  In this study, we introduce a unified neural network architecture, the Deep
Equilibrium Density Functional Theory Hamiltonian (DEQH) model, which
incorporates Deep Equilibrium Models (DEQs) for predicting Density Functional
Theory (DFT) Hamiltonians. The DEQH model inherently captures the
self-consistency nature of Hamiltonian, a critical aspect often overlooked by
traditional machine learning approaches for Hamiltonian prediction. By
employing DEQ within our model architecture, we circumvent the need for DFT
calculations during the training phase to introduce the Hamiltonian's
self-consistency, thus addressing computational bottlenecks associated with
large or complex systems. We propose a versatile framework that combines DEQ
with off-the-shelf machine learning models for predicting Hamiltonians. When
benchmarked on the MD17 and QH9 datasets, DEQHNet, an instantiation of the DEQH
framework, has demonstrated a significant improvement in prediction accuracy.
Beyond a predictor, the DEQH model is a Hamiltonian solver, in the sense that
it uses the fixed-point solving capability of the deep equilibrium model to
iteratively solve for the Hamiltonian. Ablation studies of DEQHNet further
elucidate the network's effectiveness, offering insights into the potential of
DEQ-integrated networks for Hamiltonian learning.
