0 ( r ) .

(8)

s1(r) is a reaction coordinate whose dividing surface is at the top of the classical potential energy barrier, and it is given by s1 ( r ) = max {sα ( r ) , sβ ( r ) , sγ ( r ) , sδ ( r )} ,

(9)

with s x ( r ) ( x =α , β ,γ ,δ ) being the reaction coordinate that describes the abstraction process of one of the four methane hydrogens H x by the incident O atom, “max” stands for choosing the largest value. sx ( r ) = r ( C − H x ) − r ( H x − O ) −  r † ( C − H x ) − r † ( H x − O )  ,

(10)

where r(X-Y) denotes the interatomic distance between atoms X and Y, and r†(X-Y) is the corresponding value at the transition state geometry. s0(r) describes a dividing surface that is located in the reactant region, which is defined as

s0 ( r ) = R∝ − R .

(11)

Here R is the scattering vector that connects the incident O atom and the center of mass of the methane.

R∝

is an adjustable parameter that should be large enough so as to make the interaction between

the two reactants negligible, it value is chosen to be 7 Å in the present calculations. In this case,

( ) in Eq.(6) becomes a function of two parameters,

Cdd 0

ξa

and

ξb

,

Cdd ( 0; ξ a , ξ b ) = tr e − β H 2 ∆ ( s ( rˆ; ξ a ) ) × e − β H 2 ∆ ( s ( rˆ; ξ b ) )  ,   ˆ

ˆ

(12)

and the optimal values of ξ a and ξ b are obtained from the following stationary conditions, ∂Cdd ( 0; ξ a , ξb ) ∂ξ a

The free energy surface can be obtained from C dd

= 0,

∂Cdd ( 0; ξ a , ξb ) ∂ξb

( 0;ξa ;ξb ) , 8


= 0.

(13)

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F (ξ a , ξ b ) = −kBT log Cdd ( 0; ξ a , ξ b ) ,

(14)

and the free energy profile along the reaction path is given by

F (ξ ) = −kBT log Cdd ( 0; ξ a , ξ b ) , (ξ = ξ a = ξ b ).

(15)

All the quantities in Eq.(2) can be evaluated using imaginary time path integral Monte Carlo (PIMC)63 and adaptive umbrella sampling64 techniques, the detail descriptions have been given in previous work54. In the present calculations, the numbers of time slices are set to 120-20 over the temperature range 2001000 K, and the numbers of Monte Carlo cycles are set to (6 − 12) × 106 for obtaining converged ensemble averages.

3. RESULTS AND DISCUSSION 3.1 THERMAL RATE CONSTANTS FOR O(3P)+CH4→OH+CH3 REACTION We have calculated the thermal rate constants for O(3P)+CH4→OH+CH3 reaction with the quantum instanton method over the temperature range 200–1000 K. In the present work, all of PES-2014, PES2000 and PES-1998 constructed by Espinosa-Garcia and co-workers7,14,15 are used to calculate the QI rates, so as to compare the QI with other theoretical approaches and check the reliability of these PESs. As consequences of the Jahn-Teller effect and spin-orbit splitting of O(3P), an extra electronic partition function ratio43 is included in the QI rates,

Qe (T ) = where E (

3

P1 )

6 5 + 3e

− E ( 3 P1 )/RT

+ e− E (

3

P0 )/RT

,

(16)

and E ( 3 P0 ) are the energies of the upper spin-orbit levels of O(3P) relative to that of 3P2,

with the values of 158.29 and 226.99 cm-1, respectively. The calculated QI rates are listed in Table 1, in

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which we also tabulate some theoretical and experimental data. The corresponding Arrhenius plots are drawn in Figure 1. Table 1. Thermal rate constants for the O(3P)+CH4 reaction PES-2014

PES-2000

PES-1998

Experimental Expt.f

T(K)

QIa

RPMDb

QI

MCTDHc

RBUd

CVTe

QI

200

3.70±0.33(-19)

1.29(-19)

2.12±0.10 (-20)

7.22(-21)

1.72(-20)

8.5(-21)

1.55±0.08(-20)

300

3.55±0.14(-17)

1.50(-17)

1.36±0.07(-17)

5.81(-18)

9.34(-18)

8.7(-18)

1.42±0.08(-17)

6.64(-16)(300K)g; 2.09(-17)(297K)h; 5.48(-15)(295K)i; 6.64(-19)(293K)j 9.10(-18)(300K)k

