We report a permutationally invariant global potential energy surface (PES) for the H + CH4 system based on ∼63 000 data points calculated at a high ab initio level (UCCSD(T)-F12a/AVTZ) using the recently proposed permutation invariant polynomial-neural network method. The small fitting error (5.1 meV) indicates a faithful representation of the ab initio points over a large configuration space. The rate coefficients calculated on the PES using tunneling corrected transition-state theory and quasi-classical trajectory are found to agree well with the available experimental and previous quantum dynamical results. The calculated total reaction probabilities (Jtot = 0) including the abstraction and exchange channels using the new potential by a reduced dimensional quantum dynamic method are essentially the same as those on the Xu-Chen-Zhang PES [Chin. J. Chem. Phys. 27, 373 (2014)].

The hydrogen abstraction of methane by a hydrogen atom and its reverse (H + CH4↔H2 + CH3) have become a prototype in understanding dynamics of bimolecular reactions in the gas phase.1,2 The kinetics of this reaction have been studied both experimentally3–9 and theoretically.10–27 for decades. The dynamics of this reaction and its deuterated variants have also been the subject of experimental investigations focusing on reaction cross sections and mode specificity.28–36 On the theoretical front, the reaction dynamics has been investigated using both quasi-classical trajectory (QCT)22,37–42 and quantum dynamical (QD) methods. While many of the QD studies were based on reduced-dimensional models,43–58 full-dimensional calculations of the reaction dynamics have been reported by Manthe and coworkers at both the initial state selected59–62 and state-to-state levels,63 using the multi-configuration time-dependent Hartree (MCTDH) approach.

Central to all these studies is the potential energy surface (PES), which controls both kinetics and dynamics.1 This six-atom system (CH5) has only eleven electrons, thus amenable to high-level ab initio calculations. However, it is far from trivial to construct an accurate and efficient full 12-dimensional PES for the title reaction. The early semi-empirical PESs are not considered quantitatively accurate, despite the fact that ab initio data were often used to calibrate the transition state.14,64–66 In 2004, Manthe and coworkers developed the first ab initio based PES using the modified Shepard interpolation.15 Unfortunately, this PES was restricted to configuration space near the transition state, thus inadequate for state-resolved reaction dynamics. In addition, this PES describes only the direct hydrogen abstraction channel, neglecting the exchange channel (see Figure 1 for the schematic illustration of the abstraction and the exchange reactions). Using the permutationally invariant polynomial (PIP) method,67 Zhang, Braams, Bowman, and coworkers developed several versions of the global PES for this system, including both the abstraction and exchange channels, namely, ZBB1,68 ZBB2, and ZBB3.41 These PESs, based on ∼20 000 points at the partially spin-restricted coupled-cluster method with the augmented correlation-consistent polarized valence triple-zeta basis set (RCCSD(T)/AVTZ), are significantly more accurate than all previous PESs. In 2011, Zhou et al.54 constructed another global PES using the modified Shepard interpolation method based on ∼30 000 data points (including energies and first and second order derivatives) obtained at the level of spin unrestricted CCSD(T) (UCCSD(T)) with the AVTZ basis set for the energies and 6-311 + + G(3df, 2pd) for the derivatives, respectively. This PES, labeled as ZFWCZ, has a comparable accuracy to ZBB3 PES.54 However, the evaluation of the PES is extremely slow due to the nature of the high-dimensional interpolation method, limiting its application.54,69 In 2014, Xu, Chen, and Zhang (XCZ) greatly improved the situation with a new global PES for this benchmark system using a non-linear neural network (NN) fitting method based on ∼48 000 points calculated at the level of UCCSD(T)-F12a/AVTZ.70 Although the total root mean squared error (RMSE) is rather small, less than 4 meV, the five hydrogen atoms are not exactly treated as identical in this PES. A smoothed exchange scheme was used to adapt the permutation symmetry approximately,71,72 but problems can still occur especially at high symmetry configurations. In addition, the evaluation of the PES is quite slow due to the smoothing strategy, although significantly faster than the ZFWCZ PES.

FIG. 1.

