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We investigate a simple and robust scheme for choosing the phases ofadiabatic electronic states smoothly (as a function of geometry) so as tomaximize the performance of ab initio non-adiabatic dynamics methods. Ourapproach is based upon consideration of the overlap matrix ($\mathbf{U}$)between basis functions at successive points in time and selecting the phasesso as to minimize the matrix norm of $\log(\mathbf{U})$. In so doing, one canextend the concept of parallel transport to cases with sharp curve crossings.We demonstrate that this algorithm performs well under extreme situations wheredozens of states cross each other either through trivial crossings (where thereis zero effective diabatic coupling), or through nontrivial crossings (whenthere is a nonzero diabatic coupling), or through a combination of both. In allcases, we compute the time-derivative coupling matrix elements (or equivalentlynon-adiabatic derivative coupling matrix elements) that are as smooth aspossible. Our results should be of interest to all who are interested in eithernon-adiabatic dynamics, or more generally, parallel transport in large systems.

A Robust and Unified Solution for Choosing the Phases of Adiabatic  States as a Function of Geometry: Extending Parallel Transport Concepts to  the cases of Trivial & Near Trivial Crossings