Semiempirical Quantum Chemistry in the Age of <i>ab initio</i> Data and Differentiable Programming: I. Differentiable Molecular Orbital Theory

Semiempirical quantum chemistry (SQC) methods offer fast quantum chemical insights by constructing and solving a parametric effective minimal basis Fock matrix. Establishing suitable parametrizations has long been a challenging and time-consuming task involving tedious grid searches or costly finite-difference gradients of carefully crafted loss functions based on select experimental data. The growing availability of differentiable programming environments that leverage algorithmic differentiation to obtain complicated derivatives together with access to a wealth of reliable reference data from <i>ab initio</i> calculations offers a new and more efficient approach. In this work, we extend a previous, basic implementation of SQC methods in PyTorch [Zhou, G. <i>J. Chem. Theory Comput.</i> 2020, 16, 4951-4962] by including global algorithmic considerations in the code design. This allows for improved general applicability and establishes a robust back-end for rapid SQC parametrizations. In particular, we address the general differentiability of the eigensolver and the iterative SCF procedure. The new implementation offers dramatic improvements in both computing cost and memory footprint, while simultaneously increasing numeric stability in gradient evaluation. We highlight the importance of these advances and their improvements over existing formulations and illustrate their role in the context of SQC parametrization.