^{1}, R. Narulkar

^{1}, L. M. Raff

^{1,a)}, M. Hagan

^{1}, S. Bukkapatnam

^{1}, P. M. Agrawal

^{1}and R. Komanduri

^{1}

### Abstract

A general method for the development of potential-energy hypersurfaces is presented. The method combines a many-body expansion to represent the potential-energysurface with two-layer neural networks (NN) for each -body term in the summations. The total number of NNs required is significantly reduced by employing a moiety energy approximation. An algorithm is presented that efficiently adjusts all the coupled NN parameters to the database for the surface. Application of the method to four different systems of increasing complexity shows that the fitting accuracy of the method is good to excellent. For some cases, it exceeds that available by other methods currently in literature. The method is illustrated by fitting large databases of *ab initio* energies for clusters obtained from density functional theory calculations and for vinyl bromide and all products for dissociation into six open reaction channels (12 if the reverse reactions are counted as separate open channels) that include C–H and C–Br bond scissions, three-center HBr dissociation, and three-center dissociation. The vinyl bromide database comprises the *ab initio* energies of 71 969 configurations computed at MP4(SDQ) level with a basis set for the carbon and hydrogen atoms and Huzinaga’s (4333/433/4) basis set augmented with split outer and orbitals (43321/4321/4) and a polarization orbital with an exponent of 0.5 for the bromine atom. It is found that an expansion truncated after the three-body terms is sufficient to fit the system with a mean absolute testing set error of . Expansions truncated after the four-body terms for and provide fits whose mean absolute testing set errors are 0.0056 and 0.0212 eV, respectively. For vinyl bromide, a many-body expansion truncated after the four-body terms provides fitting accuracy with mean absolute testing set errors that range between 0.0782 and 0.0808 eV. These errors correspond to mean percent errors that fall in the range 0.98%–1.01%. Our best result using the present method truncated after the four-body summation with 16 NNs yields a testing set error that is 20.3% higher than that obtained using a 15-dimensional (15-140-1) NN to fit the vinyl bromide database. This appears to be the price of the added simplicity of the many-body expansion procedure.

We thank Professor Tucker Carrington, Jr. and Dr. Sergei Manzhos for generously providing us with a preprint of their HDMR results (Ref. 69) using NNs, exponential transfer functions, and optimized input coordinates in advance of publication. This project was funded by a grant (No. DMI-0457663) from the National Science Foundation, Division of Civil, Mechanical, and Manufacturing Innovation (CMMI). The authors thank Dr. Jocelyn Harrison, Program Director for Materials Processing and Manufacturing for the interest and support of this work. One of the authors (R.K.) also thanks the A.H. Nelson, Jr. Endowed Chair for additional financial support.

I. INTRODUCTION

II. GENERAL METHOD

III. ILLUSTRATIVE APPLICATIONS

A. clusters

B. clusters

C. clusters

D. Vinyl bromide and dissociation products

IV. DISCUSSION, SUMMARY, AND CONCLUSIONS

### Key Topics

- Databases
- 39.0
- Testing procedures
- 32.0
- Dissociation
- 15.0
- Potential energy surfaces
- 11.0
- Silicon
- 9.0

## Figures

Flow chart for the general many-body expansion/NN fitting algorithm.

Flow chart for the general many-body expansion/NN fitting algorithm.

Flow chart for the many-body expansion/NN fitting algorithm for the specific case of clusters with the expansion truncated after the three-body terms.

Flow chart for the many-body expansion/NN fitting algorithm for the specific case of clusters with the expansion truncated after the three-body terms.

Distribution of fitting errors for , , and clusters with four-body terms included in the many-body expansion. The mean absolute testing set error is 0.0056 eV. With the database covering an energy range of about 6 eV, the error corresponds to a fitting accuracy of about 0.093%.

Distribution of fitting errors for , , and clusters with four-body terms included in the many-body expansion. The mean absolute testing set error is 0.0056 eV. With the database covering an energy range of about 6 eV, the error corresponds to a fitting accuracy of about 0.093%.

Comparison of the fitted energies obtained using Eq. (1) with the NNs specified under calculation 1 in Table II with the *ab initio* DFT energies for clusters. If the fitting were perfect, all points would fall on the 45° line in the figure. The mean absolute testing set error is 0.0353 eV, which corresponds to a mean percent testing set error of 0.21%.

Comparison of the fitted energies obtained using Eq. (1) with the NNs specified under calculation 1 in Table II with the *ab initio* DFT energies for clusters. If the fitting were perfect, all points would fall on the 45° line in the figure. The mean absolute testing set error is 0.0353 eV, which corresponds to a mean percent testing set error of 0.21%.

Distribution of fitting errors for clusters when the many-body expansion is truncated after the four-body term and the NNs are those described under calculation 1 in Table II. The mean absolute testing set error is 0.0353 eV, which corresponds to a mean percent error of 0.21%.

Distribution of fitting errors for clusters when the many-body expansion is truncated after the four-body term and the NNs are those described under calculation 1 in Table II. The mean absolute testing set error is 0.0353 eV, which corresponds to a mean percent error of 0.21%.

Comparison of the fitted energies obtained using Eq. (1) with the NNs specified under fit 1 in Table III with the *ab initio* UMP4(SDQ) energies for the vinyl bromide database. If the fitting were perfect, all points would fall on the 45° line in the figure. The mean absolute testing set error is 0.0808 eV, which corresponds to a mean percent error of 1.01%.

Comparison of the fitted energies obtained using Eq. (1) with the NNs specified under fit 1 in Table III with the *ab initio* UMP4(SDQ) energies for the vinyl bromide database. If the fitting were perfect, all points would fall on the 45° line in the figure. The mean absolute testing set error is 0.0808 eV, which corresponds to a mean percent error of 1.01%.

Distribution of testing set errors for the vinyl bromide database when the many-body expansion is truncated after the four-body term and the NNs are those described under fit 1 in Table III. The mean absolute testing set error is 0.0808 eV, which corresponds to a mean percent error of 1.01%.

Distribution of testing set errors for the vinyl bromide database when the many-body expansion is truncated after the four-body term and the NNs are those described under fit 1 in Table III. The mean absolute testing set error is 0.0808 eV, which corresponds to a mean percent error of 1.01%.

## Tables

Number of configurations of each of the clusters in the training set.

Number of configurations of each of the clusters in the training set.

NN specifications for the many-body expansions for clusters. Errors are given in eV. denotes the mean absolute testing set error. The entry labeled % error is , where range is the total energy range spanned by the database employed in the fitting of the potential. The entry labeled No. of parameters gives the total number of weight and bias parameters contained in the three NNs.

NN specifications for the many-body expansions for clusters. Errors are given in eV. denotes the mean absolute testing set error. The entry labeled % error is , where range is the total energy range spanned by the database employed in the fitting of the potential. The entry labeled No. of parameters gives the total number of weight and bias parameters contained in the three NNs.

Many-body expansion/NN/ME results for two different network architectures for fitting the vinyl bromide database. Testing set errors are given in eV.

Many-body expansion/NN/ME results for two different network architectures for fitting the vinyl bromide database. Testing set errors are given in eV.

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