Emory University] Cherry L. Emerson Center for Scientific Computation and
Department of Chemistry, Emory University, Atlanta, Georgia 30322, USA

# A Machine Learning Approach for Rate Constants III: Application to the
Cl(2P)+CH4$\rightarrow$CH3+HCl Reaction

Paul L. Houston  Department of Chemistry and Chemical Biology, Cornell
University, Ithaca, New York 14853, U.S.A. and Department of Chemistry and
Biochemistry, Georgia Institute of Technology, Atlanta, Georgia 30332, U.S.A
<EMAIL_ADDRESS>Apurba Nandi  Joel M. Bowman<EMAIL_ADDRESS>[

###### Abstract

The temperature dependence of the thermal rate constant for the reaction
Cl(3P) + CH4 $\rightarrow$ HCl + CH3 is calculated using a Gaussian Process
machine learning (ML) approach to train on and predict thermal rate constants
over a large temperature range. Following procedures developed in two previous
reports, we use a training dataset of approximately 40 reaction/potential
surface combinations, each of which is used to calculate the corresponding
data base of rate constant at approximately eight temperatures. For the
current application, we train on the entire dataset and then predict the
temperature dependence of the title reaction employing a “split” dataset for
correction at low and high temperatures to capture both tunneling and
recrossing. The results are an improvement on recent RPMD calculations
compared to accurate quantum ones, using the same high-level ab initio
potential energy surface. Both tunneling at low temperatures and recrossing at
high temperatures are observed to influence the rate constants. Recrossing
effects, which are not described by TST and even sophisticated tunneling
corrections, do appear in experiment at temperatures above around 600 K. The
ML results describe these effects and in fact merge at 600 K with RPMD results
(which can describe recrossing), and both are close to experiment at the
highest experimental temperatures.

###### keywords:

Machine learning, thermal rate constants, Eckart tunneling, recrossing

##  1 Introduction

Svante Arrhenius nearly turned away from chemistry twice during his early
studies because of poor performance, but he went on to earn a Nobel Prize in
the field for his work on electrolytic solutions. Of more relevance to the
current topic is his paper in 1889 on the temperature dependence of reaction
rates,1 where he found that $k(T)=Ae^{-E_{a}/kT}$ where $A$ is a constant, $T$
is the Kelvin temperature, and $E_{a}$ is an activation energy. A major
subsequent advance came from the development in 1935 by Henry Eyring, and by
Merideth Evans and Michael Polyani of transition state theory (TST),2, 3, 4
which associated the activation energy with the location of a dividing surface
that separated the reactants from the products in the space of energy vs.
configuration. For reactions that have a saddle point between reactants and
products, the dividing surface typically includes the saddle point. The
constant $A$ is associated in TST with a ratio of partition functions.

The near-linear relationship between the log of the rate constant with $(1/T)$
predicted by both the Arrhenius equation and TST is actually not observed for
many reactions, particularly at low temperatures. This is because quantum
mechanical tunneling is found to be important at low temperatures and results
in an increase in the rate constant from the linear prediction. Early
proposals for including tunneling effects in the theory of chemical reactions
were based on one-dimensional potentials. Of particular relevance to what
follows in this paper is Eckart’s early work,5 which continues to be used,6,
7, 8 because it gives an analytical and often accurate result.

However, to get agreement with exact quantum methods or with experiment, we
need more sophisticated quantum mechanical (QM) approaches, even though they
are more computationally demanding. These methods have mostly been applied to
collinear reactions, as summarized in a useful compilation by Allison and
Truhlar.9 Three-dimensional calculations, notably for the H + H2 reaction,10
have also been studied, and these demonstrate shortcomings of the one-
dimensional Eckart model that include, for example, recrossing of the TST
dividing surface, corner cutting tunneling paths, and vibrationally adiabatic
effects. To incorporate these effects, a variety of QM approaches have been
employed, including instanton methods,11, 12 VPT2-based semiclassical
transition state theory,13, 14, 15, 8, 16, 17 reduced-dimensionality quantum
methods,18, 19, 20, 21, 22 a large-curvature, corner-cutting reaction path,23,
9 ring polymer molecular dynamics (RPMD),24, 25 and ring polymer instanton
methods.26 Exact direct methods have also been developed to obtain the rate
constant27, for example using multi-configuration time-dependent Hartree
(MCTDH) theory, at least for zero total angular momentum,28, 29 and then
augmented by $J$-shifting20.

