Accelerating SCF Convergence Through Equivariant Density-Matrix Learning and Analytic Refinement

We present dm-PhiSNet, a physically constrained PhiSNet-based equivariant model that predicts one-electron reduced density matrices (1-RDMs) directly from molecular geometries in an atomic-orbital (AO) basis to accelerate self-consistent-field (SCF) convergence. Training follows a two-stage schedule with progressively introduced physically motivated objectives, and the resulting predictions are refined by a lightweight analytic block. This block enforces electron-number conservation, drives the 1-RDM toward generalized idempotency with respect to the AO overlap matrix $S$ , and regularizes the occupation spectrum of the density matrix in an orthogonalized AO representation. Across six closed-shell systems- $H_{2}O$ , ${\mathrm{CH}}_4$ , ${\mathrm{NH}}_3$ , HF, ethanol, and ${\mathrm{NO}}_3^{-}$ -the refined 1-RDMs provide SCF initial guesses that substantially reduce iteration steps by 49%-81% relative to standard initializations. Beyond SCF acceleration, the learned 1-RDMs yield accurate one-shot total energies and Hellmann-Feynman atomic forces without force supervision, indicating that the model captures chemically meaningful electronic structure. These results demonstrate that combining equivariant learning with analytic constraint enforcement provides a simple, general route to solver-ready density-matrix initializations and accelerated SCF calculations.