Spectrally Constrained Density Matrix Learning

There is growing interest in learning the electronic structure matrices produced within Density Functional Theory (DFT) calculations, which simultaneously contain rich electronic-level information and enable direct evaluation of total energies and atomic forces. Machine learned Hamiltonian models (MLHs) and machine learned density matrix models (MLDMs) are two such approaches that target either the Hamiltonian ($\mathbf{H}$) or the density matrix ($\mathbf{D}$). Despite having several numerical advantages, MLDMs have typically underperformed MLHs. Here, we show that this discrepancy originates from misalignment in spectral properties inherent to $\mathbf{D}$. We thus introduce a spectrally constrained density-matrix learning framework, combining a Cayley-transform parameterization with an occupied-virtual (OV) loss. The Cayley parameterization preserves electron number, occupation spectrum, and idempotency by construction, while the OV loss supervises occupied-subspace alignment. On QH9, our model achieves a one-shot energy error of $0.168$ meV with no outliers above $100$ meV, outperforming both standard MLDM approaches and state-of-the-art MLH baseline.