Submitted to J. Chem. Phys., 11/7/2015, revised, 11/28/2015 Communication: An accurate full fifteen dimensional permutationally invariant potential energy surface for the OH + CH 4 H 2 O + CH 3 reaction Jun Li, 1,* and Hua Guo 2,* 1 School of Chemistry and Chemical Engineering, Chongqing University, Chongqing 400044, China 2 Department of Chemistry and Chemical Biology, University of New Mexico, Albuquerque, New Mexico 87131, United States *: corresponding authors, emails: jli15@cqu.edu.edu (JL), hguo@unm.edu (HG) 1
Abstract A globally accurate full-dimensional potential energy surface (PES) for the OH + CH 4 H 2 O + CH 3 reaction is developed using the permutation invariant polynomial-neural network (PIP-NN) approach based on ~135,000 points at the level of UCCSD(T)-F12a/AVTZ. The total root mean square fitting error is only 3.9 mev or 0.09 kcal/mol. This PES is shown to reproduce energies, geometries, and harmonic frequencies of stationary points along the reaction path. Kinetic and dynamical calculations on the PES indicated a good agreement with available experimental data. 2
An important pre-requisite for studying dynamics of chemical reactions is an accurate representation of the Born-Oppenheimer potential energy surfaces (PESs). Despite the tremendous advances in electronic structure theory, the analytical representation of multi-dimensional global PESs based on large number of ab initio points has only recently become a reality for reactive systems beyond three atoms. 1-3 Indeed, several analytical functional forms with the necessary permutation invariance have been proposed to represent the complex topography of the PES. The two main forms are the permutation invariant polynomials (PIP) 2 and neural network (NN) methods. 4, 5 Our recent work combining the two approaches led to the so-called PIP-NN method, 6, 7 which has been demonstrated to be both accurate and efficient in representing global PESs for reactive systems up to six atoms. 8-11 In this Communication, we report a globally accurate PIP-NN PES for a seven-atom reactive system based on ~135,000 high-level ab initio points, which spans unprecedented fifteen degrees of freedom. The hydrogen abstraction reaction involving the simplest hydrocarbon, methane, by the hydroxyl radical, i.e., OH + CH 4 H 2 O + CH 3 (ΔH = -13.4 kcal/mol), is of great importance in atmospheric chemistry because of its role in removing the greenhouse gas methane, which in turn provides a main sink for atmospheric OH radicals. This reaction is also relevant in combustion at high temperatures. For these reasons, its thermal rate coefficients have been measured using different techniques over a wide range (178-2025 K) of temperatures (see Ref. 12 and references therein). In addition, the state-to-state dynamics of the OH/OD + CH 4 /CD 4 /CHD 3 reactions has been investigated experimentally by Liu and co-workers. 13-15 In addition, the vibrational spectroscopy and decay dynamics of the CH 4 OH 3
16, 17 complex in the entrance channel have been studied by Lester and co-workers. Much theoretical work has also been devoted to the reaction barrier and rate coefficients at various levels of theory for this prototypical bimolecular reaction with seven atoms. 18-24 However, a complete understanding of the reaction dynamics requires a full-dimensional global PES. The first full-dimensional PES for the title reaction was constructed by Espinosa-Garcia and co-workers using a molecular mechanics functional form based on both theoretical and experimental information. 25 Several kinetics and dynamics calculations have since been performed on this so-called PES-2000, 25-30 and the agreement with available experimental data has generally been good. A refined PES, PES-2014, was later constructed by fitting exclusively to ab initio data, using essentially the same approach. 31 Additional studies have been performed on the new PES to provide insight into the kinetics and dynamics of this reaction. 31-33 Despite their immense value as the first full-dimensional global PESs, their limitations are also apparent. Due to the simple analytical forms and the limited number of ab initio points used to determine the small set of parameters, these PESs are not expected to be accurate outside the region near the minimum energy path (MEP). 31 Even for configurations along the MEP, it has been pointed out that the saddle points in these PESs are significantly different from that determined at a higher level of theory. 30 In addition, only the permutation symmetry of the four hydrogens in CH 4 was taken into account. To better understand the dynamics of this important reaction, a PES that is uniformly accurate in all relevant configurations is highly desired. To this end, we report a new globally accurate PES based on 135,000 ab initio points at the explicitly correlated coupled cluster singles, doubles, and perturbative triples level with the augmented correlation consistent 4
