An Automatic Differentiation and Diagrammatic Notation Approach for Developing Analytical Gradients of Tensor Hyper-Contracted Electronic Structure Methods

We show how the combination of automatic differentiation (AD) and diagrammatic notation can facilitate the development of analytical nuclear derivatives for tensor hyper-contraction based (THC) electronic structure methods. The automatically-derived gradients are guaranteed to have the same scaling in terms of both operation count and memory footprint as the underlying energy calculations, and the computation of a gradient is roughly three times as costly as the underlying energy. The new AD/diagrammatic approach enables the first cubic scaling implementation of nuclear derivatives for THC tensors fitted in molecular orbital basis (MO-THC). Furthermore, application of this new approach to THC-MP2 analytical gradients leads to an implementation which is at least four times faster than the previously reported, manually-derived implementation. Finally, we apply the new approach to the 14 tensor contraction patterns appearing in the supporting subspace formulation of multireference perturbation theory, laying the foundation for future developments of analytical nuclear gradients and nonadiabatic coupling vectors for multi-state CASPT2.