400

6.36±0.28(-16)

3.17(-16)

3.67±0.31(-16)

2.12(-16)

500

4.62±0.17(-15)

2.37(-15)

3.58±0.25(-15)

2.12(-15)

600

1.96±0.06(-14)

1.30(-14)

2.02±0.14(-14)

1.07(-14)

700

6.85±0.21(-14)

6.80±0.46(-14)

3.64(-14)

800

1.73±0.04(-13)

1.63±0.08(-13)

9.49(-14)

8.99(-14)

1000

7.17±0.14(-13)

6.94±0.22(-13)

3.92(-13)

3.53(-13)

a

5.41(-13)

2.80(-16)

1.13(-14)

3.3(-16)

4.13±0.23(-16)

5.95(-16)

3.57±0.18(-15)

5.44(-15)

1.78±0.07(-14)

2.59(-14)

6.41±0.25(-14)

8.37(-14)

1.5(-13)

1.49±0.04(-13)

2.11(-13)

6.3(-13)

6.81±0.14(-13)

8.43(-13)

1.7(-14)

Unit: cm3s-1, the statistical errors are included, powers of 10 are in parentheses; b From ref.43; c From

ref.41; d From ref.18; e From ref.15; f From ref.1; g From ref.65; h From ref.66; i From ref.67; j From ref.68; k

From ref.69.

In Table 1, it is seen that the statistical errors are less than 10% for all of the QI rates, and they are only 2-3% at 1000 K. We also see that all of the QI rates on PES-2014, PES-2000 and PES1998 decrease with the decreasing of temperature. Detailed comparisons show that the QI rates on PES-2000 and PES-1998 are nearly the same, this phenomenon is not surprising because PES2000 just improves the symmetry of PES-1998. Compared the QI rates on PES-2014 with those

10


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on PES-2000 and PES-1998, we can see that these three kinds of QI rates are very close to each other at high temperatures, this is because the zero-point energy corrected potential energy barriers are nearly the same for these PESs (The values are 9.96, 9.98 and 9.98 kcal/mol for PES1998, PES-2000 and PES-2014, respectively.). However, the rates on PES-2014 become much larger than those on PES-2000 and PES-1998 at low temperatures, as seen in Figure 1. For instance, the ratio of the QI rates on PES-2014 and PES-2000 is 2.61 at 300 K.

QI (2014) QI (2000) QI (1998) RPMD (2014) CVT (2000) MCTDH (2000) RBU (2000)

-12

1.0×10

-15

1.0×10

3 -1

k(cm s )

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Expt. Froben et al. Westenber et al. Cadle et al. Falconer et al. Cohen et al.

-18

1.0×10

-21

1.0×10

1

2

3

4

5

1000/T(K) Figure 1. Arrhenius plots of the O(3P)+CH4 rate constants: QI rates on PES-2014, solid line; QI rates on PES-2000, dashed line; QI rates on PES-1998, dotted line; RPMD rates on PES-201443, dash-dotted line; CVT rates on PES-200015, dash-dot-dotted line; MCTDH rates on PES-200041, short dashed line; RBU rates on PES-200018, short dotted line; Expt.1, solid squares; Froben et al.65, solid circle; Westenberg et

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al.66, solid triangle; Cadle et al.67, solid diamond; Falconer et al.68, solid sexangle. Cohen et al.69, solid

pentangle.

In order to clarify the discrepancies of the QI rates at low temperatures, we plot the free energy profiles along the reaction path at 300 K in Figure 2. It is clear that PES-2014 has the lowest free energy barrier, while the free energy barriers of PES-2000 and PES-1998 are very similar. As we know, the lower the barrier is, the larger the rate will be. Considering that the zero-point energy corrected potential energy barriers are nearly the same, the differences of the free energy barrier heights should be caused by the different shapes of the PESs. Indeed, the imaginary frequencies at the transition states for PES-2014, PES-2000 and PES-1998 are 1919, 1549 and 1507 cm-1, respectively, it is clear that PES-2014 has the thinnest potential energy barrier. It is because of the thinnest potential energy barrier that PES-2014 has the lowest free energy barrier. The underlying mechanism is that the quantum delocalization of the transferred H atom has a role to lower the free energy barrier at low temperature, and this effect is especially remarkable on a thin potential energy barrier. In addition, PES-2014 has a shallow well before the free energy barrier, which is corresponding to the intermediate complex before the transition state. These characteristics of PES-2014 mean that the quantum tunneling should be significant.

