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WIREs Comput Mol Sci
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Finite temperature quantum dynamics of complex systems: Integrating thermo‐field theories and tensor‐train methods

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Abstract This review provides the fundamental theoretical tools for the development of a complete wave‐function formalism for the study of time‐evolution of chemico‐physical systems at finite temperature. The methodology is based on the non‐equilibrium thermo‐field dynamics (NE‐TFD) representation of quantum mechanics, which is alternative to the commonly used density matrix representation. TFD concepts are extended and integrated with the tensor‐train (TT) numerical tools leading to a novel and powerful theoretical and computational framework for the study of complex quantum dynamical problems. In addition, NE‐TFD techniques are extended to enable the study of dissipative open systems via a new formulation of the hierarchical equations of motion (HEOM) fully integrated with TT methodologies. We demonstrate that the combination of the TFD machinery with computational advantages of TTs results in a powerful theoretical and computational framework for scrutinizing dynamics of complex multidimensional electron‐vibrational systems. We illustrate the validity and the computational advantages of the developed methodologies by applying them to the study of quantum coherence effects in the energy‐transfer processes in antenna systems, to the analysis of fingerprints of vibrational modes in electron‐transfer and charge‐transfer processes in various model and realistic multidimensional molecular systems, as well as to simulation of other fundamental models of physical chemistry. This article is categorized under: Theoretical and Physical Chemistry > Reaction Dynamics and Kinetics Theoretical and Physical Chemistry > Statistical Mechanics
Population dynamics of a homodimer with 14 nuclear vibrations. Bath reorganization energies are (a) 90 cm−1; (b) 300 cm−1. (c) Convergence of the norm for different TT truncation ranks as indicated in the legend. Converged results are obtained with TT ranks 115. The hierarchy level is truncated at m = 10 on each bath
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(a) Electronic population of the initially populated state (a) for different values of the TT compression ranks; (b) for different values of the temperature, TT ranks are set to 50
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Effective site spectral densities Jp(ω) and Jt(ω) describing the coupling of the physical and tilde bosonic DoFs with the electronic subsystem at different temperatures. (a,b) 77 K, (c,d) 300 K
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(a) Time evolution of the electronic populations pn(t) of seven (n = 1 − 7) BChl molecules of the FMO complex at different temperatures indicated in the panels. The initial excitation is localized on site 1. Different colors label different sites as specified in the legend. (b) Inverse participation ratio Π(t) as a function of time; () 300 K, (−−) 77 K, (−⋅) 0 K
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Effective site spectral densities Jp(ω) and Jt(ω) describing the coupling of the physical and tilde bosonic DoFs with the electronic subsystem at different temperatures. (a,b) 77 K, (c,d) 300 K
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The time evolution of the electronic population P(t), for different values of α = λ/(2ωc) and for a tunneling amplitude V = 40 cm−1; (a) λ = 20 cm−1, T = 30 K; (b) λ = 20 cm−1, T = 300 K; (c) λ = 80 cm−1, T = 30 K; (d) λ = 80 cm−1, T = 300 K
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Population P(t) of the initial electronic state as a function of time for different values of the TT compression ranks simulated for α = (20 cm−1)/(2ωc) and V = 40 cm−1. (a) T = 300 K; (b) T = 30 K
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Graphical representation of a tensor train. Each square node represents a core of the TT, and each vertical line represents an index ik of the tensor. Connecting lines represent the contractions over the indices αk
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Population P(t) of the initial electronic state at T = 300 K as a function of time, for different values of α = λ/(2ωc); (a) λ = 5 cm−1; (b) λ = 20 cm−1 (both with the tunneling amplitude V = 40 cm−1); (c) λ = 20; (d) λ = 80 (both with V = 100 cm−1). Full lines: TFD‐TT calculations. Blue dots: Numerically exact HEOM calculations of Ref. 79
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Theoretical and Physical Chemistry > Statistical Mechanics
Theoretical and Physical Chemistry > Reaction Dynamics and Kinetics

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