polarized valence triple-zeta basis set (UCCSD(T)-F12a/AVTZ). 34, 35 These points were fit using the PIP-NN approach, 6, 7 with a total root-mean square error (RMSE) of only 3.9 mev. The construction of the PES consists of two steps: sampling of the data points in the relevant configuration space and fitting them with an analytic function. To this end, a small set of data points around the vicinity of the MEP was first selected by running direct dynamics trajectories at a low theory level. Then, UCCSD(T)-F12a/AVTZ calculations were performed at these selected points, and a primitive PES was constructed. Trajectories with various initial conditions were then dispatched on this primitive PES to further explore the configuration space and to generate new points. Points were added to the data set if they are not too close to those already in the existing data set and a new PES is subsequently generated. This procedure was repeated until convergence. Finally, a total of ~135,000 points were calculated and fit using the PIP-NN method. 6, 7 They correspond to 1.62 10 6 points because of the hydrogen (5!=120 fold) permutation symmetries. The permutation symmetry of the system can be enforced by symmetry functions as the input layer of the NN and the permutation symmetry of all five hydrogens is taken into account. In PIP-NN, the PIPs are symmetrized monomials of Morse-like variables of the internuclear distances ( p exp( r ) with =1.0 Å -1 ): 36 ij ij 7 PIP( ) ˆ lij r S p (1) ij i j ij where Ŝ is the symmetrization operator and l the order. The PIP-NN PES can be formally expressed as V f (PIP( r )) (2) NN ij All 1331 PIPs up to the fifth order were used in the input layer for this A 5 BC type molecule 5
and the NN architecture was selected to be of 4 and 100 neurons for the two hidden layers, resulting in 5929 parameters. To avoid overfitting, the data are randomly divided into the training (90%), validating (5%) and testing (5%) sets. The target function is defined as the RMSE of the PES points from their ab initio counterparts. Further details of the PIP-NN fitting can be found in our recent publications. 8-11 The final PES is the average of three best PIP-NN fits, which have RMSEs for the training/validating/testing sets and maximum deviation of 4.8/6.4/5.6/187.4, 4.2/5.0/6.8/225.7, and 4.4/6.1/6.3/212.1 mev, respectively. The RMSE of the final PES is 3.9 mev with a maximum deviation 148.3 mev. The PES can be obtained from the authors upon request. Figure 1 presents the contour plot of the PIP-NN PES along the two reactive coordinates, r OH and r CH, with other internal coordinates fixed at saddle point. It is clearly seen that this reaction has a reactant-like barrier, much earlier than that on the PES-2014.{Espinosa-Garcia, 2015 #9133} The reaction energy of -13.19 kcal/mol is in reasonably good agreement with the latest theoretical values of -13.5, 20 and -13.3 kcal/mol. 31 In Figure 2, geometric parameters of the stationary points along MEP, namely the reactants (OH + CH 4 ), transition state (TS), products (H 2 O + CH 3 ), as well as van der Waals complexes in the entrance (R-vdW) and exit (P-vdW) channels are displayed. The general agreement with previous theoretical results 18, 21-23, 30 is quite good except for TS and R-vdW (vide infra). The energies and geometries of the stationary points are all reproduced well by the PIP-NN PES, as illustrated in Figure 2. The differences in energy are less than 0.02 kcal/mol. The geometries of the reactants, products, and TS agree well with the ab initio ones, and the deviations of the bond lengths and angles are less than 0.003 Å and 0.4º, respectively. The agreement for the 6
geometries of R- and P-vdW is not as good because of the floppy nature of these complexes. Similarly, the harmonic frequencies, which are collected in Supplementary Information (SI), 37 of the rigid modes are reproduced better than those of the floppy ones. In early work, both eclipsed and staggered geometries were found for TS at relative low levels of theory. 19 Later, the staggered conformation was identified as a second-order saddle point, 21 and the eclipsed conformation was thus considered as the TS, confirmed subsequently at the MCG3/3 23 and MP2/6-31G(d,p) levels. 31 However, a staggered and slightly non-linear TS is found at the UCCSD(T)-F12a/AVTZ level, as shown in Figure 2. Our TS geometry is quite different from that in PES-2014: which overestimates the breaking C-H bond by ~0.1 Å and the forming O-H bond by ~0.04 Å. The classical barrier of 6.29 kcal/mol obtained here can be compared to the literature values of 6.7, 20 6.31, 22 6.1-6.5, 24 and 6.4 kcal/mol. 31 The van der Waals complexes in the entrance and exit channels are not present in PES-2000, but included in PES-2014. At the level of UCCSD(T)-F12a/AVTZ, we did not find the o.4 kcal/mol deep H 3 CH OH well present in PES-2014, 31 but identified a C 3v H 4 C HO well (Figure 2), which is 1.22 kcal/mol lower than the reactant asymptote. This is consistent with the recent theoretical work. 16, 38 The P-vdW complex, on the other hand, was found in an H 3 C HOH configuration, stabilized by 1.84 kcal/mol with respect to the product asymptote, consistent with previous theoretical reports. 21, 31 It is interesting that some small imaginary frequencies persist and are hard to eliminate at the ab initio level, as shown in SI, due apparently to the floppy nature of the system. The thermal rate coefficients of the title reaction were computed on the PIP-NN PES 7