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300 K

12

Free energy (kcal/mol)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


PES-2014 PES-2000 PES-1998

8

4

0 0.0

0.5

1.0

ξ

Figure 2. Free energy profiles along the reaction path for O(3P)+CH4 reaction at 300 K. The solid, dashed and dotted lines stand for the results on PES-2014, PES-2000 and PES-1998, respectively.

The quantum tunneling effect can be revealed by the Arrhenius plots of the rates, as shown in Figure 1, at high temperatures, all of the QI rates follow the Arrhenius law, which results in linear Arrhenius plots, while at low temperatures, the quantum tunneling becomes important and these QI rates form curved Arrhenius plots. The Arrhenius plots for the QI rates demonstrate that the quantum tunneling effect on PES-2014 is most remarkable, which is consistent with the situation that PES-2014 has the thinnest free energy barrier. As described in Sec. 2, the QI method defines two dividing surfaces ( ξ a

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ξb ,

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in Eq.(4)), between which the quantum tunneling most probably takes place, and

therefore it has the advantage to visualize the quantum tunneling effect. At high temperatures, the quantum tunneling is negligible, these two dividing surfaces usually coalesce into one ( ξ a = ξ b ). However, at low temperatures, the quantum tunneling becomes significant, these two dividing surfaces are separated, one locates on the reactant side, the other is on the product side. Under this situation, there will be two saddle points on the free energy surface (defined in Eq.(14)), as illustrated in Figure 3.

Figure 3. Local topographies of the free energy surfaces near the top of the barrier at 200 K. The cross symbols stand for the locations of the saddle points. (a) On PES-2000; (b) On PES-2014.

In Figure 3, we can see that both PES-2000 and PES-2014 have two saddle points near the top of the barrier at 200 K, and these two saddle points can be easily transformed to two separated dividing surfaces. The corresponding dividing surfaces (ξ a , ξ b ) or (ξ b , ξ a ) are (0.98, 1.07) and (0.90, 1.07) on PES-2000 and PES-2014, respectively. The distance


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between ξ a and ξ b reveals the magnitude of the quantum tunneling, the corresponding values ( ξa −ξb ) are 0.09 and 0.17 on PES-2000 and PES-2014, respectively. It is clear that quantum tunneling effect on PES-2014 is much more remarkable than that on PES-2000. Compared with others, on PES-2014, the QI rates are always larger than those of RPMD in the whole tested temperature range. For instance, the ratio of the QI rate to the RPMD one is 1.33 at 1000 K, while it is increased to 2.37 at 300 K. Both the QI and RPMD treat all degrees of freedom on an equal footing via the imaginary time path integral formulation. However, the RPMD method involves the recrossing effect through the long-time limit of the ring polymer transmission coefficient, while the QI method is a short-time approximation to the flux-flux correlation function, it does not consider the recrossing effect. With this in mind, the differences between the QI and RPMD at high temperatures are most probably caused by the recrossing effect. At 1000 K, the QI may overestimate the rate by 30%. The much more remarkable discrepancies at low temperatures are very similar to the case70 of H+CH4. Suleimanov et al. have attributed the difference between the QI and RPMD to the choice of dividing surface. Indeed, the RPMD rate does not depend on the reaction coordinate, while the QI rate is sensitive to the two dividing surfaces. Previous studies70,71 also show that the QI is better than the RPMD for atom-diatom chemical reactions, but it becomes worse than RPMD as the dimensionality of the reaction increases. The underlying mechanism is that the quantum tunneling effect of QI depends on the two dividing surfaces, with these two dividing



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surfaces, the QI predicts much larger quantum tunneling effects than the RPMD for relatively complex reactions. On PES-2000, the QI rates are consistent with the CVT rates at high temperatures, for instance, the QI rate is only 1.10 times larger than that of CVT at 1000 K. However, with the decreasing of temperature, the QI rates become much larger than the CVT ones, for instance, the QI rate is 1.56 times as large as that of the CVT at 300 K. The consistency of the QI and CVT at high temperature confirms the accuracy of the QI, even though both of them ignore the recrossing effect. The difference between the QI and CVT at low temperature is due to the fact that these two methods have used different ways to incorporate the quantum tunneling effect. The quantum tunneling in the CVT is included by using the microcanonical optimized multidimensional tunneling approach, in which an optimized tunneling path is used. However, the quantum instanton method considers all of the tunneling paths and automatically gives each path its natural weight by the quantum Boltzmann factor, it gives a much larger quantum tunneling effect. The MCTDH is regarded as one of the most accurate methods to predict the rates for complex reactions, although the J-shifting approximation72 may underestimate the rates by about 30%. For the title reaction, the MCTDH rates are nearly 2 times smaller than the QI counterparts in the whole tested temperature range, and the difference between these two kinds of rates slightly increases with the decreasing of temperature. The MCTDH method also adopts the flux-flux correlation function, and the rate constants are obtained


