Modeling a crystal as a periodic point set, we present a fingerprint consisting of density functions that facilitates the efficient search for new materials and material properties. We prove invariance under isometries, continuity, and completeness in the generic case, which are necessary features for the reliable comparison of crystals. The proof of continuity integrates methods from discrete geometry and lattice theory, while the proof of generic completeness combines techniques from geometry with analysis. The fingerprint has a fast algorithm based on Brillouin zones and related inclusion-exclusion formulae. We have implemented the algorithm and describe its application to crystal structure prediction.

中文翻译：

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周期点集的密度指纹

通过将晶体建模为一个周期点集，我们展示了一个由密度函数组成的指纹，该指纹函数有助于高效地搜索新材料和材料特性。在一般情况下，我们证明了在对称性，连续性和完整性下的不变性，这是可靠比较晶体的必要特征。连续性证明结合了离散几何和格理论的方法，而通用完整性证明则结合了几何和分析技术。指纹具有基于布里渊区和相关包含/排除公式的快速算法。我们已经实现了该算法，并描述了其在晶体结构预测中的应用。