with the canonical variational transition-state theory (CVTST) 39 using POLYRATE. 40 Motions orthogonal to the reaction path were treated using quantum mechanical vibrational partition functions under the harmonic approximation, except for the torsional mode of the OH group with respect to CH 3 group. The hindered rotor model was used to account for the anharmonicity in that mode. 23 Quantum effects in the reaction coordinate were included by using the micro-canonical optimized multidimensional tunneling (μomt) approach, 39 in which, at each total energy, the larger of the small-curvature (SCT) and large-curvature (LCT) tunneling probabilities was taken as the best estimate. The rotational partition functions were calculated classically. The symmetry factor of 12 and the OH spin-orbit splitting (140.0 cm -1 ), were taken into account in the reactant electronic partition function. As shown in Figure 3(a), the calculated rate coefficients on the current PIP-NN PES and PES-2014{Espinosa-Garcia, 2015 #9133} are both in good agreement with experimental data, 12 reproducing the tunneling facilitated non-arrhenius feature at low temperatures. As discussed by Ellingson et al.,{ellingson, 2007 #7618} the treatment of the torsional problem and the coordinate system have a significant effect on the kinetics, and they remain open questions. We plan to perform a systematic study on the kinetics of the title reaction, including the kinetic isotope effects, in the near future. The kinetics of the reaction is mostly sensitive to the TS region of the PES. To further test the PIP-NN PES in a wider range of configurations, quasi-classical trajectory calculations were performed using VENUS 41 for the experimentally studied OH + CD 4 HOD + CD 3 reaction at collision energies ranging from 4 to 20 kcal/mol. At each collision energy, 10 5 or 2 10 5 trajectories were calculated starting from the ro-vibrational ground states of OH and 8
CD 4, and the maximal impact parameter (b max ) was determined using small batches of trajectories with trial values. The trajectories were initiated with a reactant separation of 8.0 Å and terminated when products or reactants reached a separation of 8.0 Å. Other scattering parameters including the spatial orientation of the initial reactants, vibrational phases, and impact parameter were determined according to the Monte Carlo approach as implemented in VENUS. 41 The gradient of the PES is obtained numerically by a central-difference algorithm. The propagation time step was 0.05 fs. Almost all trajectories conserved energy to within a chosen criteria (10-4 kcal/mol), which testifies the smoothness of the PES. Figure 3(b) presents the reaction cross sections, compared to the experiment 14 and recent calculations. 31 Since only the relative cross sections were reported experimentally, the theoretical QCT values on the PIP-NN PES and PES-2014 are both normalized at the collision energy of 16 kcal/mol. Overall, they agree well with each other, and the PIP-NN PES appears slightly better than PES-2014 in reproducing the experiment. To summarize, we have developed a full fifteen-dimensional highly accurate global PES for the OH + CH 4 H 2 O + CH 3 reaction based on ~135,000 ab initio points calculated at the level of UCCSD(T)-F12a/AVTZ. Full permutation symmetry is enforced, which not only reduces computational costs, but also provides correct description of the PES, particularly at high symmetry regions. The preliminary kinetic and dynamical calculations on this PES are very promising. It is our expectation that this PES will provide a reliable platform for dynamical studies of this important bimolecular reaction in the future. Acknowledgments: This work was supported by the Hundred-Talent Foundation of Chongqing University (project no. 0220001104420 to JL) and National Natural Science 9
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Figures Captions Figure 1. Contour plot for the OH + CH 4 H 2 O + CH 3 reaction along the two reactive coordinates, r OH and r CH, (in Å) with other internal coordinates fixed at the transition state. The energies are in kcal/mol relative to the reactant asymptote. The comparison of the minimum energy path on the PIP-NN PES and by the ab initio calculation is also shown in the inset. Figure 2. Schematic illustration of the reaction pathway for the reaction OH + CH 4 H 2 O + CH 3. All energies are in kcal/mol relative to the OH + CH 4 asymptote. The geometries (distances in Å and angels in deg.) of the stationary points are also shown. The first value (in black) is obtained on the present PIP-NN PES and the second (in red) is determined at the level of UCCSD(T)-F12a/AVTZ. The third entry in blue corresponds to those on PES-2014. 31 The geometries of TS on PES-2014 are also included (the third entry in blue) for comparison. The numbers are not meant to indicate the significant figures, but are used here to compare with other theoretical values. Figure 3. (a) Comparison of thermal rate coefficients for the title reaction OH + CH 4 H 2 O + CH 3. (b) Relative reaction cross section as a function of the collision energy (E c ) for the reaction OH + CD 4 HOD + CD 3. The experimental values (only the ground state CD 3 was considered) are adopted from Ref. 14, and the QCT results on the PES-2014 are adopted from Ref. 31. 12
Fig. 1 13
Fig. 2 14
Fig. 3 15