^{1}School of Physics and Optoelectronic Engineering, Anhui University, Hefei, China^{2}State Key Laboratory of Physical Chemistry of Solid Surfaces, College of Chemistry and Chemical Engineering, Fujian Provincial Key Laboratory of Theoretical and Computational Chemistry and Innovation Laboratory for Sciences and Technologies of Energy Materials of Fujian Province (IKKEM), Xiamen University, Xiamen, Fujian, China^{3}Department of Physics and Astronomy, University of Delaware, Newark, DE, United States

## Introduction

The simulation of the inherently quantum mechanical dynamics underlying charge, energy, and coherence transfer in the condensed phase is one of the most difficult challenges in computational physics and chemistry. The exponential scaling of the computational cost with system size makes the quantum-mechanically exact simulations of such processes in complex systems infeasible. With the exception of a few model Hamiltonians whose form makes the numerically exact quantum dynamics simulations possible, any simulation of general condensed-phase systems must rely on approximations [1–33]. Data-driven machine learning (ML) methods for quantum dynamics emerged as an attractive alternative to physics-based approximations due to their low computational cost and high accuracy [34–51]. Development and testing of new simulation methodologies, both physics- and ML-based, would be greatly facilitated if high-quality reference quantum dynamics data for a diverse set of quantum systems of interest were available.

Here, we present a QD3SET-1 database, a collection of eight datasets of time-evolved population dynamics of the two systems: the spin-boson (SB) model and the Fenna–Matthews–Olson (FMO) light-harvesting complex. The datasets are given in Table 1. The SB model describes a (truncated or intrinsic) two-level quantum system linearly coupled to a harmonic bath [52]; [53]. The physics of both the ground state and dynamics of the SB model is very rich. This has been a continuous subject of study during past decades. SB has become a paradigmatic model system in the development of approximate quantum dynamics methods, and nowadays, it is becoming a popular choice for the development of ML models [38]; [45]; [35].

**TABLE 1**. Summary of all datasets. More details are given in the main text. Here, “SB” stands for the spin-boson model. ^{a}modified-quantum_HEOM is the quantum_HEOM package with some local modifications to make it compatible for larger Hamiltonians. ^{b}In the parameter space ^{c}In the parameter space *λ*, *γ*, and *T*) based on farthest-point sampling. The parameter range for each dimension is *λ* = {10, 40, 70, …, 310} cm^{−1}; *γ* = {25, 50, 75, …, 300} fs rad^{−1}; and T = {30, 50, 70, …, 310} K. ^{d}In the parameter space *λ*, *γ*, and *T*) where *λ* = {10, 40, 70, …, 520} cm^{−1}; *γ* = {25, 50, 75, …, 500} cm^{−1}; and T = {30, 50, 70, …, 510} K. ^{e}In the parameter space ^{f}In the case of the FMO complex, this column lists the initially excited sites of the FMO complex in each subset of the corresponding dataset.

The FMO system has become one of the most extensively studied natural light-harvesting complexes [54–60]. Under physiological conditions, the FMO complex forms a homotrimer consisting of eight bacteriochlorophyll-a (BChla) molecules per monomer. The biological function of the FMO trimer is to transfer excitation energy from the chlorosome to the reaction center (RC) [61]. An interest in this light-harvesting system sparked when two-dimensional electronic spectroscopy experiments detected the presence of quantum coherence effects in the FMO complex [62]; [63]; [64]. These observations triggered intense debates about the role this coherence might play in highly efficient excitation energy transfer (EET).

Early studies of the FMO complex considered only seven-site FMO models comprising BChla 1–7. Until BChla 8 was discovered, BChla 1 and 6 were both assumed to be possible locations for accepting the excitation from the chlorosome because they are believed to be the nearest pigments to the antenna, which captures sunlight [65]; [66]; [67]. From there, the energy is subsequently funneled through two nearly independent routes: from site 1 to 2 (pathway 1) or from site 6 to sites 7, 5, and 4 (pathway 2). The terminal point of either route is site 3, where the exciton is then transferred to the RC [68].

