The hierarchical equations of motion theory for Drude dissipation is optimized, with a convenient convergence criterion proposed in advance of numerical propagations. The theoretical construction is on basis of a Pad{e} spectrum decomposition that has been qualified to be the best sum-over-poles scheme for quantum distribution function. The resulting hierarchical dynamics under the {em apriori} convergence criterion are exemplified with a benchmark spin-boson system, and also the transient absorption and two-dimensional spectroscopy of a model exciton dimer system.
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