Ring-polymer molecular dynamics: Rate coefficient
calculations for energetically symmetric (near
thermoneutral) insertion reactions (X + H[subscript 2]) HX
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Citation
Suleimanov, Yury V., Wendi J. Kong, Hua Guo, and William H.
Green. “Ring-Polymer Molecular Dynamics: Rate Coefficient
Calculations for Energetically Symmetric (near Thermoneutral)
Insertion Reactions (X + H[subscript 2]) HX + H(X =
C([superscript 1]D), S([superscript 1]D)).” The Journal of
Chemical Physics 141, no. 24 (December 28, 2014): 244103.
As Published
http://dx.doi.org/10.1063/1.4904080
Publisher
American Institute of Physics (AIP)
Version
Author's final manuscript
Accessed
Wed May 25 22:14:54 EDT 2016
Citable Link
http://hdl.handle.net/1721.1/93151
Terms of Use
Creative Commons Attribution-Noncommercial-Share Alike
Detailed Terms
http://creativecommons.org/licenses/by-nc-sa/4.0/
Ring-Polymer Molecular Dynamics: Rate Coefficient Calculations
for Energetically Symmetric (Near Thermoneutral) Insertion
Reactions (X+H2)  HX + H (X=C(1D), S(1D))
Yury V. Suleimanov1,2, Wendi J. Kong1, Hua Guo3, William H. Green1
1Department
of Chemical Engineering, Massachusetts Institute of Technology, Cambridge,
Massachusetts 02139, Unites States
2Computation-based Science and Technology Research Center, Cyprus Institute, 20 Kavafi Street, Nicosia
2121, Cyprus
3Department of Chemical and Chemical Biology, University of New Mexico, Albuquerque, New Mexico
87131, United States
Abstract
Following our previous study of prototypical insertion reactions of energetically
asymmetric type with the RPMD (Ring-Polymer Molecular Dynamics) method [Y. Li,
Y. Suleimanov, and H. Guo, J. Phys. Chem. Lett. 5, 700(2014)], we extend it to
two other prototypical insertion reactions with much less exothermicity (near
thermoneutral), namely, X + H2  HX + H where X = C(1D), S(1D), in order to assess
the accuracy of this method for calculating thermal rate coefficients for this class of
reactions. For both chemical reactions, RPMD displays remarkable accuracy and
agreement with the previous quantum dynamic results that makes it encouraging
for the future application of the RPMD to other barrier-less, complex-forming
reactions involving polyatomic reactants with any exothermicity.
Introduction
Rate coefficients of chemical reactions are a key input in kinetic modeling of various
chemical processes that include, but are not limited to combustion, atmospheric
chemistry, and chemistry in the interstellar media. Experimental measurements of
rate coefficients can sometimes be troublesome, expensive, inaccurate or even
impossible under certain conditions, which is why it is indispensible to develop
alternative theoretical methods of estimating them.1
Unfortunately, a rigorous computation of rate coefficients using the quantum
mechanical formalism via the Schrödinger equation scales exponentially with the
system size and is only achievable for systems with just a few atoms.2 Therefore,
there is a considerable interest in developing various approximations to quantum
dynamics, which would allow rate calculations for much larger systems with
reasonable accuracy.
Of particular concern is the proper treatment of the quantum mechanical (QM)
effects of nuclear motion, such as zero point energy (ZPE) and tunneling. The proper
inclusion of these effects is important especially if the reaction involves the transfer
of light atoms (such as hydrogen) and takes place at room temperature (or lower)
1
where the QM effects can change the rate coefficient of a chemical reaction by
several orders of magnitude.1,3
Numerous transition-state theory (TST) methods for rate calculations were
developed in the past,4-7 which are based on various static (short-time)
approximations to the classical real-time correlation functions used to describe
chemical reactions. Various approximations were developed to include quantum
tunneling and ZPE effects which transformed methods based on classical TST to an
inexpensive and simple way for determining theoretical rate coefficients,
particularly for thermally-activated reactions with significant activation barriers.3
Many TST-based methods require only a small portion of the PES, which greatly
reduces the computational effort needed to compute a rate coefficient, and so these
practical approximate methods are widely used.
