Dunn Ian S, Tempelaar Roel, Reichman David R
Department of Chemistry, Columbia University, 3000 Broadway, New York, New York 10027, USA.
J Chem Phys. 2019 May 14;150(18):184109. doi: 10.1063/1.5092616.
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.
分层运动方程（HEOM）为计算与浴线性耦合的量子系统的约化动力学提供了一种数值精确的方法。我们发现HEOM包含随时间呈指数增长的与温度相关的不稳定性。在连续浴模型的情况下，这些不稳定性可能通过增加分层维度而延迟到更晚的时间；然而，对于与离散、非色散模式耦合的系统，增加分层维度对缓解问题作用不大。我们表明，通过投影到稳定本征模空间，这些不稳定性也可以以潜在更低的成本完全消除；此外，我们发现对于零温度下的离散浴模型，用很少分层水平计算的剩余投影动力学与精确动力学基本相同，否则精确动力学可能需要难以处理的大量分层水平才能收敛。认识到本征模的计算可能令人望而却步，例如对于大型或强耦合模型，我们提出了一种 Prony 滤波算法，当对角化成本过高时，该算法可作为完成此投影的替代方法。我们展示了通过对角化和 Prony 滤波投影的 HEOM 的有效性。我们还讨论了与 HEOM 的非正态性相关的问题。
