In molecular dynamics or molecular statics (MD/MS) multi-body potentials empirically capture the energetic interactions in atomistic systems enabling the computation of the corresponding atomistic forces as energetic conjugates to the atomistic positions. We distinguish here between spatial and material atomistic positions and consequently between the corresponding spatial and material atomistic forces. In quasi-statics, i.e. MS, the former, also denoted as deformational atomistic forces, contribute to the classical deformational mechanics (i.e., equilibrium) problem that seeks to minimise the total potential energy of an atomistic system with respect to the atomistic positions relative to the ambient space. The latter, also denoted as configurational atomistic forces, contribute to the configurational mechanics (i.e., non-equilibrium) problem that determines the release of total potential energy of an atomistic system upon variation of the atomistic positions relative to the ambient material, i.e., due to perturbations of the material (initial) atomistic configuration. The importance of material atomistic forces is that they drive energetically favourable re-organisations of the material atomistic configuration, thereby characterising the tendency of generic atomistic defects to propagate. In this contribution we focus on two-, three-, and four-body potentials, whereby we distinguish between novel stretch- and classical angle-based potentials for the two latter cases. Taken together, as the main contribution, we derive expressions for the corresponding spatial and, for the first time, material atomistic forces and highlight their striking formal similarity. The derivations are detailed but the final expression compact and well-suited for numerical implementation.

在分子动力学或分子静力学 (MD/MS) 中，多体势能凭经验捕获原子系统中的能量相互作用，从而能够将相应的原子力计算为原子位置的能量共轭。我们在这里区分空间和材料原子位置，从而区分相应的空间和材料原子力。在准静态学中，即 MS，前者，也表示为变形原子力，有助于经典变形力学（即平衡）问题，该问题旨在最小化原子系统相对于原子位置的总势能环境空间。后者，也表示为构型原子力，导致构型力学（即非平衡）问题，该问题决定了原子系统相对于环境材料的原子位置发生变化时原子系统总势能的释放，即由于材料的扰动（初始) 原子配置。材料原子力的重要性在于它们在能量上推动了材料原子构型的有利重组，从而表征了一般原子缺陷传播的趋势。在这个贡献中，我们专注于二体、三体和四体势，由此我们区分了后两种情况的新型基于拉伸和经典角度的势。总之，作为主要贡献，我们推导出相应空间的表达式，并且第一次，物质原子力并突出它们惊人的形式相似性。推导很详细，但最终表达式紧凑且非常适合数值实现。