However, several persistent issues with the TST assumption limit the accuracy of
these rate calculation methods. The main problem is an exponential sensitivity of
short-time limits of correlation functions to the choice of the dividing surface that
separates reactants from products. Within the TST framework, it is assumed that we
can always find a dividing surface that will provide an accurate approximation to
the long-time limit of the real-time correlation functions (no recrossings
assumption). However, making a proper choice of such a dividing surface is not easy
for many complex systems and there is no unambiguous way to treat recrossings
within the TST framework as it is inherently a real-time dynamical feature. For
some thermally-activated reactions, recrossing plays a crucial role, for instance, in
the case of heavy-light-heavy mass combinations10-13. Finding an accurate dividing
surface is also challenging for reactions with no or low barriers, and for systems
with multiple large-amplitude vibrational modes. Several methods have been
developed to try to deal with these problems, e.g. multi-structural methods [ref] and
variational transition state theory3, including variable reaction coordinate
approaches which allow significant flexibility in constructing improved dividing
surfaces.8,9 In these cases, simple methods for approximating ZPE and tunneling
effects also tend to become less accurate, the computations become more complex,
and larger portions of the PES need to be computed. In the present paper we focus
on barrierless insertion-scission reactions, where it is challenging to define the
dividing surface and the reaction proceeds through a deep well.14,15
As an alternative to TST, the quasi-classical trajectory (QCT) method was developed
in an attempt to treat properly recrossing events.14,16 QCT is a full dimensional
approach that treats nuclear motions classically but includes the ZPE effect of the
products and reactants. However, QCT does not take into account the change in ZPE
along the reaction coordinate and completely ignores the tunneling effect.14,16 The
problem with ZPE is particularly significant in the case of the barrier-less reactions:
trajectories exiting the potential well may violate the ZPE and lead to substantial
errors.14,17,18
2
Recently, another alternative method based on the Ring Polymer Molecular
Dynamic (RPMD) theory has been developed19-21 which exhibits several distinctions
from both TST and QCT methods. RPMD is a full dimensional approach that uses the
classical isomorphism22 between the statistical properties of a quantum particle and
a ring polymer consisting of harmonically connected classical copies of the original
particle to approximate the real-time quantum dynamics in the system.19-21 Though
introduced in an ad hoc manner, this approach turned out to possess several
desirable features for calculating chemical reaction rate coefficients. First, at high
temperatures where the ring polymer collapses to a single bead, RPMD rate
becomes exact. Second, the RPMD method has a well-defined short-time limit
related to TST-methods, providing an upper bound on the RPMD rate. Finally,
because the RPMD formalism is based on the long-time limit correlation functions,
the rate coefficient is independent of the choice of the transition state dividing
surface used to compute it. This is an extremely advantageous feature of RPMD,
especially when it is challenging to determine the optimal dividing surface of a
complex multidimensional reaction.
In addition to avoiding the need to define a good dividing surface, RPMD
automatically includes effects of large-amplitude motions, anharmonicity, and multidimensional tunneling. However, it is more computationally expensive than TSTbased methods, primarily because it requires more of the PES, and it is more
computationally expensive than QCT because it propagates the motion of a ring of
many beads to represent each atom. It is therefore important to determine how
accurate this new method is, so users can decide whether it is worth the
computational cost.
In a series of recent studies13,15,17,20,21,23-26,28-32,34 it has been shown that the RPMD
method provides accurate rate coefficients when applied to systems with various
dimensionalities starting from one-dimensional Eckhart barriers,20,21 atomdiatom13,15,26,32 to more complex polyatomic reactions.17,23-25,28-31 The RPMD rate
coefficient captures the ZPE effect almost perfectly, and is within a factor of 2-3 of
the exact rate at low temperatures in the so-called deep tunneling regimes, where
tunneling dominates. At low temperatures, the RPMD rate theory exhibits
systematic behavior: it consistently underestimates rates for energetically
symmetric (thermoneutral) reactions, and conversely overestimates rates for
significantly asymmetric (endo or exothermic) reactions.26,27 A recent
comprehensive test of various quantum transition-state theories and other ratecalculation methods suggested that the RPMD approach is the most accurate and
well behaved.32,33
More recently, the RPMD method was applied to insertion chemical reactions.15
Using two prototypical reactions of this type, namely X + H2  HX + H, (where X =
N(2D),O(1D)) it has been shown that RPMD can provide very accurate results when
compared to the exact quantum mechanical solution. These two very exothermic
insertion chemical reactions are highly asymmetric (ΔErxn,0 = -1.25 eV for X =N(2D)
and -1.88 eV for X = O(1D), see Fig. 1). In the previous studies of thermally-activated
3
chemical reactions, it has been well established that RPMD provides over- and
underestimations of rate coefficients depending on the reaction energy symmetry.26
It is therefore important to assess the accuracy of RPMD for energetically symmetric
(or thermoneutral) chemical reactions of the insertion type, which represent the
main purpose of this work. To this end, we employ two prototypical complexforming reactions, namely X + H2  HX + H, where the difference in energies
between reactants and products is 0.2 and 0.3 eV for X = C(1D) and S(1D),
respectively (see Fig. 1). We believe that the results of the present work will provide
insights into the performance of the RPMD approach and will be useful in future
studies of polyatomic complex-forming chemical reactions.