Ever since the discovery of the eighth BChla, the role of this pigment in EET has been extensively investigated [69]; [70]; [71]; [72]; [58]; [73]; [74]; [68]; [75]. In particular, it was shown that while the population dynamics of the eight-site FMO model is markedly different from that of a seven-site configuration, the EET efficiencies in both models were predicted to remain comparable and very high [68]. BChla 8 has also been suggested as a possible recipient of the initial excitation.

The dynamics of the FMO model has been a subject of numerous computational studies, primarily focusing on understanding the role of the protein environment in the efficiency of EET (see, e.g., [76]; [77]; [78]; [79]). Numerical simulations typically employ one of the several parameterized or fitted into the experimental data FMO model Hamiltonians that differ in the BChla excitation energies and the couplings between different BChla [61]; [80]; [54]; [81]; [82]; [83]; [84]. Simulations of the full FMO trimer containing 24 BChlas have also been performed [85]; [86].

Accompanying this data report, the QD3SET-1 database contains seven datasets of time-evolved population dynamics of FMO models with different system Hamiltonians and initial excitations for several hundreds of bath and system–bath parameters. The hierarchy of equations of motion (HEOM) approach [5,7] was used to simulate the population dynamics of SB and FMO models, in one of the seven FMO datasets. HEOM is a numerically exact method that can describe the dynamics of a system with a non-perturbative and non-Markovian system–bath interaction. The high computational cost of HEOM, however, limits the number of FMO simulations that can be performed with this method. To generate the other six FMO datasets, an approximate method—the local thermalizing Lindblad master equation (LTLME) [87]; [88]—was used.

Some of our data were already used in previous studies developing and benchmarking ML models for quantum dynamics simulations [37,38]; [35]; [36]. Here, we regenerate one of the datasets to augment it with more data and provide many new datasets generated from scratch (Table 1). To facilitate their use, we organized the datasets in a coherently formatted database and provided metadata and extraction scripts. We expect that our database that accompanies this data report will serve as a valuable resource in the development of new quantum dynamics methods.

## Methods

### Spin-boson dataset

This dataset is regenerated with the same settings and parameters as in on our previous SB dataset [38] in order to include all the elements (populations and coherences) of the reduced density matrix (RDM) of the system. The populations and population differences were published and used previously [38]; [35]. In the following section, we provide a brief summary of the self-containing presentation of the dataset.

#### Spin-boson model

The spin-boson model comprises a two-level quantum subsystem (TLS) coupled to a bath of harmonic oscillators. The Hamiltonian has the following standard system–bath form: *ℏ* = 1)

where *ϵ* is the so-called energy bias and Δ is the tunneling matrix element. The harmonic bath is an ensemble of independent harmonic oscillators

where *N*_{b} independent harmonic bath modes with masses {*m*_{j}} and frequencies {*ω*_{j}}. The TLS and bath are coupled through the additional term

where {*c*_{j}} is the coupling coefficients.

The effects of the bath on the dynamics of the TLS are collectively determined by the spectral density function [89]

In this work, we choose to employ the Debye form of the spectral density (Ohmic spectral density with the Drude–Lorentz cutoff) [90]

where *λ* is the bath reorganization energy, which controls the strength of system–bath coupling, and the cutoff frequency *γ* = 1/*τ*_{c} (*τ*_{c} is the bath relaxation time).

All dynamical properties of the TLS can be obtained from the RDM

where *α*, *β* ∈ {| + ⟩, | − ⟩},

The initial state of the total system is assumed to be a product state of the system and bath in the following form:

In Eq. 7, the bath density operator is an equilibrium canonical density operator *k*_{B} is the Boltzmann constant. The initial density operator of the system is chosen to be

#### Data generation for the spin-boson model

The dataset for the spin-boson model was generated as described previously [38]. We also summarize it here. The following system and bath parameters were chosen: *t*_{max}Δ = 20, and the HEOM integration time step was set to *dt*Δ = 0.05. In total, 1,000 HEOM calculations, 500 for symmetric (*ϵ*/Δ = 0) and 500 for asymmetric (*ϵ*/Δ = 1) spin-boson Hamiltonians, were performed. The dataset contains a set of RDMs from *t*Δ = 0 to *t*_{max}Δ = 20, saved every *dt*Δ = 0.05, for every combination of the parameters

### Fenna–Matthews–Olson complex datasets

In this section, we first describe the general theory behind the FMO model Hamiltonian, and later, for each dataset, we provide specific technical details. Table 1 provides an overview of each dataset.

