Fig. 1: HamGNN architecture and the illustration of its subnetworks.

a The overall architecture of HamGNN. This neural network architecture predicts the Hamiltonian matrix through five steps. The prediction starts from the initial graph embedding of the species, interatomic distances, and interatomic directions of molecules and crystals. The atomic orbital features with angular momentum l in the local environment are included in the l-order components of the E(3) equivariant atom features and are refined through T orbital convolution blocks. In the third step, pair interaction features \({\omega }_{l,p,c,m}^{ij}\) of atomic orbitals are constructed by pair interaction blocks. In the fourth step, the IST representations of on-site and off-site Hamiltonian matrices are constructed by passing the features of atomic orbitals and pair interactions through the on-site layer and off-site layer, respectively. The final step is to construct the on-site and off-site Hamiltonian matrices block-by-block via the parameterized Hamiltonian given by Eq. (12). b Orbital convolution block. The equivariant atomic features that include the features of atomic orbitals of each angular momentum l are refined by the equivariant message passing and update functions. c Pair interaction block. This block is used to construct the pair interaction features between the orbitals of two adjacent atoms by equivariant tensor product. d On-site layer. The equivariant features of atoms are transformed into ISTs of on-site blocks \({\mathbf{\Omega}}_{i}^{on}\) by the on-site layer. e Off-site layer. The pair interaction features between atomic orbitals are transformed into ISTs of the off-site block \({\mathbf{\Omega}}_{ij}^{off}\) by the off-site layer. f Residual block. The residual block is used in the on-site layer and off-site layer to perform a nonlinear equivariant transformation of input features.