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 http://dx.doi.org/10.1016/j.chemphys.2013.09.007 CHEMPH 8959 To appear in: Chemical Physics Received Date: 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: http://dx.doi.org/10.1016/j.chemphys.2013.09.007 This is a PDF file of an unedited manuscript that has been accepted for publication As a service to our customers we are providing this early version of the manuscript The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain 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@hotmail.com (Hung M Le) Preprint submitted to Chemical Physics August 2, 2013 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 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 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 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+ 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 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 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 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 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 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 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 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 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 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 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 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 + , R1 R2 (1b) 170 x2 = 171 x3 = logsig (R1 ) + logsig (R2 ) , 172 (1c) and the bending angle is then converted to be the fourth signal: x4 = cos θ (1d) 173 174 175 176 177 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 = 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 xi − xmin i − 1, xmax − xmin i i (2a) and E − E − 1, (2b) Es = max E − E where xmin and xmax are the minimum and maximum values of signal xi , i i respectively; E and E max are the minimum and maximum values of the energy E, respectively For convenience, we choose E 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 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: a1j wi,j pi + b1j =f , j = 1, , 35 (3) i=1 193 194 195 196 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: 35 wj2 a1j + b, Es = (4) i=1 197 198 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 228 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 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 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) 229 Finding BBS→BSB transition state 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 230 231 232 233 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 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 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 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 Figure 1: 16 Figure 2: 17 Energy (eV) 1.5 0.5 B3LYP NN 0 10 20 30 Figure 3: 18 40 50 Spin state Triplet B3LYP NN Singlet Figure 4: 19 Potential Energy (eV) 1.5 0.5 10 15 Reaction Coordinate Figure 5: 20 20 1.8 3.5 1.75 1.65 R (˚ A) R (˚ A) 3.2 2.9 2.6 1.55 2.3 1.45 50 10 20 30 40 50 60 1.5 30 20 10 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 0 436 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 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(1988) 5680 526 530 531 [55] K Hornik, M Stinchcombe, H White, Neural Networks (1989) 359– 366 29 Research highlight: • B2S can adopt either the B-B-S (singlet) or B-S-B (triplet) conformation • A neural network potential energy surface for B2S transformation is developed • The transformation pathway is predicted using the nudged elastic band method • Molecular dynamics of the transformation is investigated with various excitations ... 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.. .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,∗... 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