#### FMO model Hamiltonian

The FMO complex in this work is described by the system–bath Hamiltonian with the renormalization term

where |*n*⟩ denotes that only the *n*th site is in its electronically excited state and all other sites are in their electronically ground states, *E*_{n} is the transition energies, and *V*_{nm} is the Coulomb coupling between *n*th and *m*th sites. The couplings are assumed to be constant (the Condon approximation). It should be noted that the overall electronic ground state of the pigment protein complex |0⟩ is assumed to be only radiatively coupled to the single-excitation manifold, and as such, it is not included in the dynamics calculations. Analogous with the SB model, the bath is modeled by a set of independent harmonic oscillators. The thermal bath is coupled to the subsystem’s states |*n*⟩ through the system–bath interaction term

where each subsystem’s state is independently coupled to its own harmonic environment and *c*_{nj} is the pigment–phonon coupling constants of environmental phonons local to the *n*th BChla.

The FMO model Hamiltonian contains a reorganization term that counters the shift in the minimum energy positions of harmonic oscillators introduced by the system–bath coupling. In the case that each state |*n*⟩ is independently coupled to the environment, the renormalization term takes the following form:

where

Analogous to the SB dataset, the initial state of the total system is assumed to be a product state of the system and bath. The initial electronic density operator given by

#### FMO-Ia, FMO-Ib, and FMO-II datasets: seven-site FMO models using the local thermalizing Lindblad master equation approach

We generated datasets for the two seven-site system (*N*_{e} = 7) Hamiltonians. The FMO-I dataset was generated for the system Hamiltonian parameterized by Adolphs and Renger [54], and is given by (in cm^{−1})

The FMO-Ia dataset comes directly from our previous studies [37,49], and the FMO-Ib dataset was generated here for a broader parameter space described as follows.

The FMO-II dataset was generated for the Hamiltonian parameterized by Cho et al. [81], which takes the following form (in cm^{−1}):

The diagonal offset of 12,210 cm^{−1} is added to both Hamiltonians. Each site is coupled to its own bath characterized by the Drude–Lorentz spectral density, Eq. 5, but the bath of each site is described by the same spectral density.

For the FMO-Ia dataset, the following spectral density parameters and temperatures were employed: *λ* = {10, 40, 70, …, 310} cm^{−1}; *γ* = {25, 50, 75, …, 300} fs rad^{−1}; and T = {30, 50, 70, …, 310} K. For the FMO-Ib and FMO-II datasets, the spectral density parameters and temperatures were *λ* = {10, 40, 70, …, 520} cm^{−1}; *γ* = {25, 50, 75, …, 500} cm^{−1}; and T = {30, 50, 70, …, 510} K. Our choice of temperatures is relevant for most of the experiments performed on the FMO and other photosynthetic complexes [92]; [93]; [56].

For FMO-Ia, FMO-Ib, and FMO-II datasets, farthest-point sampling [94] was employed to select the most distant points in the Euclidean space [37] of parameters, which typically more efficiently covers relevant space than random sampling [94]. We choose the top 500 (most distant) combinations of (*λ*, *γ*, and *T*) based on farthest-point sampling. For each selected set of parameters, the system RDM was calculated using the so-called local thermalizing Lindblad master equation (LTLME) approach [95]; [88]. Implemented in the quantum_HEOM package [5,96], the LTLME method is based on the Lindblad quantum master equation, which is the commonly used approach to study the dynamics of open quantum systems [97]; [98]; [99]. Specifically, in the LTLME approach, for each unique frequency gap between eigenstates of the system Hamiltonian and for every possible combination of site *n*, Lindblad operators are constructed. A sum is carried out over all transitions with a unique frequency, and the contribution of each population transfer is weighted by *c*_{n}(Ω) is the *n*th site coefficient of exciton (eigenstate) Ω, while

Coming to data generation, two subsets of the dataset were generated, one for the initial electronic density operator

#### FMO-III and FMO-IV datasets: eight-site FMO models using the local thermalizing Lindblad master equation approach

Using the same LTLME-based approach, we generated a dataset for two different Hamiltonians for the eight-site FMO model. The first Hamiltonian (FMO-III dataset) was parameterized by Jia et al. [75]. The electronic system Hamiltonian is given by (in cm^{−1})

with a diagonal offset of 11,332 cm^{−1}.

