  Deep learning for predicting the electronic-structure Hamiltonian of quantum
systems necessitates satisfying the covariance laws, among which achieving
SO(3)-equivariance without sacrificing the non-linear expressive capability of
networks remains unsolved. To navigate the harmonization between equivariance
and expressiveness, we propose a deep learning method synergizing two distinct
categories of neural mechanisms as a two-stage encoding and regression
framework. The first stage corresponds to group theory-based neural mechanisms
with inherent SO(3)-equivariant properties prior to the parameter learning
process, while the second stage is characterized by a non-linear 3D graph
Transformer network we propose, featuring high capability on non-linear
expressiveness. The novel combination lies in the point that, the first stage
predicts baseline Hamiltonians with abundant SO(3)-equivariant features
extracted, assisting the second stage in empirical learning of equivariance;
and in turn, the second stage refines the first stage's output as a
fine-grained prediction of Hamiltonians using powerful non-linear neural
mappings, compensating for the intrinsic weakness on non-linear expressiveness
capability of mechanisms in the first stage. Our method enables precise,
generalizable predictions while capturing SO(3)-equivariance under rotational
transformations, and achieves state-of-the-art performance in Hamiltonian
prediction on six benchmark databases.
