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In this paper, we present a variational treatment of the linear dependencefor a non-orthogonal time-dependent basis set in solving the Schr\"odingerequation. The method is based on: i) the definition of a linearly independentworking space, and ii) a variational construction of the propagator over finitetime-steps. The second point allows the method to properly account for changesin the dimensionality of the working space along the time evolution. Inparticular, the time evolution is represented by a semi-unitary transformation.Tests are done on a quartic double-well potential with Gaussian basis functionwhose centers evolve according to classical equations of motion. We show thatthe resulting dynamics converges to the exact one and is unitary byconstruction.

A variational approach for linearly dependent moving bases in quantum dynamics: application to Gaussian functions