Schematic illustration of the reaction pathways, namely, the abstraction and the exchange channels, on the CH5 ground state PES. All energies are in eV relative to the H + CH4 asymptote. The geometries (distances in Å and angles in deg) of the stationary points are also shown. The first value (in black) is obtained on the PIP-NN PES, while the second (in red) is from the UCCSD(T)-F12a/AVTZ ab initio calculation.

FIG. 1.

Schematic illustration of the reaction pathways, namely, the abstraction and the exchange channels, on the CH5 ground state PES. All energies are in eV relative to the H + CH4 asymptote. The geometries (distances in Å and angles in deg) of the stationary points are also shown. The first value (in black) is obtained on the PIP-NN PES, while the second (in red) is from the UCCSD(T)-F12a/AVTZ ab initio calculation.

Close modal

In this work, we report a new full-dimensional global PES based on ∼63 000 ab initio points at the level of UCCSD(T)-F12a/AVTZ. The data set consists of the original points used for the XCZ PES,70 with approximately 15 000 more points generated using the same method. This larger data set was fit using the recently proposed PIP-NN method.73,74 The resulting PIP-NN PES has a total RMSE of 5.1 meV, and the permutation invariance of the PES is strictly enforced. This new PES is also efficient, much faster to evaluate than the XCZ PES.

To test the accuracy of the new PES, the reaction path was analyzed and rate coefficients computed using QCT and the transition-state theory with semi-classical tunneling corrections. Furthermore, QD calculations were performed to compute the reaction probabilities. This publication is organized as follows. In Sec. II, methods used in these calculations are briefly outlined. The PES and the kinetic and dynamical results are given in Sec. III. Final conclusions are given in Sec. IV.

The XCZ PES consists of three parts connected together with switching functions.70 In particular, 16 835, 20 762, and 10 186 points (totaling 47 783) were used for the H + CH4 channel, CH5 interaction region, and H2 + CH3 channel, respectively. Please see details for the separation of different regions in Refs. 70 and 71. In this work, this ab initio data set was augmented by additional 15 258 points, totaling 63 041 points. The additional points were calculated with the same UCCSD(T)-F12a/AVTZ protocol,75,76 using Molpro.77 These 15 258 points were added to patch up high-energy regions of the PES and configurations in the H2 + CH3 channel. All the 63 041 points were used in the PIP-NN fitting. Unlike the separate fitting in the XCZ PES, the present PES does not use switching functions.

In the PIP-NN approach,73,74 the input layer of the NN was replaced by a special set of symmetry functions. Particularly, PIPs of Morse variables (yij = exp(−αrij) with α = 1.0 Å−1 and i or j = 1–6) defined in terms of internuclear distances67,78 were used as the symmetry functions. Note that in the PIP-NN fitting of PESs with four or more atoms, the number of PIPs should be larger than the number of internal coordinates in order to ensure the correct permutation invariance.74 The PIP-NN method has been quite successfully in constructing reactive PESs of molecular systems involved three,73 four,74,79–81 and five atoms,82,83 with fitting errors of several meVs or less. The CH5 system represents the first application of the PIP-NN fitting to systems with six atoms.

The maximum total order of the PIPs in the input vector of the NN is chosen to be six, resulting in 848 terms. The NN has two hidden layers with N1 and N2 as the number of neurons in each layer. Tests with N1-N2 = 2-100, 3-100, 3-150, 4-100, and 4-150 were performed. The choice of a small number for N1 is due to the large number of the input vector, or else, the number of the parameters will be increased significantly, which makes the fitting extremely difficult and slow. In each NN fitting, the data were divided randomly into three sets, i.e., the training (90%), validation (5%), and testing (5%) sets. The “early stopping” method84 was used to avoid overfitting. The NN training generally converged fast, typically finishing within a few hundred steps. To avoid artificial extrapolation due to edge points in the randomly selected validation and test sets, only fits with similar RMSEs (defined as i = 1 N data E output,i E target,i 2 / N data ) for all three sets were accepted. Besides, the maximum deviation was also used for choosing the final PIP-NN PES. The final PIP-NN PES was selected as the average of three best fits to further minimize random errors. The PES is available from the corresponding authors upon request.

