Malshe M, Narulkar R, Raff L M, Hagan M, Bukkapatnam S, Agrawal P M, Komanduri R
Nanotechnology Research Group, Oklahoma State University, 218 Engineering, North Stillwater, Oklahoma 74078, USA.
J Chem Phys. 2009 May 14;130(18):184102. doi: 10.1063/1.3124802.
A general method for the development of potential-energy hypersurfaces is presented. The method combines a many-body expansion to represent the potential-energy surface with two-layer neural networks (NN) for each M-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 Si(n) (n=3,4,...,7) clusters obtained from density functional theory calculations and for vinyl bromide (C(2)H(3)Br) 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 H(2) dissociation. The vinyl bromide database comprises the ab initio energies of 71 969 configurations computed at MP4(SDQ) level with a 6-31G(d,p) basis set for the carbon and hydrogen atoms and Huzinaga's (4333/433/4) basis set augmented with split outer s and p orbitals (43321/4321/4) and a polarization f 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 Si(5) system with a mean absolute testing set error of 5.693x10(-4) eV. Expansions truncated after the four-body terms for Si(n) (n=3,4,5) and Si(n) (n=3,4,...,7) 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.
提出了一种开发势能超曲面的通用方法。该方法将多体展开用于表示势能面，并对求和中的每个M体项采用双层神经网络（NN）。通过采用部分能量近似，所需神经网络的总数显著减少。提出了一种算法，可有效地将所有耦合的神经网络参数调整到该曲面的数据库。将该方法应用于四个复杂度不断增加的不同系统，结果表明该方法的拟合精度良好至优异。在某些情况下，它超过了目前文献中其他方法所能达到的精度。通过拟合从密度泛函理论计算获得的Si(n)（n = 3,4,...,7）团簇以及溴乙烯（C₂H₃Br）的从头算能量大数据库，以及分解为六个开放反应通道（如果将逆反应视为单独的开放通道则为12个）的所有产物，包括C - H和C - Br键断裂、三中心HBr分解和三中心H₂分解，对该方法进行了说明。溴乙烯数据库包含在MP4(SDQ)水平下计算的71969种构型的从头算能量，碳原子和氢原子采用6 - 31G(d,p)基组，溴原子采用Huzinaga的(4333/433/4)基组并增加了分裂外层s和p轨道(43321/4321/4)以及指数为0.5的极化f轨道。结果发现，在三体项之后截断的展开式足以拟合Si(5)系统，测试集平均绝对误差为5.693×10⁻⁴ eV。对于Si(n)（n = 3,4,5）和Si(n)（n = 3,4,...,7），在四体项之后截断的展开式提供的拟合结果，其测试集平均绝对误差分别为0.0056和0.0212 eV。对于溴乙烯，在四体项之后截断的多体展开式提供的拟合精度，测试集平均绝对误差在0.0782至0.0808 eV之间。这些误差对应的平均百分比误差在0.98% - 1.01%范围内。使用本方法在四体求和后截断并使用16个神经网络得到的最佳结果，其测试集误差比使用15维（15 - 140 - 1）神经网络拟合溴乙烯数据库得到的误差高20.3%。这似乎是多体展开过程增加的简单性所付出的代价。
