DSpace at VNU: Theoretical investigations of BBS (singlet)- BSB (triplet) transformation on a potential energy surface obtained from neural network fitting

Accepted Manuscript
Theoretical Investigations of BBS(singlet)→BSB(triplet) Transformation on a
Potential Energy Surface Obtained from Neural Network Fitting
Hieu T. Nguyen-Truong, Cao Minh Thi, Hung M. Le
PII:
DOI:
Reference:

S0301-0104(13)00368-6
/>CHEMPH 8959

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Chemical Physics

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Accepted Date:

16 April 2013
21 September 2013

Please cite this article as: H.T. Nguyen-Truong, C.M. Thi, H.M. Le, Theoretical Investigations of
BBS(singlet)→BSB(triplet) Transformation on a Potential Energy Surface Obtained from Neural Network Fitting,
Chemical Physics (2013), doi: />
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Theoretical Investigations of

BBS(singlet)→BSB(triplet) Transformation on a
Potential Energy Surface Obtained from Neural
Network Fitting
Hieu T. Nguyen-Truonga,c , Cao Minh Thib , Hung M. Lea,∗
a

Faculty of Materials Science, University of Science, Vietnam National University, Ho
Chi Minh City, Vietnam
b
Nano-Materials Laboratory, Ho Chi Minh City University of Technology, Ho Chi Minh
City, Vietnam
c
Faculty of Electronics and Computer Science, Volgograd State Technical University, 28
Lenin Avenue, Volgograd 400131, Russia

Abstract
B2 S, the simplest Bn Sm cluster, has been shown to exhibit an interesting
ground-state structure. B3LYP/6-311G(d,p) calculations suggest that its
most stable configuration is singlet linear B-B-S. When promoted to the excited triplet state, B2 S adopts the B-S-B configuration (C2v point group). To
characterize its structural transformation, the lowest energy at each configuration is selected, and the neural network surface is developed with symmetry
exchange incorporated. The triplet potential energy is found to be 0.48 eV
above the ground state. Subsequently, the nudged-elastic-band method is
employed to locate the BBS→BSB transition state. It is found that the barrier height is 1.35 eV above the equilibrium singlet BBS energy (0.88 eV for
the reverse reaction). In addition, quasiclassical molecular dynamics with
different vibrational excitations shows that the reaction is accelerated when
the bending vibrational mode of BBS is excited, while the activation of BBS
stretching modes causes a negative effect.
Keywords: B2 S, electronic structure calculations, feed-forward neural
network, symmetry exchange, potential energy surface, molecular dynamics
∗

Corresponding author. Tel.: 84-838-350-831.
Email address: (Hung M. Le)

Preprint submitted to Chemical Physics

August 2, 2013


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1. Introduction
Boron-sulfur clusters (Bn Sm ) have attracted attention of many researchers
due to its remarkable bonding and structural properties. For a long time,
there have been significant efforts to investigate the structures of boron sulfides, which include B12 S [1], B2 S3 [2–4], (BS2 )n [4, 5], boron-sulfur ring
structures [6, 7], boron-sulfur heterocycle [8, 9]. The electronic structures
of boron-sulfur compounds are investigated based on ab initio calculations,
which include π-B8 S16 [10], the boron-sulphur heterocycle [11] and the hybrid
chalcogenoborate anions [12].
The formation of the Bn Sm cluster is fundamentally interesting and has
high potential applications. In a previous study reported by Pham et al.
[13], the simplest clustering structure, B2 S, was surprisingly shown to be
rather stable in the form of B-B-S instead of B-S-B. Recall that B2 O, a
closely-related structure to B2 S, adopts the D∞h symmetry (B-O-B) as the
most stable configuration at its singlet ground-state [14–17], while the triplet

