Rigorous Quantum Dynamics of O + O2 Exchange Reactions on an Ab Initio Potential
Energy Surface Substantiate the Negative Temperature Dependence of Rate Coefficients
Yaqin Li1,2, Zhigang Sun1,2*, Bin Jiang3, Daiqian Xie4, Richard Dawes5*, and Hua Guo3*
1
State Key Laboratory of Molecular Reaction Dynamics, Dalian Institute of Chemical
Physics, Chinese Academy of Sciences, Dalian 116023, Liaoning, China
2
Center for Advanced Chemical Physics, University of Science and Technology of China,
96 Jinzhai Road, Hefei 230026, P. R. China
3
Department of Chemistry and Chemical Biology, University of New Mexico, Albuquerque,
New Mexico 87131, USA
4
Institute of Theoretical and Computational Chemistry, Key Laboratory of Mesoscopic
Chemistry, School of Chemistry and Chemical Engineering, Nanjing University, Nanjing
210093, China
5
Department of Chemistry, Missouri University of Science and Technology, Rolla,
Missouri 65409, USA
Supporting Information
S1
S1. Quantum mechanics
S1-1. Hamiltonian and wavefunctions
For the A + BC → AB + C and AC + B reaction, the reactant and two product
channels are denoted as the α, β and γ arrangement channels, respectively. For each

channel, the corresponding Jacobi coordinates are written as (Rv, rv, θv;  v ), where v is
either α, β or γ. For v = α (v = β, γ), rv is the bond distance of the diatom molecule BC (AB,
AC), Rv is the distance from the atom A (C, B) to the center of mass of BC (AB, AC), θv is


the angle between rv and Rv,  v denote the Euler angles orienting Rv in the space-fixed
(SF) frame.
In the reactant Jacobi coordinates, the Hamiltonian for a given total angular
momentum J can be written as
Hˆ  
2
2 R
2
ˆj 2
2
 2 ( Jˆ  ˆj )2



 Vˆ ,
2
2
2
2
R 2r r 2r R 2r r
(1)
where Ĵ is the total angular momentum operator, and ĵ is the rotational angular
momentum operator of BC. The potential energy surface is that of Dawes et al.1
augmented by the long-range expression for the O + O2 asymptote.2
The wave function in the body-fixed (BF) frame can be expressed as
J 
 JM  ( R , r )   DMK
( )  ( R , r , K ; K ) ,

(2)
K
J e*
(Wa ) is the parity-adapted normalized rotation matrix depending only on the
where DMK
a
Euler angles Wa ,
J e*
DMK
(Wa ) =
a
2J +1
J +K
J*
[DMK
(Wa ) + e (-1) a DMJ *-K (Wa )],
a
a
8p (1+ dK 0 )
a
S2
(3)
where  = (-1)j+l is the parity of the system, l is the orbital angular momentum quantum
number, Kα is the projection of the total angular momentum J on the BF Z axis in the 
J e*
(Wa ) is the corresponding Wigner rotation matrix.3 The
arrangement channel, and DMK
a
wave function   ( R , r , K ; K ) , only depending on the internal coordinates (Rα, rα,
K ) and Kα, can be expanded as
  ( R , r , K ; K ) 
F
K
nmj
un ( R )m (r ) y jK ( ) ,
(4)
n,m, j
where n and m are the radial basis labels and y jK are spherical harmonics.

The initial wave packet JM
was constructed as the product of a total angular
i ji li
momentum eigenfunction, a ro-vibrational eigenfunction of molecule BC and a Gaussian
wave packet in the SF frame. To propagate the initial wave packet in the BF frame, we
need transform it from the SF to BF frame. The time propagator is approximated by a 4th
order split operator.4
To avoid the wave packet reflecting back from the boundaries, an absorbing
potential was employed with the same form as in Ref. 5:

 x  xa
D( x)  exp   t Ca 
 xb  xa


  , xa £ x £ xb ,

(5)
And

 x  xb  
D( x)  exp   t Cb 
   exp( t Ca ) , xb £ x £ xend
x

x
end
b



(6)
where x is R/r and xa, xb and xend are the first and second starting points and ending point of
the absorbing potential, which were set to 11.0/9.0, 15.0/13.0 and 16.0/14.0 au,
respectively. Ca and Cb, which control the strength of the absorbing potentials, are
0.001/0.0005, 0.003/0.003 au, respectively. The two parts in the two equations above were
S3
designed for absorbing the low and high kinetic energy components of the wave packet.
During the propagation, the fast Fourier transform (FFT) method was adopted to
evaluate the radical kinetic energy operator acting onto the wave packet on an L-shaped
grid.5 The generalized discrete variable representation (DVR)6 was used to evaluate the
action of the potential energy operator, in which the wave packet is switched between the
angular finite basis representation (FBR) and a grid representation. To extract the
state-to-state S-matrix, a reactant coordinate-based (RCB) method is adopted. For more
details, the reader is referred to Refs. 7-8. The parameters for the calculations are given in
Table S1.
S1-2 Nuclear spin statistical effects
To clarify the nuclear spin statistics effects on the studied reactions, we take the
16
O+32O2 reaction as the example. For reactions with other isotopes, similar conclusions
can be drawn. Restricted by the nuclear spin statistics, only odd rotational states are
allowed for
32
O2. Computationally, 2 ji + 1 wave packets are needed to characterize the
reaction, so the calculation of the integral cross section (ICS) for ji >0 is expensive and we
would like to make some approximations. To understand the difference in reactivity for
the ji=1 and the unphysical ji=0 states, the ICSs have been computed for both and the
results are shown in Fig. S1. It can be seen from the figure that the ICSs are essentially the
same for these two initial states. As a result, it is reasonable to use the results from ji=0 to
predict the reactivity for ji=1 in order to save computational efforts.
For reactions that producing 32O2, the same nuclear spin statistics apply. However,
our Hamiltonian has no regard to nuclear spin. As a result, both odd and even rotational
states of the product are allowed, following the Boltzmann statistics rather than the
Fermi-Dirac statistics. To enforce the Fermi-Dirac statistics, the post-antisymmetrization
procedure of Kuppermann and co-workers9 (also see Chao et al.10 and Zhang and Miller11),
which assumes that the spins are essentially uncoupled to the reaction dynamics, was used.
In our calculations, the O atom and diatomic reagent O″O′ are distinguishable and
S4
we label the three channels, O+O″O′, O′+O″O and O″+OO′, as the α, β and γ channels,
α→γ
respectively. The computed Boltzmann S-matrix elements are denoted as 𝑆𝐵
(𝑣, 𝑗 →
𝑣′, 𝑗′). For a given spin state, the correct Fermi-Dirac spatial scattering wavefunctions,
|𝜓𝐹 ±⟩, has even and odd symmetries with respect to atomic exchange. Thus, |𝜓𝐹 ±⟩ is a
superposition of the α → β and β → γ Boltzmann wavefunctions:
α→γ
|𝜓𝐹 ±⟩ = (|𝜓𝐵
α→γ
Obviously, |𝜓𝐵
β→γ
⟩ ± |𝜓𝐵
β→γ
⟩ and |𝜓𝐵
vector, r, is replaced by –r.
⟩) /√2
(7)
⟩ are physically identical except that the O2 internuclear
Asymptotically, in the γ channel, the scattering amplitudes
are thus equal for 𝑗′=even, and equal in magnitude but opposite sign for 𝑗′=odd. This
implies for a single I=0 state of 32O2 (ji=odd),
16
𝑆𝐹 ( 𝑂 +
32
𝑂2 →
16
𝑂+
α→γ
32
𝑂2 ) = √2 × {
𝑆𝐵
(𝑣, 𝑗 → 𝑣 ′ , 𝑗 ′ ),
0,
𝑗 ′ = 𝑜𝑑𝑑
(8)
𝑗 ′ = 𝑒𝑣𝑒𝑛
Correspondingly, the Fermi-Dirac cross sections are given by