Starting from the saddle point geometry, the minimum energy path (MEP) to both the reactant and product sides was determined in redundant curvilinear internal coordinates85 using the Euler steepest-descent (ESD) method86 with a step size of 0.000 94 amu1/2 bohrs and with the Hessian calculated every nine steps. The MEP was determined from s = − 1.9 amu1/2 bohrs on the reactant side to s = + 1.9 amu1/2 bohrs on the product side and the scaling mass for all coordinates is set to 1 amu. A generalized normal-mode analysis was performed using a redundant curvilinear projection formalism85 by projecting these modes along the MEP as well as overall rotations and translations. With such information, vibrationally adiabatic potentials were determined by the formula Va(n, s) = VMEP(s) + εint(n, s), where VMEP(s) is the classical potential energy along the MEP with its zero at the reactant asymptote, and εint(n, s) is the vibrational energy at s. Va(n = 0, s) corresponds to the ground vibrational state adiabatic potential, V a G ( s ) .

The reaction path curvature κ(s) consists of components BmF(s), the coupling terms between the mth generalized normal mode and reaction coordinate, mode F, κ ( s ) = m = 1 F 1 B m F ( s ) 2 1 / 2 , and these coupling terms control and reflect the non-adiabatic flow of energy amongst the generalized vibrational modes and the reaction coordinate. These coupling terms were used in the calculation of transmission coefficients that include the effects of the reaction path curvature.

The rate coefficients of the abstraction reaction were then computed using the canonical variational transition-state theory (CVTST or CVT).87 Quantum effects for motions orthogonal to the reaction path were included by using quantum mechanical vibrational partition functions under the harmonic approximation, while quantum effects in the reaction coordinate were included by using the micro-canonical optimized multidimensional tunneling (μOMT) approach,87 in which, at each total energy, the larger of the small-curvature tunneling (SCT) and large-curvature tunneling (LCT) probabilities was taken as the best estimate. The rotational partition functions were calculated classically.

The reactive rate constants of the abstraction reaction at temperatures 3000, 2500, and 2000 K were calculated by the standard QCT implemented in VENUS,88 and at each temperature, about 105 trajectories were calculated to make the statistical errors less than 5.0%. No QCT calculations were attempted at lower temperatures due to the low reactivity. The initial ro-vibrational energies of CH4 and translational energies between H and CH4 were sampled according to the Boltzmann distribution at each specific temperature. The trajectories were initiated with a reactant separation of 8.0 Å and terminated when products or reactants reached a separation of 6.0 Å. The maximal impact parameter (bmax) is determined using small batches of trajectories with trial values. The gradient of the PES is obtained numerically by a central-difference algorithm with the propagation time step 0.01 fs. Almost all trajectories conserved energy to within a chosen criteria (10−4 kcal/mol), which testifies the smoothness of the PES. Note that the exchange reaction becomes viable at these high temperatures, but representing only a minor channel. The rate constant was calculated as follows: k = 8 k B T π μ π b max 2 N r N total , in which Nr and Ntotal are the numbers of the reactive and total trajectories and μ is the reduced mass for the reactant channel, namely, μ = μ H μ CH 4 / μ H + μ CH 4 .

A 7-dimensional time-dependent wave packet (7D TDWP) method was used to calculate the total reaction probabilities, in which the non-reactive CH3 group was constrained in a C3v symmetry. For details of the reduced dimensional TDWP method, please refer to our early works.48,55,89 Numerical parameters for the wave packet propagations are listed in Table I. To calculate the reaction probabilities of H atom with ground ro-vibrational state CH4 at Jtot = 0, a total of 120 sine basis ranging from 0.5 to 15.0 a0 were used for the translational coordinate, in which 55 were in the interaction region. The number of vibrational basis function describing the reactive C–H stretch is 6 for the asymptotic region and 30 for interaction region ranging from 1.0 to 5.0 a0. The bond length of non-reactive CH3 group was fixed at its equilibrium geometry of 2.0558 a0, and the umbrella motion was described by 17 basis functions ranging from 70.0° to 125.0°. The size of the rotational basis functions is controlled by the parameters Jmax = 69, lmax = 45, jmax = 24, and kmax = 6. The size of the rotational basis functions is 57 458, and the number of the total basis functions is about 1.993 × 109.

TABLE I.