structure of B-B-O has been shown to be less stable. Therefore, we believe
that it is significant to carry out an investigation to inspect the relative phase
stability of B2 S structures in both singlet and triplet spin states. The detailed approaches in this study are as following: we first construct a global
potential energy surfaces (PES) based on electronic structure calculations using the neural network (NN) fitting [18] method, then we employ a numerical
optimization method to find the intermediate pathway of BBS→BSB transformation. Finally, quasiclassical molecular dynamics (MD) simulations are
executed to estimate the statistical reaction probability of BBS→BSB transformation in correspondence with different vibrational excitations.
During the past two decades, artificial NN has become an advanced fitting
tool because of its robustness and accuracy in numerical fitting. Indeed,
NN has been widely applied in PES construction for gas phase molecules
and condensed-phase systems [19–29]. The development of such a NN PES
for a particular molecular system is simple, which is produced by fitting
(training) a set of energy data obtained from electronic structure calculations
as a function of input parameters (which adequately describe the molecular
configuration). The resulted NN is then employed to predict energies and
gradients (forces) during classical MD simulations with high accuracy. There
are extensive reviews of NN applications in PES fitting, which are available
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in the literature [24, 30, 31].
In fact, when developing NN PESs, we realize that in many molecular
systems, exchange symmetry (permutation) of atoms of similar identity become very critical. In many cases, the permutation of two atoms of similar
identity (or equivalently, a specific set of input parameters are exchanged
with another set) would not cause any changes in the overall energy. Thus,
improving NNs to handle such a critical issue is an important and challenging

task which has a crucial effect on fitting accuracy. In order to incorporate
symmetry exchange, Prudente et al. [19] employed a so-called “symmetric
neurons” in their multi-layer perception to study the vibrational levels of
H+
3 . In another approach, Lorenz et al. [20] proposed the symmetry-adapted
coordinates as inputs to the NN and illustrated the accuracy and efficiency
of their method by studying H2 interacting with the (2×2) potassium covered Pd(100) surface. Behler and Parrinello [21] introduced a new type of
symmetry adaptation and used a cutoff function of interatomic distances to
define the energetically relevant local environment. Consequently, in their
approach, the output of each symmetry function reflects the energy of a
particular local environment. For years, this technique has been applied in
various studies for molecular-surface interaction PESs [22–24, 32]. An alternative and simple solution to incorporate symmetry exchange is multiplying
the training database, as employed in several previous studies of HOOH [26],
ClOOCl [33], BeH3 [27], and ozone [28]. However, this treatment has a major
limitation, which is further extending the database, and consequently lower
fitting accuracy.
Recently, we have presented a novel method to deal with symmetry exchange by modifying the structure of the first layer in a feed-forward NN
model [29]. This new approach was successfully employed to construct PESs
for the H2 O and ClOOCl molecular systems. In this work, we will employ
such a technique [29] to develop a global NN for B2 S that fully characterizes
the BBS→BSB transformation. The strategy to construct a NN PES is simple. First, electronic structure calculations at singlet and triplet for every
configuration are performed, then the lowest energy at each configuration is
simply seleted. Then, we employ a NN with symmetry exchange to fit the
PES.

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2. Developing a PES for B2 S
To construct a PES for B2 S, we first perform electronic structure calculations to obtain a database of 3,456 points. The PES is built with two S-B
bonds varying in the range of 1.373 ˚
A to 3.609 ˚
A and the BI -S-BII angle vary◦
◦
ing in the range of 0 to 180 . Then, the PES is fitted using a two-layer feedforward NN with symmetry adaptation [29]. Based on the resulted PES, we
will employ the Nudged Elastic Band (NEB) algorithm [34, 35] to locate the
intermediate transition state between BBS and BSB configurations. Subsequently, MD investigations at various levels of internal energy are performed
to statistically determine the reaction probability of BBS→BSB transformation.
2.1. Electronic structure calculations
All electronic structure calculations in our study are carried out using
the GAUSSIAN 03 package [36] with the B3LYP [37–40] level of theory and
the 6-311G(d,p) [41, 42] basis sets. Each B2 S structure is carefully examined
to find the most stable spin state by comparing the singlet, triplet, and
quintet energies. Therefore, for each geometric configuration, three different
calculations are executed with various multiplicity states (1, 3, 5).
Surface crossing (hopping) in molecular dynamics of nonadiabatic systems
has become an important issue for a long time. There are several treatments
developed to handle this issue, which include the wave function approach
suggested by Martinez et al. [43, 44] and electronic-state switching probability
approach by Tully et al. [45, 46]. In this study, to simplify the PES, the spinorbit coupling is omitted, and we assume that the molecule of interest can
switch instantaneously to the most favourable spin state (lowest in energy)