Until recently, none of these methods had been applied to the important
Cl(${}^{3}P_{3/2}$) + CH4 $\rightarrow$ HCl + CH3 reaction, a reaction
important in assessing the effect of chlorofluorocarbons in stratospheric
chemistry. Many experimental measurements are available, mostly in very good
agreement with one another. Collectively these measurements span a large
temperature range from roughly 180 to more than 1000 K. We will refer to these
by the names of the first authors of the reports: Clyne,30 Manning,31
Michael,32 Whytock,33 Keyser,34 Zahniser,35 Ravishankara,36 Heneghan,37
Seeley,38 Pilgrim,39 and Bryukov.40. High-level, CCSD(T)-based potential
energy surfaces for this reaction have been reported by Czakó and Bowman
(CB),41 and, quite recently, by Li and Liu (LL).42 Substantial theoretical
investigations have also been reported. Barker, Nguyen, and Stanton43 have
reported kinetic isotope effects in the reaction using ab initio semiclassical
transition state theory, while Georgievskii and Klippenstein (GK)44 have
recently studied how tunneling is affected by the non-separability of the
reaction coordinate from other modes. The approach of the latter report
provides a method that substantially improves on the one-dimensional
estimation of tunneling; the comparison with experiment in that paper was
limited to temperatures below 300 K, where agreement is very good. We return
to this recent work in the Discussion section below. Li and Liu42 used their
potential to perform RPMD calculations for estimation of the temperature
dependent rate constant, though over a limited range and with only modest
agreement with experiment. Subsequently, a “first principles” theory for the
reaction was recently reported by Hoppe and Manthe (HM),45 who used the LL
potential in a fully QM method to calculate the rate constant as a function of
temperature up to 540 K. Going to higher temperatures was evidently not
computationally feasible. Their results are essentially in exact agreement
with the experimental data up to that temperature. Thus, from the perspective
of theory, this reaction has acquired an especially important role for both
the potential energy surface and the reaction dynamics/rate constant theory.

One aspect of this reaction that was not directly addressed by these
theoretical approaches is the extent of recrossing at high temperature. Recall
that recrossing refers to the reduction of the reaction flux predictions of
TST, which, by assumption, ignores recrossing. Recrossing is due to
multidimensional dynamical effects where flux that crosses the transition
state dividing surface recrosses but back in the direction of reactants. We
return to this effect here where we apply our Machine Learning (ML) approach
to obtain bimolecular thermal rate constants over a large temperature
range.46, 47 The motivation for this approach was given in detail in those
papers and so is just briefly reviewed below. Other ML methods have
subsequently been reported for reaction rates by Valleau and co-workers48, 49,
activation energies in an article by Lewis-Atwell, Townsend and Grayson,50 and
for activation energies and reaction mechanisms in publications by Green and
colleagues.51, 52