Page 17 of 39

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directly from the important eigenvalues of the thermal flux operator. It captures both the quantum tunneling and recrossing, while the QI is under the no-recrossing assumption, so we can infer that the recrossing may reduce the rates by less than one half for the present reaction. Indeed, our inferred recrossing effect is in line with that of RPMD42 (the ring polymer transmission coefficient is considered as an approximate recrossing factor of RPMD at high temperature, its value is 0.719 at 900 K). The four dimensional RBU (rotating bond umbrella model18) rates on PES-2000 are also tabulated in Table 1. It is seen that the RBU calculation gives very similar rates to the QI at low temperatures, however, the RBU rates are much lower than those of QI at high temperatures, for instance, the QI rate is larger than the RBU one by a factor of 1.97 at 1000 K. The perfect agreement at low temperature is most probably due to error cancellation because both the QI and RBU contain approximations, such as the norecrossing assumption in the QI and the lack of overall rotation in the RBU. The tunneling factor of RBU is approximately 2 times larger than that of CVT in the whole tested temperature range, which is consistent with the situation that the QI also gives a much larger quantum tunneling effect than the CVT. The RBU rates are close to those of the MCTDH at high temperatures, this phenomenon demonstrates that this reduced dimensional calculation is effective at high temperatures. Experimentally, Baulch et al.1 analyzed the available data, they found that the rate constants were well defined above 1000 K, while none of the data was considered reliable



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below 400 K, they also recommended an expression for the rate constants over the temperature range 400-2500 K. The corresponding rate constants are tabulated in Table 1, it is seen that both the QI and CVT rates are close to the recommended values, this is not surprising because both the CVT and experimental rates have been used in the fitting process of the analytical PESs. However, the rates of RPMD, MCTDH and RUB are significantly low, this is mainly due to the fact that all of these methods have considered the recrossing effect, which lowers the rates. Below 400 K, the experimental data is − 19 −15 − 5 .48 × 10 scattered, the measured rates around 300 K are in the range of 6.64 × 10

cm3s-1. It is difficult to decide which potential energy surface is better by the rates, because all of the theoretical rates on these potential energy surfaces are included in this range, although only PES-2014 can reproduce the experimental OH product rotovibrational

distribution14.

More

precise

experimental

measurements at

low

temperatures are needed, so as to give a benchmark for both the potential energy surfaces and dynamical methods. 3.2 KINETIC ISOTOPE EFFECTS FOR THE O(3P)+CH4 REACTION The kinetic isotope effect (KIE) is defined as the ratio of the rate constants for the O(3P)+CH4 and heavier isotopically substituted reactants. The KIE results from the quantum tunneling and zero-point energy, both of which depend on the masses of reactants, this characteristic makes the KIE a good candidate for checking the quantum tunneling effect.


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Table 2. H/D kinetic isotope effects for the O(3P)+CH4/CD4 reaction O(3P)+CH4/O(3P)+CD4 PES-2014

PES-2000

PES-1998

T(K)

QI

RPMDa

CUSa

QI

CUSb

QI

200

389.07

290.4

136.6

50.97

64.8

38.51

300

45.60

30.8

18.6

18.04

18.9

16.64

29.1

400

15.69

10.7

6.2

8.01

8.4

7.67

12.0

500

8.16

5.1

3.4

4.46

4.30

6.6

600

5.25

3.8

2.4

3.88

3.38

4.2

700

3.92

3.09

800

3.24

2.85

2.2

2.36

1000

2.24

2.23

1.4

2.16

a

2.0

1.43

3.4

CUSc

2.89

1.78

From ref.43; b From ref.15; c From ref.7.

