An automatic differentiation and diagrammatic notation approach for developing analytical gradients and non-adiabatic couplings of molecular systems

Analytical nuclear derivatives and non-adiabatic couplings are important for computational studies of excited state properties of molecular systems . However, its development is increasingly challenging as the complexity of the electronic structure method grows, especially for non-variational methods that preclude the use of the Hellman-Feynman theorem. We show how the combination of automatic differentiation (AD) from computer algebra and diagrammatic notation from quantum circuit models can facilitate the development of analytical nuclear derivatives. In particular, automatically-derived gradients are guaranteed to have the same scaling as the underlying energy calculations, and the computation is roughly three times as costly. The new AD/diagrammatic approach has been applied to our recently proposed supporting subspace multi-reference perturbation theory, providing a balanced treatment of static and dynamic correlation for excited states. We will present an application to green fluorescent protein that show how the selection of QM region affects the excitation energies and shapes of the potential energy surfaces around conical intersections.