Numerical parameters used in the wave packet calculations.

NR 120/55
(Rmin, Rmax (0.5, 15.0)a 
Nr  6/30 
(rmin, rmax (1.0, 5.0)a 
NU  17 
(Umin, Umax (70.0, 125.0)b 
Jmax  69 
lmax  45 
jmax  24 
kmax 
Nrot  57 458 
Ntot  1.993 × 109 
NR 120/55
(Rmin, Rmax (0.5, 15.0)a 
Nr  6/30 
(rmin, rmax (1.0, 5.0)a 
NU  17 
(Umin, Umax (70.0, 125.0)b 
Jmax  69 
lmax  45 
jmax  24 
kmax 
Nrot  57 458 
Ntot  1.993 × 109 
a

Unit in a0.

b

Unit in degree.

The final PIP-NN PES employed a NN with two hidden layers each with 4 and 100 neurons, resulting in 3997 parameters. The total RMSEs and the maximum deviation of the three best PESs are 6.8/632.7, 6.4/578.2, and 6.7/970.8 meV, respectively. The final PES has an overall RMSE of 5.1 meV and a maximum deviation of 434.1 meV for all points, which reduce to 4.6 meV and 209.9 meV, respectively, for points of energy less than 4.0 eV. The fitting errors as a function of the ab initio energy are shown in Figure 2(a). As shown, the small errors are evenly distributed in the entire energy range. Figure 2(b) shows the distribution of the fitting errors. One can see that ∼44 000 points are of fitting errors less than 2.5 meV, and ∼11 000 points possess fitting errors within 2.5-5.0 meV.

FIG. 2.

(a) Fitting errors (EfitEab, in eV) as a function of the ab initio energy (eV) relative to the H + CH4 asymptote; (b) distributions of fitting errors (defined as |EfitEab|) for the selected points.

FIG. 2.

(a) Fitting errors (EfitEab, in eV) as a function of the ab initio energy (eV) relative to the H + CH4 asymptote; (b) distributions of fitting errors (defined as |EfitEab|) for the selected points.

Close modal

Figure 1 presents the geometries and energies (in eV relative to the H + CH4 asymptote) of the stationary points along the abstraction and exchange reaction coordinates. One can see that the stationary point energies on the PES reproduce ab initio data well: the differences between the two are less than 6.9 meV. The stationary point geometries and harmonic frequencies on the PES are also in excellent agreement with the ab initio counterparts, as shown in Figure 1 and Table II. The contour plot of the PES along the two reactive bonds, namely, rCH and rHH, is shown in Figure 3, with all other coordinates constrained at the transition state TS1. It is clear from the contours that this reaction has a late barrier.

TABLE II.

Comparison of the harmonic frequencies (cm−1) of the stationary points, including the abstraction (TS1) and exchange (TS2) saddle points.

Frequencies (cm−1)
Species Method 1 2 3 4 5 6 7 8
H + CH4  PIP-NN PESa  1345(t)b  1567(e)  3029  3158(t)         
  UCCSD(T)-F12ac  1344(t)  1569(e)  3033  3156(t)         
H2 + CH3  PIP-NN PESa  4401  507  1426(e)  3115  3294(e)       
  UCCSD(T)-F12ac  4401  518  1420(e)  3120  3303(e)       
TS1  PIP-NN PESa  1447i  527(e)  1073  1116(e)  1440(e)  1790  3078  3228(e) 
  UCCSD(T)-F12ac  1464i  527(e)  1074  1115(e)  1440(e)  1782  3080  3231(e) 
TS2  PIP-NN PESa  1869i  848(e)  1295  1353(e)  1388(e)  1503  3014  3201(e) 
  UCCSD(T)-F12ac  1797i  841(e)  1306  1349(e)  1381(e)  1540  3020  3208(e) 
Frequencies (cm−1)
Species Method 1 2 3 4 5 6 7 8
H + CH4  PIP-NN PESa  1345(t)b  1567(e)  3029  3158(t)         
  UCCSD(T)-F12ac  1344(t)  1569(e)  3033  3156(t)         
H2 + CH3  PIP-NN PESa  4401  507  1426(e)  3115  3294(e)       
  UCCSD(T)-F12ac  4401  518  1420(e)  3120  3303(e)       
TS1  PIP-NN PESa  1447i  527(e)  1073  1116(e)  1440(e)  1790  3078  3228(e) 
  UCCSD(T)-F12ac  1464i  527(e)  1074  1115(e)  1440(e)  1782  3080  3231(e) 
TS2  PIP-NN PESa  1869i  848(e)  1295  1353(e)  1388(e)  1503  3014  3201(e) 
  UCCSD(T)-F12ac  1797i  841(e)  1306  1349(e)  1381(e)  1540  3020  3208(e) 
a

