Supporting information:
State-to-state Dynamics of the Cl + H2O → HCl + OH Reaction: Energy Flow into
Reaction Coordinate and Transition-state Control of Product Energy Disposal
Bin Zhao1, Zhigang Sun,2 and Hua Guo1,*
1
Department of Chemistry and Chemical Biology, University of New Mexico,
Albuquerque, New Mexico 87131, USA
2
Center for Theoretical and Computational Chemistry, and State Key Laboratory of
Molecular Reaction Dynamics, Dalian Institute of Chemical Physics, Chinese Academy
of Sciences, Dalian 116023, China
* Corresponding author: hguo@unm.edu
1
S1. Theory
S1.1. Hamiltonian and Wavefunctions
To study the title reaction at the state-to-state level, both the atom-triatom and diatomdiatom Jacobi coordinates are required. As shown in Fig. S1, the A+BCD atom-triatom
and the AB+CD diatom-diatom Jacobi coordinates are denoted as ( R, r1 , r2 ,1 ,2 ,  ) and
( R, r1, r2,1,2 ,  ) , respectively. The R and R vectors are defined as the z-axis of the
body-fixed (BF) frame for the A+BCD and AB+CD systems, respectively, and the vector
r1 and r1 lies in the corresponding x-z planes, respectively.
The Hamiltonians of both A+BCD and AB+CD systems have formally the same form
( =1 hereafter),1
ˆji2 
1  2 ( Jˆ  ˆj12 )2 2  ˆ
Hˆ  


h
(
r
)


 i i
  V ( R, r1 , r2 ,1 ,2 ,  ) ,
2 R 2
2 R 2
2i ri 2 
i 1 
(S1)
and the wavefunction can be expanded in terms of BF bases,
 JM  ( R, r1 , r2 , t ) 


 ˆ
ˆ ˆ
FnJMjK (t )un1 ( R)1 (r1 )2 (r2 ) y JM
jK ( R, r1 , r2 ) ,
(S2)
n, , j , K
where the detailed description of Hamiltonian and wavefunction can be found in Ref. 1.
S1.2. Thermal Flux Operator
 Hˆ /2 kBT ˆ  Hˆ /2 kBT
Fe
The definition of the thermal flux operator FˆT  e
is based on flux
operator Fˆ  i[ Hˆ  h ] with h as the Heaviside function.2 The flux operator F̂ is singular,
and its eigenvalues diverge with respect to the largest momentum for the grid spacing and
2
its eigenstates oscillate rapidly in the coordinate space. The Boltzmann operators in FˆT
remove the singularity and renders the thermal flux eigenstates smooth in the coordinate
representation. As a result, higher accuracy can be achieved for the coordinate
transformation.
The thermal flux operator can be expanded in the spectral representation,3
FˆT   fTn fTn
fTn 
(S3)
n
where fTn and fTn are the eigenvalues and eigenfunctions of the low-rank thermal flux
operator, respectively. The real eigenvalues fTn come in pairs with same absolute value but
different signs,4 and the corresponding eigenstates fTn come in complex conjugate pairs
and they are interpreted as the ro-vibrational eigenstates of the activated complex at the
transition-state region.
The eigenstates fTn are localized near the transition state region and calculated
on a diving surface, which are defined on the R  r1 plane as follows
q  (r1  r10 )  ( R  R0 ) tan   0
(S4)
where  is the angle of the line to the R axis and ( R0 , r10 ) is a point on the R  r1 plane.
The flux operator then can be expressed as
Fˆ  i[ Hˆ  h(q)]  FˆR  Fˆr1 ,
(S5)
where
3
i
 