Computational Details
All calculations were performed using the RPMDrate code developed by Suleimanov
and co-workers.34 The detailed explanation of the RPMD rate theory and
computational procedure implemented in this code can be found in Ref. 34. Here we
only describe the key steps of the rate calculations. First, we make use of the
Bennett-Chandler factorization and express the RPMD rate coefficient in the
following form:
kRPMD (T )  kQTST (T ; # ) (t  ; # )
(1)
The first term, kQTST (T ; # ) , designates the static contribution, which depends on the
position of the dividing surface and is completely determined by the static
equilibrium properties. When the dividing surface is defined in terms of the centroid
variables (which is the present case), it coincides with the centroid-density quantum
transition-state theory (QTST) rate coefficient.4,5,35
In practice, kQTST (T ; # ) is calculated from the centroid potential of mean force
(PMF) along a given reaction coordinate  as:34
1
 1  2   W ( # )W ( 0) 
 e
kQTST (T ,  # )  4R2 
 2 R 
(2)
where μR is the reduced mass between the two reactants,   (k BT ) 1 and
W ( # )  W (0) represent the free energy difference between the dividing surface and
the separated reactants at R , which is usually obtained using umbrella integration
along  ,36,37 and R is an adjustable parameter chosen to be sufficiently large to
make the interaction between the reactants negligible.23,34
The second term in Eq.(1),  (t  ;  # ) , is the dynamic factor (transmission
coefficient) which accounts for recrossings at  # , counterbalancing kQTST (T ;  # ) and
4
securing the independence of the RPMD rate coefficient from the choice of the
dividing surface. Usually, the position of  # is chosen to correspond with the
maximum value of W ( ) to try to minimize the recrossings and accelerate the
calculations.
The reaction coordinate  for the reactions X + H2 (X = C(1D), S(1D)) is determined
using two standard dividing surfaces used in all our previous atom-diatom studies
(see Eqs.(27),(70), and (77-79) in Ref.13.). Although these dividing surfaces may not
be an optimal choice for these particular reactions, they will have no effect on the
final result due to the “long-time dynamics” nature of RPMD.
For this particular work, the thermal rate coefficients for the C(1D) + H2 and S(1D) +
H2 reactions were calculated for three temperatures: 200, 300, and 400 K. The input
parameters are outlined in the Table 1 and are similar to previous ones used in the
recent RPMDrate study of exothermic insertion chemical reactions15 as well as
several previous studies of thermally-activated chemical reactions. RPMDrate
provides a convenient parallel computing option that speeds up the recrossing
factor calculations.15,34 These calculations require more effort to converge in the
case of complex-forming chemical reactions, due to much longer times of the
propagation of the recrossings.
The RPMD calculations were carried out using 128 ring polymer beads. For the PMF
calculations, we divided the interval of the reaction coordinate  , between -0.05
and 1.05 in windows of .01. The end of umbrella integration corresponded to the
intermediate configuration along the reaction path between reactants and products
(see Table 1). In each window 200 constrained RPMD trajectories of 100 ps were
run, in which the first 20 ps were used for thermalization in the presence of an
Andersen thermostat.38 The time step was set to 0.0001 ps. For each window  i, a
bias is applied to its center by adding a harmonic potential of the type k(  -  i)2 to
the Hamilton function of the system. In doing so, the trajectory is biased to explore
in the vicinity of ξi.. Therefore, the force constant must be large enough to constrain
the sampling to a region about the vicinity of ξi, but small enough to allow
overlapping between adjacent ξ distributions.36,37
To obtain the transmission coefficient,  (t  ;  # ) , a long “parent” trajectory of 2
ns is run after a thermalization period of 20 ps in the presence of an Andersen
thermostat with the constraint of having its centroid pinned at  # , which
corresponds to the maximum value of the PMF. Once every 2 ps, configurations are
taken from the parent trajectory as initial conditions for 100 unconstrained
“daughter” trajectories. All daughters’ trajectories are run for 2 ps (C + H2) or 3 ps (S
+ H2) with different initial momenta and without either the constraint or the
thermostat to accumulate the transmission coefficient. The parameters used in our
calculations are summarized in Table 1 and they were checked to be sufficient to
converge RPMD rate coefficients to within a statistical error of ∼10%.
