Dimensionally consistent Gaussian basis set bridging bulk solids and nanomaterials

Gaussian basis sets are highly appealing for large-scale electronic structure calculations of quantum dots (QDs) due to their smaller basis set size and ease of computing multicenter integrals. Yet the prevailing Gaussian basis sets are commonly designed for atoms or molecules and contain highly diffusive functions, which are not transferable for electronic structure calculations of large-size QDs and bulk solids, thus lacking dimensional-consistency. Herein, we resolve this issue by exploring an inverse design strategy for novel Gaussian basis sets, i.e., we design Gaussian basis set for periodic bulk solid and inversely apply it to confined nanostructures. The key to success is to optimize the exponents of polarization functions in the Gaussian basis set to best reproduce the reference lattice constant and band structure of a given bulk solid calculated by using the plane-wave basis set. While applying this strategy to ten traditional semiconductors of groups IV, II-VI, and III-V, the resulting novel Gaussian basis sets exhibit excellent transferability to nanostructured materials, including a two-dimensional (2D) quantum well (QW) down to the monolayer limit and zero-dimensional (0D) QD down to ultrasmall sizes ($\ensuremath{\sim}1\phantom{\rule{0.16em}{0ex}}\mathrm{nm}$), thus achieving the desired dimensional-consistency. Meanwhile, the resulting relative error in the lattice constant or averaged bond length is mostly below 2%, and the band-gap error below 0.1 eV compared to plane-wave references, thus offering competing accuracy. This work provides an alternative pathway of designing novel Gaussian basis sets capable of treating periodic solids and confined nanomaterials on an equal footing, which is broadly applicable for highly accurate basis set optimization of the existing Gaussian basis sets.