Solver-Oriented Electron Density Learning with Equivariant Gaussian Shells

Learning electron density from molecular geometry supports electronic-property recovery and self-consistent field (SCF) initialization. We introduce GEM-FIELD, an equivariant Gaussian-shell representation with learnable radial parameters, nonlinear angular distributions, and exact electron-number conservation. Its Gaussian expansion provides analytical Coulomb integrals for target atomic-orbital bases, connecting predicted densities to property evaluation and SCF solvers. On QM9-PBE, GEM-FIELD achieves 0.226% dense-grid density NMAE; its compact variant achieves 0.237% while reducing evaluation time by 96.3% relative to ChargE3Net. On larger OE62 molecules, density-derived frontier-orbital errors are 86.7%–92.8% lower than task-specific predictions. On SCFBench OOD molecules with up to 60 atoms, GEM-FIELD achieves 100% convergence, reduces SCF iterations by 40.5%, and lowers end-to-end runtime by 8.98% relative to MINAO. Further evaluations demonstrate transferable initialization across target AO bases and exchange–correlation functionals.