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This paper introduces a general atoms-in-molecules (AIM) framework to decompose second-harmonic generation (SHG) in nonlinear-optical crystals, partitioning momentum matrix elements into atomic contributions while satisfying sum rules. The framework is applied to six UV and deep-UV nonlinear-optical crystals, revealing the hierarchy of atomic contributions to SHG. Results show that two-center terms are the leading contributors, with significant secondary contributions from three-center terms, and highlight the role of anionic frameworks and cation sublattices in different crystals.
Unveiling the atom-level origins of second-harmonic generation in crystals reveals that two-center atomic interactions dominate, challenging the assumption that complex multi-center effects are always key.
In this work, we develop a new framework for computing atom-resolved contributions to optical properties based on atoms-in-molecules (AIM) schemes. The formalism is independent of the specific AIM method and is made rigorous by partitioning momentum matrix elements into atomic contributions while exactly satisfying the relevant sum rules. We apply it to second-harmonic generation (SHG) in six representative UV and deep-UV nonlinear-optical crystals, namely $\beta$-\ce{BaB2O4} (BBO), \ce{LiB3O5} (LBO), \ce{CsB3O5} (CBO), \ce{CsLiB6O10} (CLBO), \ce{KBe2BO3F2} (KBBF), and \ce{LiCs2PO4} (LCPO). The atom-triplet decomposition reveals a clear hierarchy for the largest SHG component of each crystal. In general, two-center terms provide the leading contribution, one-center terms remain comparatively small, and fully three-center terms supply an important secondary contribution. A motif-triplet decomposition further indicates behavior dominated by the anionic framework in KBBF and LBO. In BBO, CBO, and CLBO, contributions from the anionic framework and the cation sublattice act cooperatively, although the cation contribution is crystal dependent. Moreover, cooperative contributions from the phosphate framework and the Cs sublattice are also observed in LCPO, where the O-Cs contribution is particularly significant. These results may provide a new perspective for understanding the microscopic origin of SHG in nonlinear-optical materials.