during a MD trajectory. In other words, the molecule is assumed to choose
the spin state with lowest energy on the PES. This assumption was also
adopted in two previous studies of SiO2 [47] and ozone [28].
As mentioned earlier, we have performed a grid scan on three internal
coordinates. By adopting the above assumption, we have constructed a
database with 3,456 geometric configurations with their corresponding energies.
The BBS linear structure is the most stable configuration with singlet
spin-state and two B-S distances being 1.613 ˚
A and 3.371 ˚
A. When B2 S is
promoted to the excited state (triplet spin state), the structure adopts C2v
symmetry with two equivalent B-S bonds being 1.813 ˚
A and the BI -S-BII
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bending angle being 52.78◦ . According to our B3LYP/6-311G(d,p) calculations, the singlet structure is more stable than the triplet state by 0.48 eV.
For illustration, the equilibrium structures of singlet BBS (ground state) and
triplet BSB (excited state) are shown in Fig. 1. Observationally, the expectation value of spin (S2 ) in the singlet BBS and triplet BSB cases are 0.00 and
2.01, respectively. To validate the resulted equilibrium singlet and triplet
structures given by B3LYP calculations, we also perform ab initio calculations at a higher level of theory (U/UCCSD(T) [48–50], unrestricted openshell calculations) with the aug-cc-pVTZ basis set [51–53]. At convergence,

the resulted singlet structure is shown to be more stable than the equilibrium
triplet state by 0.64 eV. In geometry configuration, we also conceive good
agreements between CCSD(T)/aug-cc-pVTZ and B3LYP/6-311G(d,p) calculations. In the singlet linear structure, two B-S distances are found to be
1.620 ˚
A and 3.386 ˚
A, while in the triplet state with C2v symmetry, two B-S
bonds are identical and have the same length of 1.814 ˚
A, and the resulted
◦
bending angle is 53.20 . The geometry parameters and energies from B3LYP
and CCSD(T) calculations are given in Table 1.
In addition, vibrational analysis is performed to compute the theoretical
wavenumbers of all vibrational modes. In the linear BBS structure, three
vibrational wavenumbers are observed, one of which is the pi linear bending
of B-B-S angle, while the other two describe the symmetric and asymmetric
stretching of BBS. It should be noticed that the B-B-S bending is very low in
energy (only 25 cm−1 ) compared to the other two stretching modes. In the
triplet BSB non-linear structure (of C2v point group), we also observed three
vibrational modes, which include symmetric B-S stretching, B-S-B bending,
and asymmetric B-S stretching. The resulted wavenumbers from our electronic structure calculations are shown in Table 1. The vibrational wavenumbers of both singlet and triplet configurations are also predicted by CCSD(T)
calculations (shown in Table 1), and we observe good consistency with previous B3LYP calculations, except for the case of B-B-S pi linear bending. For
convenience, the equilibrium energies of singlet BBS, triplet BSB, and the
transition state are provided in Table 2.
It is revealed in our database that the quintet-state energy is always higher
than singlet and triplet energies. Therefore, quintet B2 S is not energetically
stable compared to singlet and triplet state configurations. Overall, we have
classified 2,668 configurations to favour the singlet spin state, while the remaining configurations (788 configurations) are more energetically stable at
the triplet spin state.
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In order to accurately determine the BBS→BSB transformation, we stepwisely examine the spin state of each B2 S structure during an MD trajectory.