This work, fitted PIP-NN PES.

b

The “t” and “e” denote the triple or double degeneracy of the mode, respectively.

c

This work, UCCSD(T)-F12a/AVTZ.

FIG. 3.

Contour plots (0–3 eV with an interval of 0.1 eV) on the present PIP-NN PES for the H + CH4 → H2 + CH3 abstraction reaction along the two reactive coordinates, rHH and rCH distances, with all other coordinates constrained at the transition state TS1.

FIG. 3.

Contour plots (0–3 eV with an interval of 0.1 eV) on the present PIP-NN PES for the H + CH4 → H2 + CH3 abstraction reaction along the two reactive coordinates, rHH and rCH distances, with all other coordinates constrained at the transition state TS1.

Close modal

Figure 4 shows (a) the classical MEP (VMEP) and vibrationally adiabatic ground state energy ( V a G ), (b) generalized normal mode frequencies, and (c) the curvature (κ) as a function of the abstraction reaction coordinate s in the range ±1.9 amu1/2 bohrs, which were calculated by the reaction path analysis using POLYRATE 9.7.90 The classical barrier on the PES is 0.64 eV at the transition state s = 0.0 amu1/2 bohr, but the maximum of V a G ( s ) (0.57 eV with respect to the reactant asymptote) is shifted to s = + 0.10 amu1/2 bohrs. This is indicative of a small variational effect for the system. From Figure 4(b), one can see that the symmetric stretching (labeled 4) frequency of the methane reactant gradually decreases to 1458 cm−1 at s = − 0.11 amu1/2 bohrs, as it is approached by the H atom, signaling mode softening as the activated complex is formed. There are two peaks in the reaction path curvature κ before and after the saddle point, as shown in Figure 4(c), as often seen in thermo-neutral reactions.

FIG. 4.

Reaction path analysis of the abstraction reaction H + CH4 → H2 + CH3: (a) the MEP (denoted as VMEP relative to the H + CH4 classical energy) and the adiabatic ground state energy ( V a G relative to the energy of the H + CH4 ground ro-vibration state) along the reaction coordinate s (amu1/2 bohr) on the PIP-NN PES. The unit of the energy is kcal/mol. (b) The 11 generalized normal mode frequencies (103 cm−1) along s. (c) The reaction path curvature κ (amu−1/2 bohr−1) along s.

FIG. 4.

Reaction path analysis of the abstraction reaction H + CH4 → H2 + CH3: (a) the MEP (denoted as VMEP relative to the H + CH4 classical energy) and the adiabatic ground state energy ( V a G relative to the energy of the H + CH4 ground ro-vibration state) along the reaction coordinate s (amu1/2 bohr) on the PIP-NN PES. The unit of the energy is kcal/mol. (b) The 11 generalized normal mode frequencies (103 cm−1) along s. (c) The reaction path curvature κ (amu−1/2 bohr−1) along s.

Close modal

Figure 5 plots the calculated rate coefficients by the CV T/μOMT and QCT methods on the present PIP-NN PES as a function of 1000/T (T, temperature, 200–3000 K) along with experimental7,9 and theoretical results.27,69 Note that we did not consider the issue of the zero point energy leaking in QCT. One can see that at the temperature above 500 K, all the rate constants are consistent with each other, while for low temperatures, the present CV T/μOMT rate constants are slightly larger than the experimental results by Sutherland et al.7 and the recommended ones by Baulch et al.9 The rate constants calculated on the revised ZFWCZ-WM PES by the full-dimension (12-D) MCTDH approach69 and on the XCZ PES or on the scaled XCZ PES by an eight-dimension (8D) QD method27 agree well with the experiment although only in the temperature range of 300-600 K.

