We investigate the numerical stability of the hierarchical equations of motion (HEOM) method applied to systems with the Brownian oscillator (BO) and multimode BO (MBO) spectral densities. It is shown that, with the increase in the system–bath coupling strength, the standard HEOM may become unstable, and a simple increase in the truncation depth of the HEOM cannot remove the instability at long times. To solve this problem, we first show that the high-temperature approximation of the HEOM with the BO spectral density is equivalent to the celebrated quantum Fokker–Planck equation (QFPE). By starting from the HEOM, we then derive a new multidimensional phase space differential equation that generalizes the QFPE to arbitrary temperature. It is further shown that the numerical instability can be removed if the new low-temperature QFPE is expanded in a basis set different than the one that leads to the conventional HEOM. The matrix product state method is also employed to propagate the new equation based on the low-temperature QFPE and to resolve the numerical instability problem for an electron transfer model with the MBO spectral density presented in the recent literature.
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14 February 2022
Research Article|
February 11 2022
A low-temperature quantum Fokker–Planck equation that improves the numerical stability of the hierarchical equations of motion for the Brownian oscillator spectral density
Tianchu Li
;
Tianchu Li
Beijing National Laboratory for Molecular Sciences, State Key Laboratory for Structural Chemistry of Unstable and Stable Species, Institute of Chemistry, Chinese Academy of Sciences
, Zhongguancun, Beijing 100190, China
and University of Chinese Academy of Sciences
, Beijing 100049, China
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Yaming Yan
;
Yaming Yan
a)
Beijing National Laboratory for Molecular Sciences, State Key Laboratory for Structural Chemistry of Unstable and Stable Species, Institute of Chemistry, Chinese Academy of Sciences
, Zhongguancun, Beijing 100190, China
and University of Chinese Academy of Sciences
, Beijing 100049, China
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Qiang Shi
Qiang Shi
b)
Beijing National Laboratory for Molecular Sciences, State Key Laboratory for Structural Chemistry of Unstable and Stable Species, Institute of Chemistry, Chinese Academy of Sciences
, Zhongguancun, Beijing 100190, China
and University of Chinese Academy of Sciences
, Beijing 100049, China
b)Author to whom correspondence should be addressed: qshi@iccas.ac.cn
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b)Author to whom correspondence should be addressed: qshi@iccas.ac.cn
J. Chem. Phys. 156, 064107 (2022)
Article history
Received:
December 12 2021
Accepted:
January 26 2022
Citation
Tianchu Li, Yaming Yan, Qiang Shi; A low-temperature quantum Fokker–Planck equation that improves the numerical stability of the hierarchical equations of motion for the Brownian oscillator spectral density. J. Chem. Phys. 14 February 2022; 156 (6): 064107. https://doi.org/10.1063/5.0082108
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