Supplementary Material for
Kinetic study on the H+SiH4 abstraction reaction using an
ab initio potential energy surface
Jianwei Cao, 1,2,a) Zhijun Zhang, 1,2,a) Chunfang Zhang, 1,2 Wensheng Bian, 1,b)
and Yin Guo 3,c)
1
Beijing National Laboratory for Molecular Sciences, State Key Laboratory of
Molecular Reaction Dynamics, Institute of Chemistry, Chinese Academy of Sciences,
Beijing 100190, China.
2
Graduate University of the Chinese Academy of Sciences, Beijing 100049, China
3
Department of Physics, Oklahoma State University, Stillwater, Oklahoma 74078,
USA.
a)
These authors contributed equally to this work.
b)
Electronic mail: bian@iccas.ac.cn.
c)
Electronic mail: yin.guo@okstate.edu.
The construction of the PES
The present PES is constructed using a dual-level strategy, which contains two
parts: the lower-level reference surface V0 and the difference surface V , thus the
final potential V is the sum of the two, V  V0  V .
The lower-level reference surface V0 is constructed with the modified Shepard
interpolation scheme devised by Collins and co-workers, 1-4 in which the PES is given
by an interpolation of second-order Taylor expansions centered at data points
scattered throughout the configuration space of the system. This scheme has been
presented in detail elsewhere, 1-3, 5 and only a brief description is presented here.
For a molecular system with N atoms at a specific electronic state, we can use
( N ( N  1) / 2) inverse interatomic distances Qi  1 Ri to define the system, where
Ri is the ith interatomic distance. Thus, any configuration of the system can be
described by a vector Q  Q1 , Q2 , K , QN ( N 1) / 2  . A vector of 3N  6 independent
internal coordinates,  (Q) , can be defined in terms of the vector Q via a singular
value decomposition:
n 
N ( N 1) / 2

k 1
(n  1 , 2K, N, 
3
U nk Qk
. 6)
(1)
The potential energy at a given configuration, Q , in the vicinity of any data
point Q(i ) , can be expanded as a Taylor series to second-order, Ti (Q) ,
Ti (Q)  V [Q(i)] 

3 N 6
 [
k 1
k
  k (i)]
V
 k
Q Q ( i )
1 3 N 6 3 N 6
 2V
[



(
i
)]