In our first paper46 we proposed using ML to find a correction to the Eckart
tunneling correction to TST. The great appeal of the Eckart correction is two-
fold. First, it is often quite accurate and second it requires no more
information than is needed for a standard TST calculation. Of course this is
not an exact correction and more accurate methods (though short of “exact”)
are available, as mentioned above. However, these methods require
substantially more information about the potential energy surface (PES),
ranging from an expansion of the PES around the saddle-point beyond the
harmonic expansion needed in TST to a global PES needed in RPMD and the MCTDH
quantum calculations. Our goal was to explore whether ML can be applied to
correct the Eckart correction and achieve a substantial improvement in
accuracy at virtually no additional cost and with no additional information
needed about the PES. The ML approach we took followed the standard protocol
of training the desired quantity on a known database. For us the quantity is
the correction to the Eckart tunneling correction and the database is a large
set of (ideally) exact quantum rate constant calculations for a variety of
chemical reactions. We discuss below the database we used. For the ML model we
used Gaussian process (GP) regression, which we briefly review in the next
section. We used the database of rate constants for 13 reaction/potential
surface combinations at typically 8 different temperatures. We created this
database by data mining the large compilation of tables of rate constants
reported by Allison and Truhlar in 1998. The tables contained TST and exact
quantum results for numerous collinear reactions, of which we selected a total
of 52. Also, all parameters needed for an Eckart tunneling correction were
given. From the TST, Eckart correction and exact results the correction to
Eckart were calculated. Then GP was trained on the 52 reactions and testing
was then done on a set of 39 reaction/potential surface combinations. The GP
method, when averaged over all test reactions, was within 80% of the accurate
answer, considerably better than TST (330%) and Eckart corrected TST (ECK)
(110%). The test reactions included 5 one-dimensional symmetric A + BA
reaction/surface combinations; 16 one-dimensional asymmetric A + BC
reaction/surface combinations; 15 three-dimensional reaction/surface
combinations, both symmetric and asymmetric; and three polyatomic reactions:
O(${}^{3}P$) + CH4, H + CH4, and HH + OH. A challenge for future predictions
is that the dataset is relatively small. A strategy, then, is to examine other
reactions that are studied by accurate methods, to determine to what extent GP
is effective in predicting the new rate constant, and then to add the new
reaction to the dataset. In our second paper47 we focused on the O(${}^{3}P$)
+ HCl reaction on the 3A′ and 3A′′ potential energy surfaces. In that paper we
took a different approach. We trained on the entire original dataset and the
predicted the rate constant of this reaction. As described in detail in that
paper we split the entire data set into two datasets, as briefly reviewed in
the next section in given in more detail in Supplementary material (SM). Here
we simply note that at low temperatures the corrections to the Eckart
tunneling correction are larger than one. This is expected, as the Eckart
correction generally underestimates tunneling, especially in the deep
tunneling region. At high temperatures where there is virtually no Eckart
correction, i.e., the correction is about unity, the correction to Eckart is
less than one. This accounts for recrossing and where in general TST
overestimates the rate constant.

In the next section we briefly summarize the computational approach we use. We
then present new predictions of the rate constant for the Cl(2P) + CH4
reaction. This reaction is a challenge for our ML approach both because its
parameters are at the edge of those of the training set and because it brings
up possible issues, such as spin-orbit coupling and Van der Waals wells in
both the entrance and exit channels. Detailed comparisons are made with the GP
results of this work and the results from GK44 as well as with the QM results
from HM45 and the RPMD results from LL.42 We then incorporate “learning” this
reaction into the correction datasets and verify that when used with ML we do
get good agreement with the experiment, as expected.

##  2 Computational Approach

The specific approach we proposed46 is summarized next. Let $k(T)$ be an exact
rate constant calculated over a temperature range and using a given PES, and
let $k^{TST}(T)$ be the conventional TST rate constant using the same PES. The
approach we took earlier is to represent the exact rate constant by the
equation

$k(T)=[\kappa_{ECK}(T)k^{TST}(T)]\chi(T),$ (1)

where $\kappa_{ECK}(T)$ is the Eckart transmission coefficient.6, 7, 53 and
$\chi(T)$ is the correction to the Eckart correction to produce agreement with
the accurate rate constant. We reasoned that this approach is better than
using ML on the exact $k^{TST}(T)$ directly because $\kappa_{ECK}(T)$ is both
easy to obtain and generally gives realistic tunneling corrections. Thus, the
hope is that the GP correction to the Eckart transmission coefficient is small
and easy to “learn”. To proceed with machine learning of $\chi(T)$ we need the
usual three elements, namely a database of exact rate constants, a set of
descriptors, and a general machine learning method to train on the database.
We described these in detail previously46, so we just briefly summarize these
key elements here.