We have calculated the QI rates for the O(3P)+CD4→OD+CD3 (R2) reaction over the temperature range 200-1000 K. The corresponding KIEs are listed in Table 2. We can see that all of the QI KIEs on PES-1998, PES-2000 and PES-2014 are much larger than 1, this is because the zero-point energy has a role to lower the classical potential energy barrier, and it reduces the barrier of O+CH4 much more than that of O+CD4 (It should be mentioned that both the O+CH4 and O+CD4 have the same classical potential energy barrier). Our normal mode analysis reveals that the zero-point energies can decrease the classical potential energy barriers of O+CH4 and O+CD4 by 3.87 and 2.64 kcal/mol, respectively. It is clear that the zero-point energy corrected potential energy barrier of O+CD4 is higher than that of O+CH4 by 1.23 kcal/mol, and therefore the rates of O+CD4



are smaller than those of O+CH4. The value 1.23 is in line with the free energy difference (1.40 kcal/mol) between the O+CH4 and O+CD4 at 1000 K, as shown in Figure 4, this demonstrates that the KIE is dominated by the zero-point energy at high temperatures.

PES-2014

18 Free energy (kcal/mol)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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O + CH4 (200 K) O + CD4 (200 K) O + CH4 (1000 K)

12

O + CD4 (1000 K)

6

0 0.0

0.5

1.0

ξ

Figure 4. Free energy profiles along the reaction path on PES-2014. The solid, dashed, dotted and dash-dotted lines correspond to the results of O+CH4 (200 K), O+CD4 (200 K), O+CH4 (1000 K) and O+CD4 (1000 K), respectively.

From Table 2, we can see that all of the QI KIEs on PES-1998, PES-2000 and PES2014 have a rapid increase with the decreasing of temperature, which can be explained by the quantum tunneling effect. Due to the small masses, both H and D atoms have remarkable quantum tunneling effects, however, with the decreasing of temperature, the


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quantum tunneling effect of H atom increases more rapidly than that of the heavier D atom, which makes the difference between the rates of O+CH4 and O+CD4 larger and larger. This phenomenon can be clarified by the free energy difference between the O+CH4 and O+CD4, as shown in Figure 4, substituting D for H increases the free energy barriers by 1.40 and 2.13 kcal/mol at 1000 and 200 K, respectively, the increase of the free energy difference with the decreasing of temperature demonstrates that the H atom has a more remarkable quantum effect. In Table 2, it is also seen that the QI KIEs on PES-1998 and PES-2000 are nearly the same, this is due to the fact that these two potential energy surfaces are very similar. The QI KIEs on PES-2014 are much larger than those on PES-2000 and PES-1998 at low temperatures, although they are close to each other at high temperatures. This situation is caused by the quantum tunneling, which has a role to increase the KIE, as discussed in Section 3.1, PES-2014 has the most remarkable quantum tunneling effect due to the thinnest barrier. Compared to others, on PES-2014, the QI KIE is close to the RPMD one at 1000 K, however, it becomes larger than that of RPMD at low temperature. For instance, the ratio of the QI KIE to the RPMD value is 1.5 at 300 K. This situation is consistent with our above conclusion that the predicted quantum tunneling effect of QI is larger than that of the RPMD. Compared to the CUS, both the QI and RPMD predict much larger KIEs, the values of QI/CUS and RPMD/CUS are 2.5 and 1.7 at 300 K, respectively. These



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Page 22 of 39

situations result from the discrepancies in treating the quantum tunneling effect. As discussed in Section 3.1, the QI reports the largest quantum tunneling effect, so it gives the largest KIE. On PES-2000 and PES-1998, the QI KIEs are in line with those of CUS, this situation reflects that the shape of potential energy barrier may play a dominant role in the KIE. On PES-2000 and PES-1998, the tops of the barriers are broad, the QI and CUS predict similar KIEs. However, on PES-2014, the barrier is thin, the QI reports a much larger KIE than the CUS. The underlying mechanism is that the KIE is controlled by the quantum tunneling at low temperature, while the quantum tunneling is affected by the shape of the barrier. Up to now, no experimental KIE is available, so it is uncertain which PES gives a better KIE. We have also calculated the rates of the O(3P)+CHD3 reaction. For this reaction, two processes must be considered, one is O(3P)+CHD3→OH+CD3 (R3), the other is O(3P)+CHD3→OD+CHD2 (R4). In the first process, only one hydrogen can be abstracted, while in the second process, there are three possible transition states, which can be obtained by permuting the three deuterium atoms. The total rate constant of O(3P)+CHD3 is the sum of these four channels. The corresponding KIEs are tabulated in Table 3.