FIG. 5.

Rate constants k of the abstraction reaction H + CH4 → H2 + CH3 as a function of 1000/T. The available experimental results by Sutherland et al.7 and the recommended ones by Baulch et al.9 are compared. The 12-D MCTDH on the revised ZFWCZ-WM PES by Welsch and Manthe69 and 8D QD on the XCZ PES or the scaled XCZ PES by Zhou and Zhang27 are also included.

FIG. 5.

Rate constants k of the abstraction reaction H + CH4 → H2 + CH3 as a function of 1000/T. The available experimental results by Sutherland et al.7 and the recommended ones by Baulch et al.9 are compared. The 12-D MCTDH on the revised ZFWCZ-WM PES by Welsch and Manthe69 and 8D QD on the XCZ PES or the scaled XCZ PES by Zhou and Zhang27 are also included.

Close modal

The reaction probabilities of H + CH4 (v = 0, j = 0) at Jtot = 0 are calculated on the present PIP-NN PES and the smoothed XCZ PES using the 7D TDWP method at collision energies up to 2.5 eV. It is noted that the exchange reaction channel opens at 1.4 eV (the ZPE corrected barrier for the exchange channel is 1.59 eV within the harmonic approximation) and makes up a large proportion at 2.5 eV. Because the two channels are difficult to separate with the present TDWP basis set functions, only the total reaction probabilities of both channels are presented in Figure 6, and details of the complicated channels and mechanisms will be discussed elsewhere. The comparison of three individual fittings composing the present PIP-NN PES is shown in Figure 6(a). The largest relative deviation of the three probabilities is only 2% located at 1.0 eV. The excellent agreement between the three fittings indicates that the fitting error is small enough, and no obvious over-fitting occurs. Figure 6(b) compares the total reaction probabilities on the current PIP-NN PES and the smoothed XCZ PES. Despite the fact that more than 15 000 ab initio points were added in the present fittings, the two results differ only 1.7% at 1.0 eV and 1.2% at 2.5 eV. As a result, this PIP-NN PES can be considered fully converged with regard to the fitting process as well as number of ab initio geometries and is reliable for the reactions up to a collision energy of 2.5 eV.

FIG. 6.

Total reaction probabilities for the H + CH4 → H2 + CH3/H′ + CH4 channels: (a) results from the three individual fittings of the present PIP-NN PES. (b) Results from the XCZ and the present PIP-NN PES.

FIG. 6.

Total reaction probabilities for the H + CH4 → H2 + CH3/H′ + CH4 channels: (a) results from the three individual fittings of the present PIP-NN PES. (b) Results from the XCZ and the present PIP-NN PES.

Close modal

In this work, we present a new, accurate, and efficient global PES for the ground electronic state of H + CH4 system. Both abstraction and exchange reaction channels are included using the PIP-NN fitting approach based on ∼63 000 high-level ab initio points. The total RMSE is only 5.1 meV, suggesting a faithful representation of the ab initio points. As a result, this PES provides a reliable platform on which the reaction dynamics can be determined. Indeed, the calculated rate constants by both CV T/μOMT and QCT methods agree well with experiment and the available literature theoretical data for the abstraction reaction. It is also shown that the rigorous adaptation of the permutation symmetry and the additional 15 000 points have a small impact on the total reaction probabilities computed by the 7D wave packet quantum dynamic approach. Finally, it is noted that some small corrections, such as the core correlation and the relativistic effect, are not taken into account in the UCCSD(T)-F12a/AVTZ method; therefore, an error of about 90-180 cm−1 is expected on the relative energies, as discussed by Czakó.91 These effects should be considered if really high accuracy is required.

This work was supported by the Hundred-Talent Foundation of Chongqing University (Project No. 0220001104420 to J.L.), National Natural Science Foundation of China (Grant Nos. 21133006, 21403104, and 91221301 to D.X.), the Ministry of Science and Technology (Grant No. 2013CB834601 to D.X. and D.H.Z.), the National Natural Science Foundation of China (Grant No. 91221301 to D.H.Z.), and the U.S. Department of Energy (Grant No. DE-FG02-05ER15694 to H.G.).

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