Relativistic Fermionic Neural Networks Based on Zeroth-Order Regular Approximation: ZORANet

In this paper, we introduce ZORANet, a Fermionic neural network framework for the relativistic calculations based on scalar zeroth-order regular approximation (ZORA), which greatly expands the scope of deep neural network-based electronic structure methods. To address the numerical difficulties specific to the relativistic calculations, the electron-nucleus cusp correction scheme and an importance sampling technique have been carefully implemented to ensure training stability and reduce statistical errors. In addition, two variants, namely, scaled ZORANet and atomic ZORANet, have been developed to mitigate the notorious gauge-dependent error of ZORA. For benchmarks, we first study the hydrogen-like systems, in which both the original and scaled ZORA energies closely agree with the exact ZORA and Dirac results, respectively. Second, we calculate the atomic ionization potentials and electron affinities for the first two rows of elements. The ZORANet results show overall improvement compared to FermiNet, particularly for the second-row elements, with average deviations within chemical accuracy of the experimental values. For the dissociation energies of simple hydrides, atomic ZORANet again outperforms FermiNet, and reaches the same level of accuracy as state-of-the-art relativistic effective core potentials. Our results demonstrate that ZORANet is a highly accurate all-electron method capable of simultaneously treating electron correlation and relativistic effects, while maintaining computational costs comparable to FermiNet. Lastly, we give some prospects on the future development of ZORANet, which helps to further build it as a more integral <i>ab initio</i> relativistic quantum chemistry method.