Construction of Arbitrary-Order Internal Coordinate Transformations to Improve Studies of Large-Amplitude Motions

Internal coordinates and their derivatives underpin the efficient treatment of geometry optimizations, high-resolution spectroscopic simulation, and the fitting of potential surfaces in quantum chemistry. Existing descriptions of the construction of internal coordinate derivatives generally either lack simplicity or generality. In this paper, we provide a simple framework for evaluating any internal coordinate derivative to any order and an automatic approach to obtain the corresponding inverse transformation. Through further extension to transformations between internal coordinate systems, this approach provides a complete, generic method for handling a wide variety of molecular problems. The utility of this construction is demonstrated by investigations into the behavior of internal coordinate interpolations for studying isomerizations, quantifying the coupling between carbonyl stretches and a complex stretch coordinate in an organometallic system, and analysis of the performance of a machine learned interatomic potential in computing anharmonic frequencies as a function of low-frequency coordinate distortions. This approach is shown to be numerically efficient as well as general, and a reference implementation is provided.