5
Similar to the previous theoretical studies, we assume that the reaction occurs in a
low pressure gas and we do not consider pressure dependence (thermal bath
effects). For the C + H2 reaction, the potential energy surface (PES) of Banares et al.39
was used. For the S + H2 reaction, the PES of Ho et al.40 was used. These PESs were
the same used in the earlier quantum dynamics calculations of rate coefficients.
The final rate coefficients were corrected by electronic partition functions: Qel = 5
was used for both reactions.
Discussion
The PMFs for the C + H2 and S + H2 reactions are plotted in Figures 2a and 2b,
respectively. These PMF profiles demonstrate several important features. First, they
show that these reactions do indeed proceed through complex formation, which
corresponds to deep wells of the free energy along the reaction coordinate. Second,
they also confirm that the free energy does not change much until the reaction gets
in the complex formation zone. This is because there is no significant bottleneck in
the PESs. Finally, the shapes of the free-energy profiles do not change much with the
temperature and do not exhibit any significant activation free-energy barrier. This
situation is very similar to the O(1D) + H2 reaction discussed in our previous work.15
A small increase of the free energy (below 0.04 eV for C(1D) + H2 and below 0.02 eV
for S(1D) + H2) occurs prior to the complex (deep well); this is due to the entropic
factor and is weakly temperature dependent as expected. For the second step in the
RPMDrate calculations, the recrossings calculations, we selected the positions of the
free-energy barriers for both systems as the initial conditions.
The recrossings factor profiles are plotted for both reactions in Figures 3a (C(1D) +
H2) and 3b (S(1D) + H2). Comparing the two reaction profiles, one can first notice
that both reach the plateau regime only after several picoseconds. This extremely
long time for the convergence of the transmission coefficients is a characteristic of
the complex-forming reaction mechanism.14,15 For reactions over barriers, the
plateau region is typically reached about an order of magnitude faster. Although the
recrossing factors reach practically the same plateau values for all temperatures, the
profiles for various temperatures are quite different. They exhibit short-time
oscillations apparently due to entering and re-entering in the complex formation
zone at short times, which are more active at higher temperature thus taking less
time to reach the plateau. Note that these short-time oscillations are due to our
particular choice of a transition-state dividing surface which has no effect on the
final rates. In all cases the recrossing factors are significantly below 1 (in the range
of 0.30-0.34 for S + H2 and ~0.47 for C + H2), confirming the importance of the
inclusion of this factor in the total rate coefficient calculation.