Critical for the success of the ML approach are the parameters that
characterize a reaction. A starting choice includes the three parameters that
are needed to obtain the Eckart transmission coefficient,6 which, in reduced
units, are

$\displaystyle\alpha_{1}=V_{1}/\omega_{im},$ (2)
$\displaystyle\alpha_{2}=V_{2}/\omega_{im},$ (3) $\displaystyle
u^{*}(T)=\omega_{im}/(0.69307\hskip 2.84544ptT).$ (4)

$V_{1}$ ($V_{2}$) is the saddle point barrier height in cm-1 relative to the
reactants (products), omitting the zero-point energy in both cases, and where
the energy of the reactants is zero. In the case of the Cl(2P) + CH4 reaction,
the zero of energy is lowered by the spin-orbit stabilization of the Cl(2P3/2)
atom, about 0.84 kcal/mol. Using transition state (TS) value of 6.97 kcal/mol
from the CCSD(T)/CBS value in Table 3 of GK, this gives a value of $V_{1}$ =
(6.97 - (-0.84)) kcal/mol = 2733 cm-1. We, as well as GK and HM, assume no
spin-orbit splitting other than the reactants. Thus, again from the
CCSD(T)/CBS value in Table 3 of GK, $V_{2}$ = (6.97-5.04) kcal/mol = 675.3
cm-1. Note that in both the entrance and exit channels we have assumed that
the presence of Van der Waals wells does not affect the $V_{1}$ and $V_{2}$
parameters used for the Eckart correction. In short, we ignore these wells. We
defer comments about the possible effect of these wells to the Discussion
section. The value $\omega_{im}$ is the magnitude of saddle point imaginary
frequency (in cm-1), which, from Table 2 of GK is 1194 cm-1. The value 0.69307
is an energy conversion factor, equal to $k_{B}/(hc)$, and $T$ is the
temperature in Kelvin. Improvements were obtained by including an additional
parameter, the skew angle, $\beta$, which identifies important mass and corner
cutting and recrossing effects. In the case of the Cl(2P) + CH4 reaction, this
value is $\beta=17.3\textdegree=0.302$ radians. More details, particularly the
values of these parameters for the training dataset, can be found in our
recent paper.46.

As noted previously,47 it is a challenge to obtain high precision in fitting
the entire $\chi$ dataset of approximately 400 $\chi$ values because they have
a very large range, from 0.2 to 25. As in the O(3P) + HCl paper,47 we thus
decided to split the data into two clusters in a manner that would permit more
precise GP training on each. The dividing line between the two clusters is at
$\chi$ equal to 3.0. The cluster with $\chi$ greater than 3.0 is denoted the
Large-$\chi$ set and the other cluster is denoted Small-$\chi$ set. Figure S1.
in the Supplementary Material (SM) shows these $\chi$ groups as a function of
$u^{*}$.

Having now a set of descriptors, $\alpha_{1}$, $\alpha_{2}$, $\omega_{im}$,
and $\beta$, as well as a dataset from ref. 46, we now turn to the learning
method. Gaussian Process (GP) regression is a machine learning method whose
goal is to produce a smooth interpolation between known data.54 The working
equation is given by

$\chi(\bm{x})=\textbf{K}_{x}^{\top}\textbf{K}^{-1}\bm{\chi}_{0},$ (5)

where $\bm{x}$ represents the set of descriptors. The known values of $\chi$
are collected in the column vector denoted $\bm{\chi}_{0}$.
$K(\bm{x}_{i},\bm{x}_{j})$ is the kernel matrix with elements at the database
values of $\bm{x}$ and where $\textbf{K}_{x}=[k(\bm{x},\bm{x}_{1})\ \cdots\
k(\bm{x},\textbf{x}_{N})]^{\top}$. In this expression $\bm{x}$ is the value of
the descriptors where $\chi$ is to be evaluated. A popular choice for the
kernel matrix is54, 55, 56

$K(\bm{x}_{i},\bm{x}_{j})=\sigma^{2}\exp\left(-\frac{d_{ij}^{2}}{2}\right)+\delta_{ij}\sigma_{noise},$
(6)

where $d_{ij}$ is the distance between the two vectors $\bm{x}_{i}/l$ and
$\bm{x}_{j}/l$, where the hyper-parameters are $l$ and $\sigma$. The length-
scale parameter, $l$ can be single length or one that depends on the
descriptor; we use the latter. $\delta_{ij}\sigma_{noise}$ is the noise term
that is added to the diagonal of the covariance matrix. In principle, this
term is not necessary for fitting, because the data are not noisy. However,
adding noise can avoid ill-conditioning of the matrix, and, more generally
enters parametrically into the optimization of the hyper-parameters according
to maximization of log-marginal-likelihood .57, 54

$\log\mathcal{L}=-\frac{1}{2}\left(\textbf{g}^{\top}\textbf{K}^{-1}\textbf{g}+\log|\textbf{K}|+N\log
2\pi\right)$ (7)

Once the optimal hyper-parameters are determined, the GP model can predict the
value of $\chi$ using Eq. 5. In a practical sense, the hyper-parameters govern
the smoothness of the interpolation, also known as prediction.