𝜎𝐹 ( 16𝑂 + 32𝑂2 →
16
𝑂 + 32𝑂2 ) = 2 × {
α→γ
𝜎𝐵
(𝑣, 𝑗 → 𝑣 ′ , 𝑗 ′ ),
0,
𝑗 ′ = 𝑜𝑑𝑑
𝑗 ′ = 𝑒𝑣𝑒𝑛
(9)
Since the total cross sections of products in odd and even rotational states are roughly the
same,12 especially at higher collision energies, as shown in Fig. S2, the nuclear spin
statistics need not be considered in calculating the total ICSs and thermal rate coefficients.
To understand the slight effects of the nuclear statistics on the ICS, the Fermi-Dirac
rotational distributions for the even and odd rotational states are shown in Fig. S3.
Another issue is the adoption of the symmetry of the intermediate complex O3*, as
discussed in the work on H + H2 by Zhang and Miller.11 However, the usage of symmetry
of the system in a practical calculation usually aims at reducing computational effort and
is optional. This is not an issue for our calculations.
S5
S1-3. Total integral cross sections of rotationally excited states
Table S2 lists the O2 rotational energies and the weights due to the rotational
partition function. The rotational energies were obtained using a sine-DVR on the O3 PES
with one O-atom fixed at large separation. The asymptotic rOO dependence of the PES is
that of Bytautas et al. so the levels are the same as reported by those authors.13 It is clear
that many rotational states of O2 contribute to the rate constant. To accurately compute the
thermal rate coefficients, thus, it is important to assess the influence of rotationally
excited O2 molecules on reactivity.
In this work, the initial state-selected ICS for R3 were explicitly computed for ji=
0, 5, 9, 21 using the quantum wavepacket method. Due to the fact that the total reaction
probabilities with different but adjacent total angular momentum follow the J-shifting
rule well,13,14,15,16 only total reaction probabilities for every ten partial waves were
calculated to save computational costs. The total reaction probabilities of remaining
partial waves were calculated by an interpolation method based on the J-shifting rule. All
of the necessary helicity numbers, which is about 2J/3, were considered in the
calculations. The ICSs for other initial rotational states were calculated by interpolating
with those of the initial states ji= 0, 5, 9, 21. In this way, the total ICSs and corresponding
rate coefficients can be estimated efficiently with reasonable accuracy.
S-2. Quasi-classical trajectory
The VENUS code14-15 was used to run QCT trajectories at collision energies
corresponding to 100, 200 and 300 K for each ji for R1 with significant weight according
to the rotational partition function, as shown in Table S2. Reactive ICSs were determined
for each reactant ji separately (to be weighted by the rotational partition function). The
maximum impact parameter bmax = 8.0 Å was determined from test batches. A
propagation time-step of 0.2 fs was found to conserve energy to typically better than 1
cm-1 over the course of the trajectories, using forces related to a numerical
(central-difference) gradient of the PES. An initial center-of-mass distance of 10 Å was
S6
used and trajectories were terminated when the products again reached this separation.
The cross-section for exchange for each reactant initial ji-value was evaluated as:
2
 j (T )   bmax
Fj (T ) ,
i
i
(10)
where Fji are the fraction of trajectories for a particular ji that undergo exchange and the
temperature dependence is only reflected in the collision energy. In order to compute rate
coefficients, the cross-sections for each ji are first combined by weighting according to
the temperature dependent rotational partition function and then the following formula is
applied:
k (T )  Qel1
8k B T

