Energy-based fragmentation methods approximate the potential energy of a molecular system as a sum of contribution terms built from the energies of particular subsystems. Some such methods reduce to truncations of the many-body expansion (MBE); others combine subsystem energies in a manner inspired by the principle of inclusion/exclusion (PIE). The combinatorial technique of M\"obius inversion of sums over partially ordered sets, which generalizes the PIE, is known to provide a non-recursive expression for the MBE contribution terms, and has also been connected to related cluster expansion methods. We build from these ideas a very general framework for decomposing potential functions into energetic contribution terms associated with elements of particular partially ordered sets (posets) and direct products thereof. Specific choices immediately reproduce not only the MBE, but also a number of other existing decomposition forms, including, e.g., the multilevel ML-BOSSANOVA schema. Furthermore, a different choice of poset product leads to a setup familiar from the combination technique for high-dimensional approximation, which has a known connection to quantum-chemical composite methods. We present the ML-SUPANOVA decomposition form, which allows the further refinement of the terms of an MBE-like expansion of the Born-Oppenheimer potential according to systematic hierarchies of ab initio methods and of basis sets. We outline an adaptive algorithm for the a posteori construction of quasi-optimal truncations of this decomposition. Some initial experiments are reported and discussed.

翻译：基于能量的碎片化方法将分子系统的势能近似为特定子系统能量构建的贡献项之和。部分此类方法可归结为多体展开（MBE）的截断形式；另一些方法则受包含/排除原理（PIE）启发，以特定方式组合子系统能量。莫比乌斯反演这一组合数学技术——作为PIE的推广形式——通过偏序集求和实现，已知可为MBE贡献项提供非递归表达式，并与相关簇展开方法存在理论关联。基于这些思想，我们构建了一个通用框架，将势函数分解为与特定偏序集（及其直积）元素相关联的能量贡献项。通过具体选择，该框架不仅可重现MBE，还能衍生多种现有分解形式，例如多级ML-BOSSANOVA架构。此外，采用不同的偏序集积构造可得到高维近似组合技术中常见的配置，该技术与量子化学组合方法存在已知关联。本文提出ML-SUPANOVA分解形式，该形式允许根据从头算方法和基组系统层级，对类MBE展开的玻恩-奥本海默势能项进行精细化重构。我们概述了一种自适应算法，用于后验构建该分解的准最优截断方案。文中报告并讨论了初步实验成果。