We performed this GP regression on the Small and Large-$\chi$ datasets using
the routines contained in the Python Scikitlearn library, which includes
optimization of the hyperparameters.58 The inputs are the descriptors and the
output is the trained $\chi$. As a reminder, the database for exact $\chi$
values is developed largely from a 1998 compilation of rate constants9 as well
as from results from several more recent quantum calculations of rates
constants for polyatomic reactions, such as OH + H2, H + CH4, and O + CH4, as
described in detail previously.46

The Small-$\chi$ data-set has a total 360 data points, and for the work we
described below we took all the points to fit the data-set using GP
regression. The RMS error of this fitting is 0.14461. The Large-$\chi$ data-
set has total 37 data points, which for ML is very small. In order to avoid
over-fitting of this limited set, we used a noise value (see Eq. 6) of 50,
larger than that used for the previously studied O(${}^{3}P$) + HCl reaction.
Noise values of 20 and 30 gave nearly identical results, as shown in Figure
S4. of the SM.

##  3 The Cl(2P_3/2) + CH4 $\rightarrow$ HCl + CH3 Reaction

The Cl(2P_3/2) + CH4 reaction is a challenge for our ML approach, given the
range and sparsity of our relatively small dataset. Recall that the dataset is
largely based on collinear A+BC reactions, with only a few three-dimensional
reactions. In particular, for this reaction, using the most accurate
electronic energies, $\alpha_{1}=2.61$ and $\alpha_{2}=0.645$ so that
$\alpha_{1}-\alpha_{2}=1.97$. Our complete dataset does have values of
$\alpha_{1}$ greater than and less than this value of $\alpha_{1}$; however,
the smallest value of $\alpha_{2}$ in our dataset is 0.70. And in that case
there are only six total entries with this value of $\alpha_{2}$. Fortunately,
for these entries $\alpha_{1}$ = 2.94 and the skew angle is 20 deg, which is
close to skew angle, $\beta$, of 17.3 deg. Our data set for skew angle ranges
from 17 to 89 deg. The value of $\beta$ for the Cl(2P_3/2) + CH4 $\rightarrow$
HCl + CH3 reaction is 17.3°, near the lower limit of the dataset. So the
values of the descriptors for this reaction are at the edges of the training
dataset. Thus, it is interesting to see if a strategy such as that used
successfully in the O(3P) + HCl reaction will work for Cl(2P_3/2) + CH4.

Then too, there is the question of when to use the results of the GP
calculation on the Large-$\chi$ set and when to use those of the Small-$\chi$
dataset. Only at the lowest temperatures are the $u^{*}$ values only found in
the former, so one must make a transition between using the two sets that does
not result in a discontinuity in the slope of the log${}_{10}(k(T))$ vs
$1000/T$ plot. Here, we use a transition between the two sets based on the
probability that a given $u^{*}$ will be associated with one set rather than
the other, as described in Section S-I of the SM along with some details of
the GP calculations using each dataset. We used a different strategy in paper
2 where a weighted average of predicted $\chi$ values was used for several
temperatures. 47

Table 1 provides a summary of the GP-predicted results along with TST and
Eckart corrected TST and experiment. The TST result is calculated either by
direct count of the rotational levels and degeneracies or by using classical
formulae, with practically identical results for the temperatures listed. Our
TST calculations are in agreement with those of HM and GK, following
discussions with HM, as described in Section S-IV of the SM. The “accurate”
value of the rate constant is taken as the result of a third-order polynomial
fit to the experimental log${}_{10}(k(t))$ vs $1000/T$ plot, shown in Fig. S1
in Section S-II. As mentioned, the Eckart correction $\kappa$ predicts only a
factor of ca. 2 correction from the TST rate constant at the lowest
temperatures and is otherwise close to unity (i.e., no correction). The GP
calculation gives a further factor of ca. 5.9 correction at the lowest
temperature, but falls somewhat short of the GP target factor, 10.2, that
would be needed to provide agreement with experiment, i.e., “accurate”.

