In this study, a variational method for the inverse problem of self-assembly, i.e., a reconstruction of the interparticle interaction potential of a given structure, is applied to three-dimensional crystals. According to the method, the interaction potential is derived as a function that maximizes the free-energy functional of the oneand two-particle density distribution functions. The interaction potentials of the target crystals, including those with face-centered cubic (fcc), body-centered cubic (bcc), and simple hexagonal (shx) lattices, are obtained by numerical maximization of the functional. Monte Carlo simulations for the systems of particles with these interactions were carried out, and the self-assembly of the target crystals was confirmed for the bcc and shx cases. However, in the many-particle system with the predicted interaction for the fcc lattice, the fcc lattice did not spontaneously form and was metastable.

中文翻译：

---

三维晶体自组装反问题的自由能泛函方法

在这项研究中，自组装逆问题的变分方法，即给定结构的粒子间相互作用势的重建，应用于三维晶体。根据该方法，相互作用势被导出为最大化单粒子和双粒子密度分布函数的自由能函数的函数。目标晶体的相互作用势，包括具有面心立方 (fcc)、体心立方 (bcc) 和简单六方 (shx) 晶格的晶体，通过泛函的数值最大化获得。对具有这些相互作用的粒子系统进行了蒙特卡罗模拟，并在 bcc 和 shx 情况下确认了目标晶体的自组装。然而，