[Paper Review] A Thermofield-based Multilayer Multiconfigurational Time-Dependent Hartree Approach to Non-Adiabatic Quantum Dynamics at Finite Temperature

Eric W. Fischer, Peter Saalfrank|arXiv (Cornell University)|Jul 1, 2021

Quantum many-body systems59 references14 citations

TL;DR

This paper introduces a thermo-field dynamics (TFD)-based multilayer multiconfigurational time-dependent Hartree (ML-MCTDH) method to simulate finite-temperature non-adiabatic quantum dynamics from a non-stochastic, wave-function perspective. By mapping thermal density operators to pure thermo-field states via the thermal quasi-particle (TQP) representation, the approach enables efficient, scalable simulations of non-adiabatic processes in molecular systems at finite temperature, validated on a 24-mode pyrazine model with excellent agreement to prior ρMCTDH results.

ABSTRACT

We introduce a thermofield-based formulation of the multilayer multiconfigurational time-dependent Hartree (ML-MCTDH) method to study finite temperature effects on non-adiabatic quantum dynamics from a non-stochastic, wave-function perspective. Our approach is based on the formal equivalence of bosonic many-body theory at zero temperature with doubled number of degrees of freedom and the thermal quasi-particle representation of bosonic thermofield dynamics (TFD). This equivalence allows for a transfer of bosonic many-body MCTDH as introduced by Wang and Thoss to the finite temperature framework of thermal quasi-particle TFD. As an application, we study temperature effects on the ultrafast internal conversion dynamics in pyrazine. We show, that finite temperature effects can be efficiently accounted for in the construction of multilayer expansions of thermofield states in the framework presented herein. Further, we find our results to agree well with existing studies on the pyrazine model based on the $ ho$MCTDH method.

Motivation & Objective

• To develop a non-stochastic, wave-function-based method for simulating non-adiabatic quantum dynamics at finite temperature.
• To extend the multilayer multiconfigurational time-dependent Hartree (ML-MCTDH) formalism to finite-temperature regimes using thermo-field dynamics (TFD).
• To enable efficient, scalable simulations of non-adiabatic processes in large molecular systems with vibrational baths at finite temperature.
• To validate the approach on the ultrafast internal conversion dynamics in pyrazine, a benchmark system in non-adiabatic dynamics.

Proposed method

• Formulates a thermo-field-based ML-MCTDH method by combining second quantization representation (SQR) of MCTDH with the thermal quasi-particle (TQP) representation of TFD.
• Maps the finite-temperature density operator to a pure thermo-field state on an extended Hilbert space, allowing time evolution via a Schr"{o}dinger-type equation.
• Constructs multilayer expansions of thermo-field states using time-dependent single-particle functions (tSPFs) for both physical and auxiliary modes.
• Applies the TQP representation to transform the vibronic Hamiltonian, including linear and bilinear coupling terms, into a form compatible with the ML-MCTDH framework.
• Performs thermal ensemble averages via expectation values in the thermo-field state, with observables like diabatic populations and vibrational occupation numbers computed directly from the wave function.
• Uses the Heidelberg MCTDH package for efficient implementation and numerical propagation of the thermo-field state.

Experimental results

Research questions

• RQ1 Can the ML-MCTDH method be extended to finite-temperature non-adiabatic dynamics using a non-stochastic, wave-function-based approach?
• RQ2 How do finite-temperature effects influence the ultrafast internal conversion dynamics in pyrazine?
• RQ3 What is the accuracy and scalability of the TFD-ML-MCTDH approach compared to existing ρMCTDH and stochastic methods?
• RQ4 How do different vibronic coupling models (linear vs. bilinear) and bath coupling affect the temperature-dependent dynamics in pyrazine?
• RQ5 Can the TQP representation efficiently capture thermal populations and correlation effects in large, high-dimensional molecular systems?

Key findings

• The TFD-ML-MCTDH method successfully captures finite-temperature effects on non-adiabatic dynamics in pyrazine with excellent agreement to prior ρMCTDH results.
• At 500 K, the number of tSPFs required for convergence increases significantly, with 37/32 tSPFs needed for the S1/S2 states in the bilinear model, indicating strong thermal effects on state mixing.
• The method accurately reproduces linear absorption spectra, with the TFD autocorrelation function matching the standard quantum mechanical result via a rigorous derivation in Appendix E.
• Thermal populations are correctly described via the TQP representation, with ⟨ˆn_k⟩_β(t) expressed as a sum of physical and auxiliary mode contributions, including the Bose-Einstein distribution term.
• The approach maintains numerical efficiency and scalability, with the number of primitive basis functions per mode set to 45 (physical) and 45 (auxiliary) for v10a, 35 for v6a, 21 for v1, and 12 for v9a.
• The method enables direct computation of observables such as diabatic populations and vibrational occupation numbers from the thermo-field state, avoiding stochastic sampling or density matrix propagation.