 (T ) ,
(11)
where μ is the reduced mass of O and O2, and Qel is the electronic partition function
used in the quantum calculations. The rates from the QCT calculations are plotted in
Figure S4 and S5. At 300 K the QCT rate for 16O + 32O2 is very similar to the quantum
result and both are well within the experimental error bars. The QCT thermal rate
coefficients is a little smaller than the quantum result, especially at 300K, but also
exhibits more pronounced negative temperature dependence.
References:
1
2
3
4
5
6
7
8
9
10
R. Dawes, P. Lolur, A. Li, B. Jiang and H. Guo, J. Chem. Phys. 139, 201103 (2013).
M. Lepers, B. Bussery-Honvault and O. Dulieu, J. Chem. Phys. 137, 234305
(2012).
R. N. Zare, Angular Momentum. (Wiley, New York, 1988).
Z. Sun, W. Yang and D. H. Zhang, Phys. Chem. Chem. Phys. 14, 1827 (2012).
Z. Sun, S.-Y. Lee, H. Guo and D. H. Zhang, J. Chem. Phys. 130, 174102 (2009).
J. V. Lill, G. A. Parker and J. C. Light, J. Chem. Phys. 85, 900 (1986).
Z. Sun, X. Lin, S.-Y. Lee and D. H. Zhang, J. Phys. Chem. A 113, 4145 (2009).
Z. Sun, H. Guo and D. H. Zhang, J. Chem. Phys 132, 084112 (2010).
A. Kuppermann, G. C. Schatz and M. Baer, J. Chem. Phys. 65, 4596 (1976).
S. D. Chao, S. A. Harich, D. Xu Dai, C. C. Wang, X. Yang and R. T. Skodje, J. Chem.
Phys. 117, 8341 (2002).
S7
11
12
13
14
15
J. Z. H. Zhang and W. H. Miller, J. Chem. Phys. 91, 1528 (1989).
D. G. Truhlar, J. Chem. Phys. 65, 1008 (1976).
L. Bytautas, N. Matsunaga and K. Ruedenberg, J. Chem. Phys. 132, 074307
(2010).
X. Hu, W. L. Hase and T. Pirraglia, J. Comp. Chem. 12, 1014 (1991).
W. L. Hase, R. J. Duchovic, X. Hu, A. Komornicki, K. F. Lim, D.-H. Lu, G. H.
Peslherbe, K. N. Swamy, S. R. V. Linde, A. Varandas, H. Wang and R. J. Wolf,
Quantum Chemistry Program Exchange Bulletin 16, 671 (1996).
S8
Table S1 Numerical parameters used in the wave packet calculation (all parameters are
given in atomic units unless specified)
Grid range and size:
R  [0.3, 16.0], N Rtot  255 , N Rint  161
r  [1.5, 14.0], N rtot  161 , N rint = 31
Initial wave packet:
R0 = 10.0, d = 0.12 , E0 = 0.25 eV
Projection plane:
R0  7.5 ,
Propagator:
Total propagation time:
4A6a(a)
400,000 a.u. with Δt = 120.0 a.u.
(a) For the parameters of 4A6a, see Ref. 4 and references therein.
S9
Table S2 Reactant rotational energy levels and rotational partition function for 16O16O
ji
0*
1
3
5
7
9
11
13
15
17
19
21
Energy (cm-1)
791.64
794.51
808.88
834.74
872.09
920.93
981.24
1053.01
1136.23
1230.90
1336.99
1454.49
Weight from rotational partition function
20 K
100 K
200 K
300 K
0.487
0.118
0.061
0.041
0.404
0.224
0.127
0.089
0.099
0.243
0.166
0.124
0.009
0.194
0.173
0.141
0.000
0.121
0.154
0.142
0.000
0.062
0.121
0.128
0.000
0.026
0.085
0.107
0.000
0.009
0.054
0.082
0.000
0.003
0.031
0.059
0.000
0.001
0.016
0.040
0.000
0.000
0.008
0.025
*forbidden (see S1-2 on nuclear spin effects)
S10
Figure S1. Comparison of the calculated integral cross sections for ji=1 and ji=0.
S11
Fig. S2 Comparison of the ICSs of product with odd and even rotational states, with initial
state as (vi, ji)=(0, 0) for reaction of 16O + 32O2.
S12
Fig. S3 Rotational state distributions of O2 in R1, R2 and R3 at collision energy of 0.1
and 0.2eV. Green line are Fermi-Dirac rotational state distributions of R1 reaction.
S13
Fig. S4. Plots of total integral cross sections for R1 as a function of reactant rotational
excitation ji obtained by QCT calculations at 100, 200, and 300 K. Exact quantum
scattering (EQS) results are shown for ji=0 for comparison and agree quite closely with
the QCT results.
S14
Fig. S5. The QCT averaged thermal rate coefficients are presented along with the data
shown in Fig.3 for comparison.
S15