We use a posteriori error estimation theory to derive a relation between local and global error in the propagation for the time-dependent Schrödinger equation. Based on this result, we design a class of h,p-adaptive Magnus-Lanczos propagators capable of controlling the global error of the time-stepping scheme by only solving the equation once. We provide results for models of several different small molecules including bounded and dissociative states, illustrating the efficiency and wide applicability of the new methods.
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