FˆR 
tan    (q)   (q)  ,
2
R 
 R
(S6a)
i 
 
Fˆr1  
  ( q )   ( q)  ,
21  r1
r1 
(S6b)
are the fluxes in the R and r1 coordinates, respectively.
S1.3. Coordinate Transformation
TSWPs are calculated in the A+BCD Jacobi coordinate and their propagations into
reactant asymptotic region are straightforward, but the propagations into product
asymptotic regions require the transform of the TSWPs to the AB+CD Jacobi coordinates.
The coordinate transformation has been discussed in detail before.5-6 Noting that the r2 and
r2 coordinates use the same basis functions, so the transformation only need to be carried
out in the other five dimensions. The transformation can be briefly divided into two steps:
In the first step, TSWPs initially prepared in the A+BCD coordinates are interpolated on
the AB+CD coordinate grids with the corresponding basis functions. In the second step,
with TSWPs represented on the AB+CD grids, they are transformed to the basis
representation by a collocation method along the r1 coordinate and by transformation
matrices in other coordinates.
S2. Numerical Details
For the sake of efficiency, an L-shaped scheme7 is adopted to define the asymptotic
and interaction regions. In this scheme, the first
N R1 out of N R2 grids along the R
coordinates are used to define the interaction region, and the remaining grids are to define
4
the asymptotic region. For the r1 coordinate, N r11 and N r21 PODVR basis functions8 are used
for the interaction and asymptotic regions, respectively. The parameters used in the
calculations for the two reactions are listed in Table SI. Calculations of the Cl + H2O → HCl
+OH reaction were based on the recently constructed LDG PES,9-10 while calculations of the
H + H2O → H2 +OH reaction used the YZCL2 PES.11
Thermal flux operator was defined in the reactant A+BCD Jacobi coordinate, and the
corresponding eigenstates were obtained by using the ARPACK/PARPACK packages with
implicitly restarted Arnoldi methods,12 and the first NT pairs of thermal flux eigenstates were
efficiently obtained by iteratively applying the thermal flux operator on to the Krylov vectors.
The imaginary time propagation in the thermal flux operator used a second-order split
operator propagator,13 which was also adopted in the following real time propagation of
thermal flux eigenstates into the reactant and product asymptotic regions.
S3. Additional results
S3.1 Convergence
Convergence of the initial state-selected total reaction probability with respect to
the number of thermal flux eigenpairs is shown in Fig. S2 for the two reactions from the
rotationless H2O(100) reactant. Convergence for R1 reaction is rather slow, and with 50
pairs of thermal flux eigenstates the result is converged to a total energy of around 0.6 eV.
400 pairs are required to converge to the total energy of 1.0 eV, which is about 0.55 eV
above threshold. For R2 reaction, only 25 pairs are required to converge to the total energy
of 1.2 eV, which is also 0.55 eV above threshold.
In Fig. S3, the accuracy of the state-to-state reaction probabilities is checked by
5
comparing the sum of state-to-state reaction probabilities with results obtained by both
initial state-specific wave packet (ISSWP) method and TSWP method with only reactant
propagation. It is seen that all the results are in very good agreement, especially at low total
energies. The small discrepancy at high energies is caused by the lack of thermal flux
eigenpairs used in the calculation, and the convergence with respect to the number of pairs
is really slow at high energy, as shown in Fig. S2. Despite the discrepancy at high energies,
we found that it is sufficient to use 400 and 200 pairs of thermal flux eigenstates for R1
and R2 reactions, respectively.
S3.2 Product Energy Disposal
Figure S4 illustrates the energy disposal in both reactions. Specifically, Ek is the
HCl / H 2
OH
averaged translational energy between the products, and Erot
and Erot
are the averaged
rotational energies for the HCl/H2 and OH products, respectively, which are defined as
HCl / H 2
Erot


 
p1 ,2 , j1 , j2 , j12 s ,b ,a , J
1 , 2 , j1 , j2 , j12

 
Ka Kc
[ E HCl / H2 (1 , j1 )  E HCl / H 2 (1 ,0)]
p1 ,2 , j1 , j2 , j12 s ,b ,a , J
1 , 2 , j1 , j2 , j12
OH
Erot


