We present a formalism for developing cyclic and helical symmetry-informed machine learned force fields (MLFFs). In particular, employing the smooth overlap of atomic positions descriptors with the polynomial kernel method, we derive cyclic and helical symmetry-adapted expressions for the energy, atomic forces, and phonons, i.e., lattice vibration frequencies and modes. We use this formulation to construct a symmetry-informed MLFF for carbon nanotubes (CNTs), where the model is trained through Bayesian linear regression, with the data generated from ab initio density functional theory (DFT) calculations performed during on-the-fly symmetry-informed MLFF molecular dynamics simulations of representative CNTs. We demonstrate the accuracy of the MLFF model by comparisons with DFT calculations for the energies and forces, and density functional perturbation theory calculations for the phonons, while considering CNTs not used in the training. In particular, we obtain a root mean square error of 1.4×10−4 Ha/atom, 4.7×10−4 Ha/Bohr, and 4.8 cm−1 in the energy, forces, and phonon frequencies, respectively, which are well within the accuracy targeted in ab initio calculations. We apply this framework to study phonons in CNTs of various diameters and chiralities, where we identify the torsional rigid body mode that is unique to cylindrical structures and establish laws for variation of the phonon frequencies associated with the ring modes and radial breathing modes. Overall, the proposed formalism provides an avenue for studying nanostructures with cyclic and helical symmetry at ab initio accuracy, while providing orders-of-magnitude speedup relative to such methods.

中文翻译：

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循环和螺旋对称性信息机器学习力场：在碳纳米管晶格振动中的应用


我们提出了一种用于开发循环和螺旋对称信息机器学习力场 （MLFF） 的形式。特别是，利用原子位置描述符与多项式核方法的平滑重叠，我们推导出能量、原子力和声子的循环和螺旋对称适应表达式，即晶格振动频率和模式。我们使用此公式为碳纳米管 （CNT） 构建对称性告知的 MLFF，其中模型通过贝叶斯线性回归进行训练，数据从零开始密度泛函理论 （DFT） 计算中执行，在代表性 CNT 的动态对称性告知 MLFF 分子动力学模拟期间执行。我们通过与能量和力的 DFT 计算以及声子的密度泛函扰动理论计算进行比较来证明 MLFF 模型的准确性，同时考虑训练中未使用的 CNT。特别是，我们在能量、力和声子频率上分别获得了 1.4×10-4 Ha/atom、4.7×10-4 Ha/Bohr 和 4.8 cm-1 的均方根误差，这完全在从头计算的目标精度范围内。我们应用这个框架来研究各种直径和手性的 CNT 中的声子，在那里我们确定了圆柱形结构独有的扭转刚体模式，并建立了与环模式和径向呼吸模式相关的声子频率变化规律。总体而言，所提出的形式主义为以 ab initio 精度研究具有循环和螺旋对称性的纳米结构提供了一条途径，同时相对于此类方法提供了数量级的加速。