Table 2 compares the RPMD rate coefficients with theoretical and experimental
results from the previous works.41-45 The title reactions are barrierless, thus,
reflecting very weak temperature dependence of the present RPMD results. Indeed,
6
such weak temperature dependence has been observed for the S(1D) + H2 reaction.45
Table 2 also shows that it is very important to take into account the recrossing effect
since the kQTST values are too high compared to the Quantum Dynamics (QD) ones. In
the case of the S + H2 reaction, the centroid-density QTST rates exhibit nonsystematic temperature dependence. This might be due to the fact that the freeenergy barrier just before the entrance into the H2S well becomes so tiny at 200 K
that the free-energy maximum in the entrance channel is attained in the asymptote
region (see Fig. 2). Correcting kQTST with  gives us the RPMD value, which turns out
to be in a very good agreement with the QD results with an error of ~ 5-10%. This
deviation is the right order of magnitude to be due to the statistical sampling
uncertainties in the RPMD calculations, but this deviation may also be due in part to
approximations made in the QD calculations: a quantum statistical model46-48 was
used and high rotational states were cut off in the quantum calculation. While the
RPMD and QD calculations agree very closely with each other, and they are within
the error bar of the experimental measurement for C(1D)+H2, they are both about
25% less than the experimental measurement for S(1D)+H2 at 300 K. It is most
likely that the discrepancy between experiment and theory is due to deficiencies in
the PES used in the present and previous41,43 dynamical calculations, but other
effects (e.g. the calculations completely ignore the low-lying triplet electronic
surface; the experiments assume there are no significant side channels) may also
contribute. The present results suggest less sensitivity of RPMD to the energetics of
insertion-type chemical reactions compared to the previously studied thermallyactivated ones.26 The two title reactions feature relatively small energy differences
between reactants and products - 0.17 eV and 0.30 eV for C(1D) + H2 and S(1D) + H2,
respectively (see Fig. 1).14 According to the reaction symmetry assignment used for
thermally-activated chemical reactions to predict whether RPMD over- or
underestimates the rates at low temperatures26,27 they would technically be
classified as “effectively symmetric” (or, alternatively, “weakly asymmetric”). For
symmetric reactions with barriers RPMD has been found to systematically
underestimate the low-temperature rates. However, our results demonstrate that
RPMD slightly overestimates them for insertion reactions with low exothermicity
and therefore this prediction doesn’t work in the present case. This is not surprising
as this prediction was developed for reactions dominated by a significant barrier,
where tunneling was significant.26,27 The present (and previous15) RPMD
calculations for prototypical insertion-type reactions suggest that RPMD slightly
overestimates the rates for this class of reactions. However, such conclusion is
rather speculative and more extensive accuracy assessment of RPMD for barrierless
reactions is required, similar to numerous recent studies of thermally-activated
chemical reactions.13,15,17,23-26,28-32,34 Nevertheless, the level of accuracy that RPMD
provides for the insertion reactions is indeed remarkable and suggests that it has
the potential to be used for polyatomic reactions involving reaction intermediates,
when high accuracy is desired.
7
Conclusions
To summarize, in this work, the RPMD method is extended to two prototype
energetically symmetric (near thermoneutral) insertion reactions, namely, X + H2 
HX + H, where X = C(1D), S(1D). Calculating rate coefficients for this type of reactions
is challenging. A significant amount of recrossing occurs along the reaction
coordinate because there is no identifiable reaction barrier, and there is a deep well.
Fortunately, the computational strategy developed for calculating RPMD rate
coefficients can effectively handle recrossing calculations for many types of
chemical reactions. In this work, we observed very similar results as to those
obtained previously for two energetically asymmetric (with more significant
exothermicity) insertion reactions (X = N(2D), O(1D)). The results are in a very good
agreement with the previous quantum dynamics calculations for these reactions
and thus encouraging for future applications to complex-forming polyatomic
chemical reactions with various energetics of reactants and products channels. The
present results also suggest that RPMD is less sensitive to the energetics of insertion
chemical reactions compared to the previously studied thermally-activated ones,
perhaps because tunneling is less important for barrierless reactions. More studies
of chemical reactions from this class of reaction with RPMD are required in order to
assess accuracy of this novel method.
Acknowledgements: Y.V.S. and W.H.G. acknowledge the support of the Combustion
Energy Frontier Research Center, an Energy Frontier Research Center funded by the
U.S. Department of Energy, Office of Basic Energy Sciences under Award Number
DE-SC0001198. WJK thanks the MIT Energy Initiative for support. HG thanks DOE
(DE-FG02-05ER15694) for support.
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10
Table 1 Input parameters for the RPMDrate calculations on the C +H2 and S + H2
reactions. The explanation of the format of the input file can be found in the
RPMDrate code manual (see Ref. 34 and
http://www.mit.edu/~ysuleyma/rpmdrate).