Table 1: Results for GP Calculation T | u∗ | TST | Eckart | GP | $k_{GP}$ | GP | $k_{expt.}$
---|---|---|---|---|---|---|---
(K) |  | (this work) | Corrctn. | Corrctn. |  | Target |
|  |  | $\kappa$ | $\chi$ |  |  |
150 | 10.00 | 6.89$\times$ 10-17 | 1.98 | 5.93 | 8.08$\times$ 10-16 | 10.20 | 1.39$\times$ 10-15
200 | 7.53 | 1.44$\times$ 10-15 | 1.62 | 2.60 | 6.07$\times$ 10-15 | 5.43 | 1.27$\times$ 10-14
300 | 5.02 | 3.48$\times$ 10-14 | 1.34 | 0.89 | 4.13$\times$ 10-14 | 2.33 | 1.09$\times$ 10-13
400 | 3.77 | 2.01$\times$ 10-13 | 1.23 | 0.64 | 1.57$\times$ 10-13 | 1.60 | 3.94$\times$ 10-13
500 | 3.01 | 6.49$\times$ 10-13 | 1.17 | 0.58 | 4.38$\times$ 10-13 | 1.28 | 9.67$\times$ 10-13
600 | 2.51 | 1.55$\times$ 10-12 | 1.13 | 0.55 | 9.70$\times$ 10-13 | 1.07 | 1.88$\times$ 10-12
700 | 2.15 | 3.06$\times$ 10-12 | 1.11 | 0.54 | 1.83$\times$ 10-12 | 0.92 | 3.12$\times$ 10-12
800 | 1.88 | 5.34$\times$ 10-12 | 1.09 | 0.53 | 3.09$\times$ 10-12 | 0.80 | 4.67$\times$ 10-12
900 | 1.67 | 8.53$\times$ 10-12 | 1.08 | 0.52 | 4.80$\times$ 10-12 | 0.70 | 6.48$\times$ 10-12
950 | 1.59 | 1.05$\times$ 10-11 | 1.07 | 0.52 | 5.83$\times$ 10-12 | 0.66 | 7.46$\times$ 10-12
1000 | 1.51 | 1.28$\times$ 10-11 | 1.07 | 0.51 | 7.00$\times$ 10-12 | 0.62 | 8.48$\times$ 10-12
1100 | 1.37 | 1.81$\times$ 10-11 | 1.06 | 0.51 | 9.73$\times$ 10-12 | 0.55 | 1.06$\times$ 10-11
1200 | 1.26 | 2.46$\times$ 10-11 | 1.05 | 0.50 | 1.30$\times$ 10-11 | 0.50 | 1.29$\times$ 10-11
1300 | 1.16 | 3.24$\times$ 10-11 | 1.05 | 0.50 | 1.69$\times$ 10-11 | 0.45 | 1.52$\times$ 10-11
1400 | 1.08 | 4.15$\times$ 10-11 | 1.04 | 0.49 | 2.13$\times$ 10-11 | 0.41 | 1.76$\times$ 10-11
1500 | 1.00 | 5.19$\times$ 10-11 | 1.04 | 0.49 | 2.63$\times$ 10-11 | 0.37 | 2.00$\times$ 10-11
1600 | 0.94 | 6.36$\times$ 10-11 | 1.04 | 0.48 | 3.18$\times$ 10-11 | 0.34 | 2.24$\times$ 10-11
1700 | 0.89 | 7.67$\times$ 10-11 | 1.04 | 0.48 | 3.80$\times$ 10-11 | 0.31 | 2.47$\times$ 10-11
1800 | 0.84 | 9.10$\times$ 10-11 | 1.03 | 0.48 | 4.47$\times$ 10-11 | 0.29 | 2.71$\times$ 10-11
1900 | 0.79 | 1.07$\times$ 10-10 | 1.03 | 0.47 | 5.19$\times$ 10-11 | 0.27 | 2.94$\times$ 10-11