Once a configuration turns to triplet (from the original singlet state at the beginning of a MD trajectory), a transformation is concluded to occur. Hence,
it is beneficial to store the favoured spin state of each configuration in the
database for later determination of product formation (triplet BSB).
2.2. Neural network architecture
In this study, a two-layer feed-forward NN [18] with modifications in its
structure to handle symmetry exchange is employed to fit the database of
3,456 configurations to construct a global PES for B2 S. Each configuration
is represented by three internal coordinates (variables), which are (R1 , R2 ,
θ), and its corresponding ab initio potential energy and spin state (singlet or
triplet). It can be easily seen that R1 and R2 variables have the same role
in the function, i.e. they can be switched without changing the potential
energy. The B-S-B bending angle, θ, ranges from 0◦ to 180◦. According to
our definition, when B2 S adopts the B-B-S linear structure, θ is 0◦ ; in the
other hand, θ is 180◦ in the B-S-B linear structure.
In the first layer of a traditional feed-forward NN, a sigmoid function
is employed as the transfer function, while a linear function is used as the
transfer function in the second layer (output layer). The use of these two
functions are believed to work efficiently in NN fitting to approximate analytic functions [55]. In our modified NN model, we only make changes on the
first layer where input parameters are first introduced into the NN.
In the NN PES, the interchange of two input parameters R1 , R2 should
not have an effect on the NN output. Therefore, in the first layer, those two
input variables need to be pre-processed to give three signals x1 , x2 , x3 :
x1 = R1 + R2 ,

(1a)

1
1
+
,

R1 R2

(1b)

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x2 =
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x3 = logsig (R1 ) + logsig (R2 ) ,
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(1c)

and the bending angle is then converted to be the fourth signal:
x4 = cos θ.
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From the above pre-processing equations, it can be easily seen that the interchange of R1 and R2 does not affect x1 , x2 , and x3 . Beside R1 and R2 , we
use cos θ as the last input parameter, which is simply denoted as x4 . Four
signals x1 , x2 , x3 , and x4 and the target energy in the database are then

scaled in range of [−1, 1] by the following expressions:
pi = 2

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xi − xmin
i
− 1,
xmax
−
xmin
i
i

(2a)

and

E − E min
− 1,
(2b)
Es = 2 max
E
− E min
where xmin
and xmax
are the minimum and maximum values of signal xi ,
i
i
min
respectively; E
and E max are the minimum and maximum values of the
energy E, respectively. For convenience, we choose E min to be 0, while E max
is 2.00 eV. Recall that when we perform vibrational analysis for the singlet BBS configuration, the zero-point vibrational energy of BBS is given as
0.137 eV, which is much lower than the chosen maximum energy. Therefore,
we believe such a selection of energy range is sensible for MD simulations.
The choice of a wide energy range would result in lower fitting accuracy.
In Table 3, the minimum and maximum values of x1 , x2 , x3 , x4 , and E
are shown. The structure of the two-layer feed-forward NN that can handle
symmetry exchange of R1 and R2 is illustrated in Fig. 2.
In this study, there are 35 hidden neurons used in the NN structure. Four
signals x1 , x2 , x3 , and x4 are converted in the first layer to become the signals
for the second (output) layer by the following expressions:
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a1j

1
wi,j
pi + b1j

=f

,

j = 1, . . . , 35.

(3)

i=1
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where wi,j and bj are the weight and bias values of the first layer, respectively;
f is a hyperbolic tangent function which is utilized to provide curvatures to
the NN function. The signals a1j are later used as an input value for the
second layer:
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wj2 a1j + b,

Es =

(4)

i=1
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wj2

where
and b are the weight and bias values of the second (output) layer,
respectively; Es is the scaled potential energy. When the fitting accuracy
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is analysed, we obtain a very good root-mean-squared error (RMSE) as
0.011 eV (0.257 kcal/mol). In terms of spin, the RMSE of the singlet PES
is 0.007 eV (0.166 kcal/mol), while fitting the triplet PES results in higher
RMSE (0.019 eV or 0.442 kcal/mol). To illustrate such fitting accuracy, a
set of 50 randomly-selected configurations is tested and shown in Fig. 3. It
is clearly shown that excellent agreements are found between the calculated
B3LYP energies and NN-predicted energies in the plot.
Beside developing a two-layer feed-forward NN to fit the analytic PES,
we also construct a pattern-recognition (PR) NN [18] for spin recognition
purposes. As discussed earlier, spin-orbit coupling is ignored during MD process, and B2 S is allowed to switch instantaneously to the most energeticallyfavoured spin state. Therefore, the objective of developing such a PR NN
is for predicting the correct spin state of a B2 S configuration. During MD
simulations, we make an assumption that if B2 S switches to the triplet state,
the configuration has entered the “excited region”, and we can conclude that
a BBS→BSB reaction just occurs.