Parameter
Reaction
C(1D) + H2
S(1D) + H2
400
300
200
128
400
300
200
128
Temperature (K)
20
20
Nbond
1
1
Nchannel
2
2
Dividing surface s1 parameter
(a )
Number of forming and
breaking bonds
Number of equivalent product
channels
X(1D)
H
H
(1.635 a , 1.302 a , 0.000 a )
(0.000 a , 0.000 a , 0.000 a )
(3.270 a , 0.000 a , 0.000 a )
(0.376 Å, 2.110 Å, 0.000 Å)
(0.000 Å, 0.000 Å, 0.000 Å)
(0.751 Å, 0.000 Å, 0.000 Å)
Cartesian coordinates (x, y, z)
of the intermediate geometry
Thermostat
Biased sampling parameters
Nwindows
ξ1
dξ
ξN
dt
ki
‘Andersen’
‘Andersen’
Thermostat option
111
-0.05
0.01
1.05
0.0001
2.72
111
-0.05
0.01
1.05
0.0001
2.72
Ntrajectory
tequilibration
tsampling
200
20
100
200
20
100
Ni
2 × 108
2 × 108
Number of Windows
Center of the first window
Window spacing step
Center of the last window
Time step (ps)
Umbrella force constant ((T/K)
eV)
Number of trajectories
Equilibration period (ps)
Sampling period in each
trajectory (ps)
Total number of sampling
points
0.00
0.604 (400 K)*
0.599 (300 K)*
0.492 (200 K)*
5000
0.00
0.886 (400 K)*
0.875 (300 K)*
0.403(200 K)*,^
5000
Start of umbrella integration
End of umbrella integration
0.0001
20
0.0001
20
Ntotalchild
200000
200000
tchildsampling
3
3
Nchild
100
100
tchild
2
3
Time step (ps)
Equilibration period (ps) in the
constrained (parent) trajectory
Total number of unconstrained
(child) trajectories
Sampling increment along the
parent trajectory(ps)
Number of child trajectories
per one initially constrained
configuration
Length of child trajectories
(ps)
Command line parameters
Temp
Nbeads
Dividing surface parameters
R∞
Explanation
Number of Beads
0
Potential of mean force
calculation
ξ0
ξ‡
Nbins
Recrossing factor calculation
dt
tequilibration
0
0
0
0
0
0
0
0
0
Number of bins
*Detected automatically by RPMDrate;
^ In the case of the chemical reaction S+H2, the maximum of the free energy at 200 K occurs before entering the complex-formation
region.
11
Table 2 Comparison between RPMD Rate Coefficients (cm3s-1) of the C(1D) + H2 and
S(1D) + H2 chemical reactions with the previous theoretical and experimental
results.
C(1D) + H2
T/K
kQTST
κ
kRPMD
QD41
Expt.42
200
3.09×10-10
0.48
1.47×10-11
---
T/K
kQTST
κ
kRPMD
QD43
Expt.44
Expt.45
200
4.64×10-10
0.34
1.60×10-10
----
300
3.35×10-10
0.47
1.57×10-10
1.50×10-10
(2.0±0.6)×10-10
400
3.56×10-10
0.47
1.66×10-10
---
300
5.32×10-10
0.30
1.62×10-10
1.49×10-10
2.1×10-10
2.07×10-10
400
4.95×10-10
0.33
1.63×10-10
----
S(1D) + H2
12
1
O( D) + H2
N(1D) + H2
C(1D) + H2
S(1D) + H2
CH + H
SH + H
0.17 eV
0.30 eV
NH + H
1.25 eV
OH + H
1.88 eV
SH2
CH2
3.90 eV
4.33 eV
NH2
5.48 eV
OH2
7.29 eV
Figure 1. Energetics of four prototypical insertion reactions X + H2  XH + H (X =
S(1D), C(1D), N(2D), O(1D)) on the lowest electronic state of the specified multiplicity.
13
0.05
W() / eV
0.00
C + H2
-0.05
T = 200 K
T = 300 K
T = 400 K
-0.10
-0.15
-0.20
(a)
-0.25
-0.30
0.0
0.1
0.2
0.3
0.4

W() / eV
0.05
0.5
0.6
0.7
0.8
0.00
S + H2
-0.05
-0.10
-0.15
(b)
-0.20
-0.25
0.0
0.1
0.2
0.3
0.4
0.5

0.6
0.7
0.8
0.9
Figure 2. Ring polymer potentials of mean force (W()) along the reaction
coordinate for the C(1D) + H2(top panel) and S(1D) + H2 (bottom panel) chemical
reactions at 200, 300 and 400 K.
14
1.0
0.9
C + H2
0.8
(t)
T = 400 K
T = 300 K
T = 200 K
0.7
0.6
(a)
0.5
0.4
1.0 0
500
1000
1500
2000
t / fs
(t)
0.8
S + H2
0.6
(b)
0.4
0.2
0
500
1000
1500
2000
2500
3000
t / fs
Figure 3. Ring polymer recrossing factors ((t)) along the reaction coordinate for
the C(1D) + H2(top panel) and S(1D) + H2 (bottom panel) chemical reactions at 200,
300 and 400 K.
15