Figure 1 shows the progression of our calculations. The blue line gives the
TST result, whereas the dashed green line gives the Eckart-corrected TST
result. As can be seen, there is very little difference between these two; the
Eckart correction is small. The black line with error bars shows the results
of the further correction from the GP calculation, whereas the solid green
line gives the result of the fit to the experiment. The GP correction
substantially improved the low-temperature rate constants but does not quite
provide quantitative agreement with the experiment. (Note that the GP
correction uses $\chi$ values from the Large and Small $\chi$ datasets as
described in detail in the SM. At the lowest three temperatures of Table 1 the
contributions from the Large-$\chi$ datasets are 87, 53, and 11 percent,
respectively. At 400 K or above the contributions are less than 3 percent; at
700 K or above they are less thatn 0.5 percent.) Neverthless, there is a major
improvement over TST and also the Eckart correction at both low and high
temperatures. At high temperatures both TST and Eckart corrected TST
overestimates the accurate result by about a factors of 2 - 5.

The representative error bars shown in Fig. 1 are caused by two sources, the
uncertainties for the ML predictions using different noise values and the
uncertainties for using differing methods for combining the small- and
large-$\chi$ data sets. The former is smaller than the latter. Fig. S4 shows
predictions for different noise values used in the large-$\chi$ set; the
curves are close except at the lowest temperatures. A larger error, shown by
the representative error bars in Fig. 1 comes from the latter error source.
The method we used (see Section S-I) was not the only method we tested, and
the representative error bars include uncertainty from the different methods.

Figure 1: Cl + CH4 results for the GP method. The method starts with the TST
rate constant (solid blue), performs a standard Eckart correction (producing
the dashed green curve), and then performs a GP correction based on the
original dataset (producing the solid black curve with representative error
bars). If the Cl + CH4 reaction is included in the dataset, the GP calculation
gives the solid red line, with representative error bars.

Of course, one goal of this study is to expand the dataset. The red line with
error bars shows the GP result following augmentation of the training dataset
with the data for the Cl(2P_3/2) + CH4 reaction under investigation. When the
reaction is included in the dataset with the target $\chi$ values, the GP
calculate reproduces the experimental data down to $1000/T=4$, but then still
underestimates slightly the needed correction at the lowest temperatures. We
note that the data, the original GP calculation, and the GP calculation made
with the augmented training dataset all predict lower rate constants at very
high temperatures, when compared to the TST rate constants, correctly
capturing the effect of recrossing trajectories.

## Discussion

Despite the small size of our training dataset, the partial coverage of it and
other challenges posed by the Cl(2P_3/2) + CH4 $\rightarrow$ HCl + CH3
reaction, the results shown above demonstrate that it is both possible to
obtain an good prediction of the temperature dependence of a new reaction and
that the prediction is improved substantially when the dataset is expanded by
including the new reaction. We now investigate how the GP prediction compares
with others.

Figure 2: Top Panel: Comparison of results for the Cl + CH4 reaction showing a
sampling of experimental results and the fit to all the experimental data
(solid green), the quantum results from ref. 37 (dashed red), the results from
ref. 44 (dotted red), our GP results using the original training set (solid
black), the RPMD results of ref. 42 (solid red), and the TST results (solid
blue). Bottom Panel: Expansion of the High-T region.

.

Figure 2. shows a comparison of recent results from several groups. Liu and
Li42 used a machine-learning approach to develop a potential energy surface
for the Cl + CH4 reaction and then performed RPMD calculations of the rate
constant for several temperatures in a limited range. Their result is shown as
the solid red line in the figure, which extends from $1000/T=0.5-5.0$. The GP
results from this work using the original training set are shown as the solid
black line with representative error bars. The agreement between the two is
good, but neither agrees perfectly with the experimental results, a sampling
of which is shown along with the green line fit to all the experimental data.
The results from the method of Georgievskii and Klippenstein44 are shown in
the dotted red line. Hoppe and Manthe45 used the LL PES to perform quantum
calculations of the rate constant as a function of temperature, and their
results are shown in the dashed red line extending from $1000/T=1.85-6.67$.
Both the GK and HH results are in relatively good agreement with the
experiment, with the HM calculation being almost in exact agreement.