The FMO-IV dataset was generated for the Hamiltonian parameterized by Busch et al. [69] (site energies) and Olbrich et al. [72] (excitonic couplings) and takes the following form (in cm^{−1}):

with a diagonal offset of 12,195 cm^{−1}.

The same set of spectral density parameters and temperatures that was used in the generation of the FMO-Ib and FMO-II datasets was used here. The LTLME method was used to propagate the system RDM from 0 to 50 ps with a 5 fs time step, and three initial states of the electronic system were considered: sites 1, 6, and 8. The dataset contains both diagonal (populations) and off-diagonal (coherences) elements of the RDM. The calculations were performed using the quantum_HEOM package [96] with some local modifications to make it compatible for the Hamiltonians with larger dimension. We refer to this as modified-quantum_HEOM implementation.

#### FMO-V dataset: FMO trimer using the local thermalizing Lindblad master equation approach

Additionally, we also generated a dataset for the FMO trimer. The overall excitonic Hamiltonian of all three subunits is given by

where *H*_{A} is the subunit Hamiltonian for which we used the same Hamiltonian as in the FMO-IV dataset (Eq. 14), while *H*_{B} is the inter-subunit Hamiltonian, which is taken from the work of Olbrich et al. [72] and is given by (in cm^{−1})

We propagate dynamics with LTLME from 0 to 50 ps with a 5 fs time step for the same parameters as were adopted in the calculations for the FMO-Ib–FMO-IV datasets. The calculations were performed with the modified-quantum_HEOM implementation for the initial excited sites 1, 6, and 8.

#### FMO-VI dataset: an eight-site FMO model using the hierarchy of equations of motion approach

The LTLME approach provides only an approximate description of the quantum dynamics of the FMO complex. Therefore, the FMO-I–FMO-V datasets are useful merely for developing machine learning models for quantum dynamics studies. For example, they can be used to train a neural network model, which can then be further improved on more accurate but smaller datasets (e.g., via transfer learning). However, LTLME dynamics cannot be used to benchmark other quantum dynamics methods. In this case, high-quality reference data are needed.

To generate a dataset with accurate FMO dynamics, we performed HEOM calculations for the eight-site FMO model with the Hamiltonian given by Eq. 14. HEOM calculations were performed using the parallel hierarchy integrator (PHI) code [100]. The initial dataset was chosen on the basis of farthest-point sampling, similar to how it was performed in the FMO-Ib–FMO-V datasets, with the only difference being that instead of the 500 most distant sets of parameters that were chosen in the preparation of FMO-Ib–FMO-V data sets, the 1,100 most distant sets of parameters were used to prepare the initial FMO-VI dataset. For certain parameters, the RAM requirements exceeded the RAM of computing nodes available to us (1 TB). Therefore, such parameter sets were excluded from the dataset. Excluded parameters correspond to low temperatures, high reorganization energies, and low cutoff frequencies. Such strong non-Markovian regimes pose significant challenges in the computational studies of open quantum systems. Approximately 20% of the initial dataset was removed because of prohibitive memory requirements. We note that even though graphics processing unit (GPU) implementations of HEOM (e.g., Kreisbeck et al. [101]) are much faster than their CPU-based counterparts, they are still limited by the small amount of memory in presently available GPUs.

For the remaining 80% of the dataset, HEOM calculations were performed for 2.0 ps. To speed up calculations, an adaptive integration Runge–Kutta–Fehlberg 4/5 [102] (RKF45) method was used, as implemented in the PHI code. Using adaptive integration reduces both the total computation time and memory requirements but can lead to artifacts if the accuracy threshold is set too large [100]. In this work, the PHI default accuracy threshold of 1·10^{−6} was used. The initial integration time step was set to 0.1 fs. In RKF45, the integration time step is varied, and therefore, the output comprises time-evolved RDMs on an unevenly spaced time grid. To obtain the RDMs on an evenly spaced time grid of 0.1 fs, cubic spline interpolation was used. The interpolation errors were examined on a few cases where 0.1 fs fixed time step integration was feasible. The errors in the populations were found to be less than 10^{−5}, which is much smaller than the convergence thresholds, as illustrated in the Technical Validation section of Supplementary Material. The final FMO-VI dataset contains 879 entries, each comprising all the populations and coherences for the RDM from 0 to 2 ps with a time step of 0.1 fs.