Page 23 of 39

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Table 3. H/D kinetic isotope effects for the O(3P)+CH4/CHD3 reaction O(3P)+CH4/O(3P)+CHD3

T(K) PES-2014

PES-2000

PES-1998

200

7.24

6.39

3.99

300

5.62

3.93

3.63

400

4.81

2.57

3.12

500

3.58

2.51

2.43

600

2.55

2.50

2.00

700

2.51

1.97

1.83

800

2.21

1.95

1.72

1000

1.67

1.71

1.55

In Table 3, it is seen that all of the QI KIEs on PES-2014, PES-2000 and PES-1998 are much larger than 1, this can be also explained by the zero-point energy. For instance, on PES-2014, the zero-point energy corrected potential energy barriers for R1, R3 and R4 are 9.98, 10.09 and 11.15 kcal/mol, respectively. Our calculations further show that R1 (O(3P)+CH4) has the largest rate, while R4 gives the smallest one. We have also drawn the free energy profiles alone the reaction path for R1, R3, R4 and R2 at 300 K in Figure 5, the corresponding free energy barriers are 11.72, 12.00, 13.69 and 13.81 kcal/mol. It is clear that the H abstraction reaction is much easier than that of D.



15 PES-2014

Free energy (kcal/mol)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 24 of 39

10

O + H-CH3

(R1)

O + H-CD3

(R3)

O + D-CHD2 (R4) O + D-CD3

(R2)

5

0 0.0

0.5

1.0

ξ

Figure 5. Free energy profiles along the reaction path for the O+H-CH3 (solid line), O+H-CD3 (dashed line), O+D-CHD2 (dotted line) and O+D-CD3 (short dashed line) reactions at 300 K on PES-2014.

All of the QI KIEs increase with the decreasing of temperature, this is due to the fact that the quantum tunneling enhances the rates of H much more than those of D at low temperatures. Detailed comparisons show that PES-2014 gives the largest KIE while PES-1998 has the smallest one, this is because the quantum tunneling increases as the width of the potential energy barrier decreases from PES-1998, PES-2000 to PES-2014. The experimental value is still not available.


Page 25 of 39

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4. CONCLUSIONS We have calculated the rate constants for the O(3P)+CH4→OH+CH3 reaction with the quantum instanton method in full dimensionality, the analytical potential energy surfaces PES-1998, PES-2000 and PES-2014 constructed by Espinosa-Garcia and co-workers have been used. The QI rates on these potential energy surfaces are close to each other at high temperatures, and they are in good agreement with the experimental data above 400 K, below which the experimental values are scattered. The shapes of PES-1998 and PES2000 are nearly the same, while the barrier of PES-2014 is much thinner than those of PES-2000 and PES-1998. This thin barrier of PES-2014 results in a lower free energy barrier due to the quantum delocalization of the transferred H atom, and causes a much larger quantum tunneling. As a result, the rates on PES-2014 are much larger than those on PES-2000 and PES-1998 at low temperatures. At high temperatures, the QI rates are close to those of CVT, however, both the QI and CVT predict much larger rates than the other approaches, such as the RPMD, MCTDH and RBU. This is because both the QI and CVT ignore the recrossing effect, indeed, the recrossing effect may reduce the rates by less than one half as reported by the RPMD and MCTDH. At low temperatures, the QI rates are close to those of RBU, but are much larger than the others. This is mainly due to the fact that these dynamical methods have



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Page 26 of 39

used different ways to incorporate the quantum tunneling, and the QI and RBU give the largest quantum tunneling effect. We have also calculated the rates of O(3P)+CD4 and O(3P)+CHD3, the corresponding kinetic isotope effects are much larger than 1, and they increase with the decreasing of temperature. These kinetic isotope effects are controlled by the zero-point energy at high temperatures, however, they are dominated by the quantum tunneling at low temperatures. Our calculations also show that the shape of the potential energy barrier gives a great effect on the kinetic isotope effect. AUTHOR INFORMATION Corresponding Author *

Wenji Wang. E-mail: [email protected]; Phone: 029-87092303

Notes

The authors declare no competing financial interest.

ACKNOWLEDGEMENTS

This research is supported by the National Natural Science Foundation of China (Grant 21203151), and Chinese Universities Scientific Fund (2452015084 and 2452015432). The authors thank Joaquin Espinosa-Garcia for providing the potential energy surfaces.


Page 27 of 39

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Thermal Rate Constants for the O(3P) + CH4 â OH + CH3 Reaction

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Thermal Rate Constants for the O(3P) + CH4 â OH + CH3 Reaction