The structure of PR NN is very similar to that of the two-layer feedforward NN employed to fit the PES. In fact, there is one major distinction,
which is the use of a hyperbolic tangent function and a round-off function
in the second layer (instead of only using a linear function in the analytic
NN). The use of a round-off function produces discrete outputs, which makes
PR NN capable of classifying categories.
At the end of the training process, the predicting accuracy of PR NN
for the training set is reported to be higher than 99%. After training the
PR NN, we also validate its classifying ability by examining the predicting
accuracy of an independent testing set. The predicting precision of such
PR NN is excellent when a set of 50 random samples is tested as shown in
Fig. 4. As a result, we believe that this PR NN can be satisfactorily utilized
to examine multiplicity change during MD simulations, and thereby predict
the formation of triplet BSB (product).

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3. Finding BBS→BSB transition state

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The NEB algorithm [34, 35] is a powerful technique that has been applied to locate theoretical transition states (saddle points) for chemical reactions/phase transformations. During the past several years, it has been
continuously developed in order to work more efficiently with gas-phase

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molecules and solid-state materials. In this study, we employ the NEB algorithm to locate the transition state for BBS→BSB transformation on our
newly-developed NN PES.
In order to accurately locate the transition state, we perform a series of 20image optimizations (excluding the initial singlet BBS and final triplet BSB
images (configurations)). From the forces given by NN PES gradients, the
NEB forces of image ith are calculated. Subsequently, intermediate image ith
is iteratively adjusted according to the resulted NEB forces. At convergence,
the transition state is located at 1.35 eV above the BBS equilibrium energy, as shown in Fig. 5. Recall that at the equilibrium singlet state (ground
state), the vibrational zero-point energy of BBS singlet configuration is found
to be 0.137 eV from our electronic structure calculations. Therefore, it can
be easily seen that a large amount of energy (much higher than the zeropoint energy) is required for the BBS configuration to overcome the barrier
and transform to the excited BSB triplet configuration. In other words, the
ground state BBS structure is very stable. While the triplet equilibrium potential energy is 0.48 eV above the singlet equilibrium potential energy (most
stable), the barrier height for the backward BSB(triplet)→BBS(singlet) reaction is predicted to be 0.88 eV.
At the transition state, two B-S chemical bonds are found to be 1.831 ˚
A
and 2.108 ˚
A. When those distances are compared to the B-S equilibrium bond
in the BBS singlet configuration (1.613 ˚
A), one B-S bond is found to be highly
stretched, while the other B-S is slightly stretched. The B-S-B bending angle
is found to be 42.6◦ , which approaches closely to the equilibrium bending
angle in the BSB triplet configuration (52.8◦ ).

Since we make an assumption to simplify the PES and omit spin-orbit
coupling, the singlet and triplet energies at the transition state must be similar. In other words, the spin state may switch instantaneously in the sensitive
transition region. However, there is uncertainty in the fit, and we notice that
the triplet energy is somewhat lower. When we re-perform electronic structure calculations for the transition state configuration, it is found that the
triplet-state energy (1.32 eV) is in excellent agreement with the NN-predicted
potential (1.35 eV), while the singlet energy is higher. In addition, an imaginary wavenumber of -270.6 cm−1 is observed when vibrational analysis is
executed for the transition structure with triplet state calculations. Similarly, when we perform vibrational analysis with singlet state calculations, an
imaginary wavenumber of -570.8 cm−1 is observed. CCSD(T)/aug-cc-pVTZ
calculations suggest that the transition state (predicted by B3LYP/NN re9


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sults) favours the singlet state, and the barrier height for BBS→BSB is reported to be 0.99 eV, while the barrier height for the reverse reaction is
0.35 eV. When triplet CCSD(T) calculations are executed, we observe that
the transition state is 1.55 eV higher than the equilibrium singlet state and
0.91 eV higher than the equilibrium triplet state.