 
1 , 2 , j1 , j2 , j12
p1 ,2 , j1 , j2 , j12 s ,b ,a , J

 
Ka Kc
a c
1 ,2 , j1 , j2 , j12
Ka Kc
[ E OH (2 , j2 )  E OH (2 ,0)]
p1 ,2 , j1 , j2 , j12 s ,b ,a , J
1 , 2 , j1 , j2 , j12
where ps ,b ,a , J K K
(S7a)
(S7b)
Ka Kc
are the state-to-state reaction probability, E HCl / H 2 (1 , j1 ) and
E OH (2 , j2 ) are the ro-vibrational energies of the HCl/H2 and OH products, respectively.
It is readily seen from the figure that the HCl product is rotationally hotter than its coproduct OH for the R1 reaction, and the relative translational motion receives more than
6
half of the available energy. Similarly, in R2 reaction, the H2 product is rotationally hotter
than its co-product OH, and the relative translational motion receives almost all the
available energy. The disposal of a large amount of energy into translational motion is in
accordance with the Sudden Vector Projection (SVP), which shows the SVP value for the
product translational mode to be 0.37 or 0.89 for R1 or R2, respectively, as shown in Fig.
1. The cold rotational distribution of the OH product in both reactions confirms its spectator
character, and the rotationally hot HCl/H2 product manifests the bended transition state
geometry. These patterns of product energy disposal also confirms the predictions of the
SVP model.
S3.3 Influence of Reactant Rotational Excitation
Figures S5 (a) and (b) show the total reaction probabilities from two rotational
states of H2O in the ground vibrational state for both reactions. Reaction probabilities of
different sub-states of the rotationally excited H2O are averaged. It is found the rotation
has little enhancement, especially for R1 reaction. Shown in Fig. S5 (c)-(f), the product
rotational state distributions are not significantly affected by the rotational excitation of the
H2O reactant.
S3-4. Contribution of Thermal Flux Eigenstates
Contributions of thermal flux eigenstates to the Cl + H2O(100,000) → HCl( 1  0, j1 )
+ OH( 2  0, j2 ) reaction at Etot = 0.915 eV are shown in Figure S6. The coupled rotational
number j12 is summed as it is not experimentally observable. It is clear that nearly all
thermal flux eigenstates contribute. The fact that the reaction probabilities are not strictly
cumulative with respect to the number of thermal flux states suggests strong interferences
7
among these states. This interference effect stems from the fact that the reaction probability
is expressed as the square of a coherent sum of all contributions from the thermal flux
states.14-15
8
Table S-I. Parameters used in the calculations of the two reactions
 (rad.)
R0
r10
R1:
Cl + H2O → HCl + OH
Thermal flux operator
0.10
4.00
2.50
R2:
H + H2O → H2 + OH
0.08
2.60
2.50
T (K)
No. of pairs
4000.0
4000.0
400
200
Parameters for Reactant Coordinate
Propagation time/step (a.u.)
24000/10
6000/10
Analyzing surface
R0 =16.0
R0 =10.0
Grid/basis range and size
R  (3.3,20.0),
R  (1.2,14.0),
R
1
2
N R =48, N R =264
N R1 =48, N R2 =120
r1  (1.1,6.0)
r1  (0.7,5.0)
N r11 =32, N r21 =8
N r11 =28, N r21 =8
r2
r2  (1.1,5.0)
N r2 =3
r2  (0.7,5.0)
N r2 =4
j2
j2  (0,24)
j2  (0,20)
j12
j12  (0,32)
j12  (0,28)
r1
Parameters for Product Coordinate
Propagation time/step (a.u.)
36000/10
8000/10
Analyzing surface
R0 =16.0
R0 =11.0
Grid/basis range and size

