Plane-wave DFT Hamiltonians become learnable: projectability-selected eigenstates cut sodium band errors to 1–3 meV

Electronic structures computed with plane-wave density functional theory (DFT) can now be converted into compact atomic-orbital Hamiltonians and used directly as training targets for equivariant neural networks. The earlier conversion route reconstructed atomic-orbital matrix elements in real space, and its formulation contains only kinetic energy, local effective potential and nonlocal pseudopotential terms, omitting exact exchange in hybrid functionals and the Hubbard correction in DFT+U. For monolayer MoS2, Hamiltonians built with the smaller Mo:2s1p1d, S:1s1p basis follow the plane-wave PBE and HSE references more closely than real-space reconstruction with the larger Mo:3s2p2d, S:3s2p1d basis, whose deviations appear mainly in the conduction bands.

The construction keeps only plane-wave eigenstates that are well represented in the atomic-orbital basis, then fills the atomic-orbital subspace with selected higher-energy states. Projectability measures how well an atomic-orbital basis represents a plane-wave eigenstate, ranging from 0 to 1, with values near 1 meaning accurate representation; the earlier method filtered out low-projectability states and added an empirical shift energy to the complementary subspace. Across the bcc, fcc, cI16, oP8 and hP4 sodium phases, band-energy mean absolute errors are 1, 3, 1, 1 and 1 meV with 13 orbitals per atom, versus 12, 26, 14, 18 and 28 meV for real-space reconstruction with 23 orbitals per atom.

The resulting Hamiltonians train an equivariant graph neural network that predicts them from atomic structure alone, avoiding repeated self-consistent plane-wave calculations. The training set contains 1,150 non-twisted bilayer MoS2 configurations covering different stackings, interlayer sliding and random atomic displacements, and includes no twisted structures. On a 42-atom twisted bilayer MoS2 supercell with a twist angle of 21.79 degrees, the model trained on these Hamiltonians gives a band-energy mean absolute error of 7.71 meV in the 2 to 2.8 eV window, compared with 16.15 meV for the model trained on real-space reconstruction.

The construction targets researchers who need atomic-orbital Hamiltonian training targets from plane-wave DFT reference calculations, works for semilocal and hybrid-functional calculations, and was benchmarked on monolayer MoS2, five high-pressure sodium phases, an Al surface slab and MgO with an oxygen vacancy through band structures, density of states and optical response. Its accuracy depends on how well the atomic-orbital basis represents the selected plane-wave states, and projectability serves as a diagnostic that can guide basis enlargement or radial-function optimization.

The construction does not exhibit linear scaling: it is 2.11 to 3.37 times faster than real-space reconstruction with the same smaller basis for 6 to 96-atom MoS2 supercells, but at 216 atoms the times are 301.10 and 290.20 s, so the advantage does not persist. Avoiding an empirical shift removes one arbitrary element of the construction but does not by itself establish the accuracy of neural-network predictions for unseen structures, which requires separate validation.