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4. Molecular dynamics of BBS→BSB transformation

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After constructing a NN PES and successfully locating the transition
state, we perform quasiclassical MD simulations to investigate transformation

probability at various levels of internal energy. During MD trajectories, the
differential equations of nuclei motions are simultaneously integrated using
the fourth-order Runge-Kutta method with a fixed step-size of 0.1018 fs. In
total, we examine the reaction probability of each trajectory for a period of
5 ps. To obtain good statistical reaction probability, 1,000 sample trajectories
are investigated for each excitation case.
In order to obtain a configuration with randomized atomic momentum,
a sample MD trajectory is performed at zero-point total energy (0.137 eV).
From the initial singlet BBS configuration, which has a linear structure with
two B-S bonds being 1.613 ˚
A and 3.371 ˚
A, vibrational energy of each mode is
introduced into the molecule using the projection method [54]. Subsequently,
a MD trajectory is performed for a random period of time (2,000-3,000 steps).
After this process, the initialization of B2 S is completely prepared, and the
structure is guaranteed to have random configuration and momenta. After providing random coordinates and momenta to all atoms, we again employ the projection method to insert an excitation energy to four vibrational
modes [54]. At this stage, the molecular system is now ready for executing
MD simulation.
After the initial stage of randomization, B2 S is still in the singlet state.
During the 5 ps investigation period, the spin state is carefully examined
using the PR NN. If a triplet state is recognized by the PR NN during the
process, we conclude that a phase transformation to BSB triplet state just
occurs, and the MD trajectory is terminated. Otherwise, the trajectory is
allowed to run for the whole period of 5 ps. In Fig. 6, a successful BBS→BSB
conversion is illustrated in a MD trajectory. Before the reaction occurs, the
BI -S bond tends to vibrate around its equilibrium position. Based on the
resulted wavenumbers of singlet BBS stretching, the vibrational periods are
expected to be 59 fs and 25 fs. We would then expect that R1 vibrates more
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rapidly with a short period. However, this is not the case for two reasons.
First, the behaviour of BBS after 20 fs (as illustrated) brings the potential
energy to about 0.9 eV, which is much higher than the zero-point energy.
Therefore, the vibrational motions at this stage are very different from the
ground-state vibrations. Moreover, most of the activation energy is consumed
to open up the B-S-B bending angle (as seen in the plot of θ, it increases
almost linearly in 30 fs, the BSB bending angle opens up and reaches a
maximum of 50◦ ). Also, we notice that the BII -S bond decreases rapidly to
2.3 ˚
A. The spin-state is maintained as singlet until the system overcomes
an energy barrier of 1.44 eV (which is somewhat higher than the minimallyrequired energy of 1.35 eV). When the potential energy decreases, we notice
a spin change signal from the PR NN, which indicates the occurrence of
phase transformation. At this stage, the singlet→triplet transition takes
place concurrently with the drop of potential energy; therefore, we conclude
that a reaction actually occurs. At the reaction state, the B2 S configuration
A, while θ = 40.82◦. Such a
A and 2.312 ˚
has two B-S bonds being 1.763 ˚
geometric configuration is relatively close to the transition state predicted
by our previous NEB optimization.

The B2 S linear structure has three vibrational modes as indicated in Table 1, and the degenerate bending mode plays an important role in the
BBS→BSB reaction. This mode represents bending behaviours of BBS,
which helps to open the B-B-S angle as well as shorten the BII -S distance.
Therefore, it is suspected that the excitation of such a low-energy vibrational mode would enhance the product formation. The other two vibrational modes (symmetric and asymmetric linear stretching of BBS), in the
other hand, is suspected to contribute less significantly in BBS→BSB conversion although they have high vibrational energies. In fact, this statement is
true according to the evidences shown in Table 4, in which various excitation
energy levels (with specific excited modes) and the corresponding BBS→BSB
probabilities are shown. For each study case, 1,000 sample trajectories are
investigated, and the reaction probability is computed based on spin-state
tracking signals given by the PR NN in a period of 5 ps.
From the obtained results, when the B-B-S bending mode is highly excited, we obtain higher reaction probabilities. In case 1, the bending mode
is least activated; as a consequence, we obtain the lowest BBS→BSB probability results. When the bending mode is further excited as in case 2, the
probability increases by a small amount. Surprisingly, the highest reaction
probability (case 3) does not come from the case with highest activation en11