R  (3.2,20.0),
R   (1.7,14.0),
R
1
2
N R =48, N R =264
N R1 =32, N R2 =112
r1
r1  (1.5,6.0)
r1  (0.7,5.0)
N r11 =32, N r21 =6
N r11 =32, N r21 =8
r2
r2  (1.1,5.0)
N r2 =3
r2  (0.7,5.0)
N r2 =4
j1
j2
j1  (0,54)
j2  (0,18)
j1  (0,38)
j2  (0,20)
9
References:
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
B. Zhao, Z. Sun and H. Guo, J. Chem. Phys. 141, 154112 (2014).
T. P. Park and J. C. Light, J. Chem. Phys. 88, 4897 (1988).
F. Matzkies and U. Manthe, J. Chem. Phys. 106, 2646 (1997).
U. Manthe, J. Chem. Phys. 102, 9205 (1995).
M. T. Cvitaš and S. C. Althorpe, J. Chem. Phys. 134, 024309 (2011).
S. Liu, C. Xiao, T. Wang, J. Chen, T. Yang, X. Xu, D. H. Zhang and X. Yang,
Faraday Disc. 157, 101 (2012).
D. H. Zhang and J. Z. H. Zhang, J. Chem. Phys. 99, 5615 (1993).
J. C. Light and T. Carrington Jr., Adv. Chem. Phys. 114, 263 (2000).
J. Li, R. Dawes and H. Guo, J. Chem. Phys. 139, 074302 (2013).
J. Li, H. Song and H. Guo, Phys. Chem. Chem. Phys. 17, 4259 (2015).
M. Yang, D. H. Zhang, M. A. Collins and S.-Y. Lee, J. Chem. Phys. 115, 174
(2001).
R. B. Lehoucq, D. C. Sorensen and C. Yang, ARPACK User Guide: Solution of
Large Scale Eigenvalue Problems by Implicitly Restarted Arnoldi Methods.
(SIAM, Philadelphia, PA, 1998).
M. D. Feit, J. A. Fleck Jr. and A. Steiger, J. Comput. Phys. 47, 412 (1982).
B. Zhao and H. Guo, J. Phys. Chem. Lett. 6, 676 (2015).
R. Welsch and U. Manthe, Mol. Phys. 110, 703 (2012).
10
Fig. S1. The A+BCD Jacobi coordinates ( R, r1 , r2 ,1 ,2 ,  ) and the AB+CD Jacobi
coordinates ( R, r1, r2,1,2 ,  ) for tetra-atomic reactions. The angle between the two BF z
axes 𝑅 and 𝑅 ′ is denoted as ∆.
11
Fig. S2. Convergence of the initial state-selected total reaction probability with respect to
the number of thermal flux eigenpairs for R1 and R2 involving the rotationless H2O (100)
reactant.
12
Fig. S3. Initial state-selected total reaction probabilities of the two reactions obtained from
three methods: (1) the ISSWP method (black solid), (2) the TSWP method with only
reactant propagation (red dashed) and (3) the sum of TSWP state-to-state reaction
probabilities over all product states (blue dash dotted). The TSWP calculations used 400
and 200 pairs of thermal flux eigenstates for the R1 and R2 reactions, respectively.
13
Fig. S4. Disposal of the reaction energy in the rotational (Erot) and translational (Ek) modes
of the products for the two reactions. The product vibrational energies are essentially a
constant close to the sum of the zero-point energy. The total energy used in Fig 3 of the
main text is marked with gray dashed lines.
14
Fig. S5. Total reaction probabilities from two rotational states of H2O in ground vibrational
state: (a) and (b), and rotational state distributions of HCl/H2 and OH products: (c), (d), (e)
and (f) for R1 reaction: (a), (c) and (e), and R2 reaction: (b), (e) and (f). Reaction
probabilities of different sub-states of the rotationally excited H2O are averaged.
15
Fig. S6. State-to-state reaction probabilities of the Cl + H2O(100,000) → HCl ( 1  0, j1 ) +
OH( 2  0, j2 ) reaction for multiple product rotational states as a function of the number
of thermal flux pairs used in the calculation. The total energy is chosen to be 0.915 eV,
which is the same as in Fig. 3 of the main text. The coupled rotational number j12 is
summed as it is not experimentally observable.
16