[



(
i
)]
 k k
j
j
2 k 1 j 1
 k  j
2
.
Q Q ( i )
(2)
If the required energy and derivatives have been evaluated at each of N data
molecular configurations, the modified Shepard interpolation gives the total potential
energy at any configuration Q as a weighted average of the Taylor series about all
N data points and their symmetry equivalents:
N data
V (Q)    wg oi (Q)Tg oi (Q) .
(3)
gG i 1
In Eq. (3), G denotes the symmetry group of the molecule, g o i denotes that the
ith data point is transformed by the group element g , g  G means that all
permutationally equivalent data points are included in the data set. Thus, the full
symmetry of the system can be accounted for by summing over the data points of the
PES data set and the symmetry equivalent points.
The weight w i in Eq. (3) gives the contribution of the ith Taylor expansion to
the potential energy at the configuration Q . It can be normalized to unity according
to
w i (Q) 
v (Q)
i
N data
v
g oi
.
(4)
(Q)
gG i 1
Here, the vi (Q) or “primitive weights” are functions of the configuration Q
with respect to the data point configuration Q(i ) . The form of the weight function
vi (Q) has been discussed in detail elsewhere. 1,3,5 In this work, we use a “two-part”
(q  2, p  15) weight function 3 which is derived via a Bayesian analysis of the data
set.
The locations and number of the reference data points are very important to the
quality of the interpolated PES. A majority of reference points were selected based on
3
classical trajectory simulations, and additionally several hundred points were selected
according to physical considerations, particularly the analysis of various potential
contour plots. Trajectory calculations were performed to check the convergence of the
interpolated PES with respect to the number of points in the data set. 6,7 Finally, a
reference data set with 3356 points was obtained, in which the first 3124 points are
the same as before, 7 and additional 232 points are added for the purpose of further
refinement of the PES for the abstraction reaction. At these 3356 reference
geometries, the spin-unrestricted Hartree–Fock (UHF) self-consistent field method
and the spin-unrestricted quadratic configuration interaction method with all single
and double excitations (UQCISD) with Dunning’s correlation consistent polarized
valence triple-zeta basis set (cc-pVTZ) are used, and calculations are performed with
the Gaussian 03 package 8 to obtain the electronic energies, gradients and Hessians for
the construction of the lower-level reference surface V0 . The spin contamination in
the above UHF-UQCISD scheme is small: the expectation value of the electron spin
angular momentum square ( or <S2>) for UHF at the abstraction transition state is
0.772; the average of <S2> values for 3356 points is smaller than 0.80 (the true value
is 0.75), which is generally regarded as acceptable. We also checked the
multireference diagnostics
9
in our calculations and concluded that they are
acceptable. In particular, the T1 diagnostic
9
for the above UHF-UQCISD scheme at
the saddle point for abstraction shows the value of 0.0164, which is much smaller than
0.025, indicating the reliability of the present single-reference correlation treatment.
4
Based on our previous calculations and convergence analysis, 7 we conclude that
the spin-restricted HF self-consistent field method, and further spin-unrestricted
coupled cluster method with single and double excitations and triple excitation
correction (UCCSD(T)) with the Dunning’s correlation-consistent polarized valence
quadruple-zeta (cc-pVQZ) basis set are suitable for the high-level energy calculations.
At the 3356 chosen reference geometries, the RHF-UCCSD(T)/cc-pVQZ calculations
are carried out using the MOLPRO 2002.6 package 10 to obtain the electronic energies.
Then the difference energies for these reference geometries are obtained:
V  V  V0  V (UCCSD(T)/cc-pVQZ)  V (UQCISD/cc-pVTZ) .
(5)
The second-degree local-IMLS method is used to fit these 3356 difference
energies. The L-IMLS method has been described in detail previously. 11,12 Briefly,
the potential energy at any configuration Q is a weighted sum of the N data local
approximants,
V (Q) 
Ndata
 w (Q) L
i
i 1
Qi
(Q) .
(6)
Here, LQi (Q) is the local approximant at the ith data point,
M
LQi (Q)   a j (Qi )b j (Q) ,
(7)
j 1
where
b (Q)
j
are polynomial basis functions, M is the number of basis functions,
and the coefficients
a 
j
are determined by minimizing the following weighted
deviation,
2
M

D   wi (Qi )  a j (Qi )b j (Qi )  V (Qi )  .
i 1
 j 1

Ndata
5
(8)
In the second-degree L-IMLS method, all polynomials up to quadratic terms are
included. Since there are 15 interatomic distances for the present six-atom system, the
number of basis functions M is 136. The weight function wi (Q) used in this study
is the same as that used in the previous work.
11,12
The parameters in the weight
function are chosen as follows. For the regular IMLS, which is used to calculate the
coefficients
a  ,
j
p  1 and N  136 . For L-IMLS, p  2 and N  40 . The
cutoff Rcut used for the IMLS is data-point dependent and defined as the distance to
the N cut th nearest point, and N cut  550 , which is approximately 4M . Since the
number of energy evaluations is not large in this study, a large fixed cutoff
Rcut  3.5  i
is used for the L-IMLS.
The error of the present dual-level PES is roughly estimated by using a random
sample of 585 configurations obtained from abstraction reactive trajectories initiated
at the H  SiH4 asymptote with a collision energy of 24.00 kcal/mol , the energy
range of which is from -11.55 kcal/mol to 29.62 kcal/mol relative to the H  SiH4
asymptote. The ab initio calculations at the UCCSD(T)/cc-pVQZ level are performed
at all configurations in this sample, and the error is computed from the difference
between the ab initio energy and the corresponding value obtained from the present
surface. The average absolute error is found to be 0.57 kcal/mol and the
root-mean-square (RMS) error is 1.04 kcal/mol . The collision energy of 24.00
kcal/mol is much higher than the classical abstraction barrier height (5.34
kcal/mol ), and thus many configurations in the above sample are far from the
6
minimum energy path (MEP) of the abstraction reaction, based on which the present
VTST calculations are performed.
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