The matrix singularity problem in the time-dependent variational method

The matrix singularity problem that arises in the time-dependent variational method when certain parameters in the trial wave-function become redundant is investigated. It is found that the equations of motion derived from the Frenkel and McLachlan variational principles continue to have solutions in the presence of the matrix singularity while the equations obtained from the least-action variational principle generally do not. A technique is developed for the practical treatment of the matrix singularity and is applied to a trial wavefunction investigated in an earlier publication. It is shown that the present method for handling the singularity becomes identical to one that was previously and successfully used for numerical calculations in that case. The present technique should make it possible to propagate wavefunctions accurately by the time-dependent variational method for a wide range of parameterizations.