Removing instabilities in the hierarchical equations of motion: Exact and approximate projection approaches

Ian S. Dunn, Roel Tempelaar, David R. Reichman

Research output: Contribution to journalArticlepeer-review

47 Scopus citations


The hierarchical equations of motion (HEOM) provide a numerically exact approach for computing the reduced dynamics of a quantum system linearly coupled to a bath. We have found that HEOM contains temperature-dependent instabilities that grow exponentially in time. In the case of continuous-bath models, these instabilities may be delayed to later times by increasing the hierarchy dimension; however, for systems coupled to discrete, nondispersive modes, increasing the hierarchy dimension does little to alleviate the problem. We show that these instabilities can also be removed completely at a potentially much lower cost via projection onto the space of stable eigenmodes; furthermore, we find that for discrete-bath models at zero temperature, the remaining projected dynamics computed with few hierarchy levels are essentially identical to the exact dynamics that otherwise might require an intractably large number of hierarchy levels for convergence. Recognizing that computation of the eigenmodes might be prohibitive, e.g., for large or strongly coupled models, we present a Prony filtration algorithm that may be useful as an alternative for accomplishing this projection when diagonalization is too costly. We present results demonstrating the efficacy of HEOM projected via diagonalization and Prony filtration. We also discuss issues associated with the non-normality of HEOM.

Original languageEnglish (US)
Article number184109
JournalJournal of Chemical Physics
Issue number18
StatePublished - May 14 2019

ASJC Scopus subject areas

  • General Physics and Astronomy
  • Physical and Theoretical Chemistry


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