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The numerical integration of the Schr\"odinger equation by discretization oftime is explored for the curved manifolds arising from finite representationsbased on evolving basis states. In particular, the unitarity of the evolutionis assessed, in the sense of the conservation of mutual scalar products in aset of evolving states, and with them the conservation of orthonormality andparticle number. Although the adequately represented equation is known to giverise to unitary evolution in spite of curvature, discretized integrators easilybreak that conservation, thereby deteriorating their stability. The CrankNicolson algorithm, which offers unitary evolution in Euclidian spacesindependent of time-step size $\mathrm{d}t$, can be generalised to curvedmanifolds in different ways. Here we compare a previously proposed algorithmthat is unitary by construction, albeit integrating the wrong equation, with afaithful generalisation of the algorithm, which is, however, not strictlyunitary for finite $\mathrm{d}t$.

Numerical integration of quantum time evolution in a curved manifold