It is interesting to note that our GP results, using the original data set,
and so a prediction, are in slightly in better overall agreement with
experiment than the RPMD ones. The former requires only 4 parameters, a simple
program, and a basic dataset; the calculation takes about a minute on a
laptop. The latter calculation requires a full PES, a much more elaborate
code, and a substantially longer calculation time. It is also curious that the
RPMD calculation is not in better agreement with full QM calculation of HM,
especially since both used the same PES. RPMD is essentially exact for the H +
CH4 reaction59 as shown in Figure. S4 of ref. 46, as well as for the O + CH4
reaction60 as shown in Figure. 3 of ref.46 and for the O(3P) + HCl reaction61
as shown in Figure. 9 of ref. 47. Whatever the answer is to this puzzle, it is
clear that the GP method is effective in making a substantial correction to
the TST or Eckart corrected TST rate constants, as we have seen in Figures. 1
and 2.

Another observation is that both our GP result and the data suggest that the
rate constant is reduced at high temperatures compared to the TST result,
likely because of recrossing. At first it seems surprising that neither the HM
nor GK result captures this effect. In the HM study, the authors specifically
commented that they did not see the trapped oscillatory motion of the H atom
due to the heavy-light-heavy mass combination. However, it seems likely from
Figure. 2 that their quantum calculation was not performed at high enough
temperatures to observe this effect; the dashed red line giving their results
stops at $1000/T=1.85$ or $T=540$ K,. The approach of GK, while effectively
accounting for the tunneling, was not designed to cover recrossing. The RPMD
calculation of LL and our GP calculation do appear to capture the effect, as
shown most clearly in the bottom panel of Figure. 2. We do note that the GP
predictions using the original datasets overestimates the extent of recrossing
at temperatures between around 340 - 600 K.

Finally, we comment briefly on the possible effects of van der Waals wells in
this reaction and in general. The importance of these for rate constants was
perhaps noted first for the O(3P)+HCl reaction.62, 63, 61 The reaction was the
focus of a “stress test” for RPMD 61, and as noted already it was one focus of
our paper II in this series.47 The fundamental effect is a quantum one, where
resonances in these well enhance the reactivity at energies below the barrier
and thus enhance tunneling. This quantum effect is not described in RPMD, and
so the absence of this enhanced tunneling was suggested as the reason RPMD
underestimates the exact quantum rate constant (on the same PES).61 We don’t
know if these effects are involved in the present reaction, which as noted
does have van der Waals wells, and it is beyond the scope of this paper to
investigate this issue. However, it is interesting to speculate on how these
wells might be approximately incorporated into a 1d tunneling correction such
as the Eckart correction. One of us presented a simple heuristic 1d model that
treats the resonances as metastable states that increase tunneling through a
1d barrier.64 That might be a first step towards such theory.

##  4 Conclusions

The Gaussian Process regression method for machine learning of thermal rate
constants developed previously46, 47 has been shown here to provide a good
estimate of the temperature dependence of the rate constant for the Cl(2P_3/2)
+ CH4 $\rightarrow$ HCl + CH3 reaction. The results suggest both that
tunneling is important at temperatures below about 250 K and that recrossing
is important at temperatures above about 500 K. The results are in fairly good
agreement with RPMD calculations by Liu and Li42, but neither method is as
accurate in predicting the experimental results as the predictions of the
method used by Georgievskii and Klippenstein44 or, especially, the quantum
calculations of Hoppe and Manthe,45 in the deep tunneling regime. However, the
present ML and RMPD results are closer to experiment at higher temperatures,
where recrossing becomes signficant. When the new reaction is included in the
training dataset, the GP results are in agreement with experiment to within
the error limits, suggesting that the new training set will be more effective
in predicting rates for new three-dimensional systems in the future. We note
that the original training basis set9 was based predominantly on the results
of one-dimensional reaction calculations. As more complex reactions are added
to the dataset, we can expect improvements in the prediction of other
reactions such as the Cl + CH4 one.

##  5 Acknowledgments

JMB thanks NASA (grant 80NSSC20K0360) for financial support. We thank Yang Liu
and Jun Li for providing their RPMD results, Hannes Hoppe and Uwe Manthe for
providing their quantum and TST results and for extensive discussions, and
Yuri Georgievskii and Stephen Klippenstein for providing the results of their
Eq. 5.46.

##  6 TOC Figure

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