373

ergy. In case 3, we can see clearly that the bending mode is highly excited,
while the other two modes are less activated. From case 4 to case 6, we keep
activating the bending mode at the same level as case 3, while the two BBS
stretching modes are further excited. Interestingly enough, the reaction probabilities in those cases tend to decrease even though an additional amount of
excitation energy is introduced into the system. Therefore, we can conclude
at this point that further activation of two stretching modes would cause a
significant negative effect on the conversion of BBS→BSB. This conclusion
can be implied again as shown by the evidence of case 2. In case 2, even the
highest excitation energy is introduced, the reaction probability is still lower
than case 3, because the stretching mode is highly excited. Although such
a bending mode is low in energy (only 25 cm−1 , its activation with a small
amount of energy is very effective in BBS→BSB conversion.

We also notice a non-mode-specific behaviour in this study as we compare the reaction probability. From the lowest excitation energy (2.413 eV)
to highest excitation energy (2.937 eV), the probabilities of complete transformations range from 0.469 to 0.620, which are not very different. Therefore,
we believe that the activation energy is spread rapidly during MD process,
and different activations do not cause huge differences on reaction probability
results.
In each investigating case, we continue to perform MD after singlet BBS is
converted to triplet BSB. Then, by employing the PR NN, the lifetime of this
triplet state is measured in each trajectory before it is converted back to the
singlet ground state. In Table 4, we also show the average lifetime of triplet
BSB in each activation case. In general, it is shown this excited state has
a less-than-11-fs lifetime, which proves that it is very unstable. As a result,
triplet BSB cannot be acquired by spectroscopy with this extremely-short
lifetime.

374

5. Summary

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In this work, we have developed a PES for the B2 S cluster that is capable of describing the BBS→BSB transformation. According to electronic
structure calculations using the B3LYP [37–40] level of theory and the 6311g(d,p) [41, 42] basis sets, the spin state of B2 S can favour either singlet
or triplet. To simplify the PES and MD simulations, we make an assumption to omit spin-orbit coupling and allow B2 S to switch instantaneously and
simultaneously to the most favoured spin state (with lowest potential en12

414

ergy). This assumption was previously adopted in the studies of SiO2 [47]
and ozone [28].
From singlet, triplet, and quintet calculations, all quintet energies are
shown to be higher than singlet and triplet energies. Thus, the quintet spin
state is believed to be unstable for B2 S. As a result, we have classified 2,668
configurations in the singlet state and 788 configurations in the triplet state
(in total, 3,456 configurations are selected with their corresponding energies
and spin-states recorded in the database).
A two-layer feed-forward NN incorporating symmetry exchange [29] is
employed to numerically fit the energies and produce an analytic NN PES.
The NN fit is shown to exhibit very good fitting accuracy when RMSE of
a testing set is analysed (which is 0.011 eV or 0.257 kcal/mol). In order to
examine the formation of product, we construct a PR NN to examine the
spin state during MD trajectories. The predicting accuracy of such a PR NN
is reported to be above 99%. Such accuracy is sufficient to provide a solid
foundation to determine product formation.
The intermediate transition state for BBS→BSB reaction is numerically
optimized using the NEB algorithm [34, 35]. At convergence, it is reported
that the transition state is 1.35 eV above the equilibrium energy of singlet
BBS. This result is in good agreement with the triplet energy at the transition
state given by B3LYP calculations.
After exploring the transition state, we perform quasiclassical MD simulations for B2 S at different levels of excitation energy. It is observed from
MD evidences that the bending vibration of B-B-S has a significant effect
on the transformation to the excited triplet BSB configuration. When the
bending mode is highly excited, we obtain higher reaction probabilities. The
excitation of two stretching modes, however, causes a negative effect on the
BBS→BSB reaction. In addition, from the resulted probability data, we do
not see a mode-specific behaviour because different levels of excitation do

not result in huge differences in the reaction probability.
The MD trajectories are further extended to examine the lifetime of triplet
BSB. The average lifetime in each activation case shows that such a excited
triplet state is very unstable and has a lifetime shorter than 11 fs.

