Where the Error Lives: Spectral Representations of Learned Kohn-Sham Hamiltonians

Kohn-Sham density functional theory (DFT) is the standard method for computing the electronic structure of molecules and materials, but its self-consistentfield (SCF) iterations make it expensive for large systems and high-throughput screening. Machine-learned Hamiltonian models offer a shortcut: an equivariant network predicts the Kohn-Sham Hamiltonian matrix from the atomic structure in one forward pass, and diagonalizing the prediction yields orbital energies, the HOMO-LUMO gap, molecular orbitals and the electron density, or an initial guess that accelerates DFT. These models are trained to match the reference matrix entry by entry, yet a small entry-wise error does not guarantee accurate eigenvalues and eigenvectors, which are what every downstream use requires. We represent the prediction error in the eigenbasis of the reference Hamiltonian, where it separates into sectors that each property, and each training signal, reads selectively. This representation explains why existing physics-informed losses help only partially: the wavefunction-alignment loss either weights all sectors uniformly or sees orbital energies only to first order, and the electron density is unchanged along about two thirds of the directions in which the prediction can be wrong, including every orbital-energy error. It also points to a representation of the Hamiltonian that sees every sector while staying in the language of the density: the cumulative local density of states (LDOS), the number of electrons at each point in space below each energy. We prove that it contains the density as a special case, responds to every component of the error, determines the Hamiltonian uniquely, and yields gradients that do not vanish for orbital energies far from their targets. Supervising a network through this representation requires no change to its architecture. Fine-tuning three Hamiltonian networks (QHNet, SPHNet and WANet) on QH9 in this way reduces the HOMO-LUMO gap error by 49-79%, 2.0-2.6 times more than the wavefunction-alignment loss, without increasing the element-wise error by more than 0.3%, and the gains carry over to out-of-distribution and molecular-dynamics splits (36-77%), where the wavefunction-alignment loss helps less and, on QHNet, makes the gap worse. These results show that how a predicted Hamiltonian is represented during training, not only the architecture that produces it, decides whether it is useful in practice.