415

Acknowledgements

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We thank the Faculty of Materials Science, University of Science, Vietnam National University for their computing supports. We also acknowledge
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cogent discussions and helps from Dr. N. N. Pham-Tran, Faculty of Chemistry, University of Science, Vietnam National University in Ho Chi Minh
City.

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FIGURE CAPTION
Fig. 1. Equilibrium structures of B2 S at the singlet ground state and the
triplet excited state.
Fig. 2. Two-layer feed-forward NN incorporating symmetry exchange at
the input layer. The output may be energy or spin-state (singlet/triplet).
Fig. 3. Testing energy fits using the NN PES for a set of 50 random
configurations.
Fig. 4. Spin prediction using PR NN with 50 random configurations. The
prediction accuracy is 100% for this testing set.
Fig. 5. Searching for the transition state using the NEB algorithm. In
this optimization, 20 intermediate images are simultaneously optimized. At
convergence, we obtain a transition structure which is 1.35 eV above the
equilibrium energy of BBS.
Fig. 6. B-S bonds, bending angle, and energy monitoring during a MD
process of B2 S: singlet – dashed line, triplet – solid line.

15


Figure 1:

16


Figure 2:

17


2

Energy (eV)

1.5

1

0.5
B3LYP
NN
0
0

10

20

30

Figure 3:

18

40

50


Spin state

Triplet

B3LYP
NN

Singlet
Figure 4:

19


Potential Energy (eV)

2

1.5

1

0.5

0

5

10
15
Reaction Coordinate
Figure 5:

20

20


1.8

3.5

1.75

1.65

R 2 (˚
A)

R 1 (˚
A)

3.2

2.9

2.6

1.55

2.3
1.45
0
50

10

20

30

40

50

60

0

1.5

30

20

10

0
0

20

30

40

50

60

10

20

30
Time (fs)

40

50

60

1.2
Potential Energy (eV)

θ (deg)

40

10

0.9

0.6

0.3

10

20

30
Time (fs)

40

50

60

Figure 6:

21

0
0


436

TABLES
Table 1: Equilibrium structure (in ˚
A or deg) and vibrational wavenumber (cm−1 ) of
singlet BBS and triplet BSB given by B3LYP/6-311g(d,p) and CCSD(T)/aug-cc-pVTZ
calculations.

B3LYP/6-311G(d,p)
CCSD(T)/aug-cc-pVTZ

B3LYP/6-311G(d,p)
CCSD(T)/aug-cc-pVTZ

Singlet BBS
Geometry
BBS pi
R1
R2
θ

linear bend
1.613 3.371 0.00
25
1.620 3.386 0.00
179
Triplet BSB
Geometry
Asymmetric
R1
R2
θ
B-S stretching
1.813 1.813 52.78
651
1.814 1.814 53.20
660

22

Symmetric
stretching
566
567

Asymmetric
stretching
1346
1330

B-S-B

bending
735
735

Symmetric
B-S stretching
1065
1045


Table 2: Potential energies (eV) of singlet BBS, triplet BSB, and the transition state from
B3LYP/6-311g(d,p) and CCSD(T)/aug-cc-pVTZ calculations.

BBS
B3LYP/6-311G(d,p)
CCSD(T)/aug-cc-pVTZ

Singlet
Triplet
Singlet
Triplet

BSB

0.00
0.48
0.00
0.64

23

Transition
state
2.27
1.33
0.99
1.55


Table 3: Minimum and maximum values of x1 , x2 , x3 , cos θ, and E.

Minimum
Maximum

x1
x2
x3
cos θ
E (eV)
3.1685 0.7724 1.6596 -1.0000 0.0000
5.6213 1.2627 1.8565 1.0000 1.9999

24