In Figure 1, we present a comparison of the dynamics of the eight-site FMO model described by Eq. 14. The calculations were performed using the LTLME and HEOM methods for two different parameter sets: *T* = 330 K, *λ* = 10 cm^{−1}, and *γ* = 475 cm^{−1} and *T* = 310 K, *λ* = 430 cm^{−1}, and *γ* = 75 cm^{−1}. The results clearly demonstrate the differences between the two methods. HEOM, being a numerically exact method, accurately captures the coherence dynamics of the FMO Hamiltonian. On the other hand, LTLME is an approximate method that does not fully account for the back-reaction from the bath to the system. As a result, it tends to underestimate the coherence dynamics in this context.

**FIGURE 1**. Population dynamics of the eight-site FMO model with the system Hamiltonian given by Eq. 14 calculated using HEOM (solid) and LTLME (dashed) methods for the following parameters **(A)**: *T* = 330 K, *λ* = 10 cm^{−1}, and *γ* = 475 cm^{−1} and **(B)** *T* = 310 K, *λ* = 430 cm^{−1}, and *γ* = 75 cm^{−1}.

## Data availability statement

All data sets can be accessed at Figshare https://doi.org/10.25452/figshare.plus.c.6389553. The data sets are stored in standard NumPy [103] binary file format (.npy) files. The following format of file names was adopted in the SB data set 2_epsilon-X_Delta-1.0_lambda-Y_gamma-Z_beta-XX.npy where X denotes the value of the energy bias (∼ε), Y is the reorganization energy ∼λ, Z is the cut-off frequency ∼γ and XX is the inverse temperature ∼β. The following format of file names was adopted in all FMO data sets X_initial-Y_gamma-Z_lambda-XX_temp-YY.npy, where X denotes the number of sites in the FMO model, Y is the initial state, Z is the value of bath frequency, XX is the value of reorganization energy, and YY is the temperature. A Python package for extracting data is provided together with the data set and can be accessed at https://github.com/Arif-PhyChem/QD3SET.

## Author contributions

PD and AU conceived the idea of creating a HEOM-based spin-boson database. AU conceived the idea of creating an LTLME-based database for the FMO complex. AK conceived the idea of creating an FMO dataset using the HEOM method. AU performed the HEOM calculations for the spin-boson, along with the LTLME calculations for the FMO complex. AU wrote the provided package for easy extraction of the data. AK and LH performed the calculations and created database files for the FMO-VI dataset. All authors analyzed the results. AK took the lead in writing the original draft of the manuscript. All authors contributed to the article and approved the submitted version.

## Funding

This work was supported by General University Research (GUR) Grants and startup funds of the College of Arts and Sciences and the Department of Physics and Astronomy of the University of Delaware. PD acknowledges funding by the National Natural Science Foundation of China [No. 22003051 and funding via the Outstanding Youth Scholars (Overseas, 2021) project], the Fundamental Research Funds for the Central Universities (No. 20720210092), and via the Lab project of the State Key Laboratory of Physical Chemistry of Solid Surfaces. This project was supported by the Science and Technology Projects of Innovation Laboratory for Sciences and Technologies of Energy Materials of Fujian Province (IKKEM) (No. RD2022070103). AK acknowledges the Ralph E. Powe Junior Faculty Enhancement Award from Oak Ridge Associated Universities.

## Acknowledgments

This research was supported in part through the use of Data Science Institute (DSI) computational resources at the University of Delaware. Calculations were also performed with high-performance computing resources provided by the Xiamen University.

## Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

## Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors, and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

## Supplementary material

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fphy.2023.1223973/full#supplementary-material

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Keywords: machine learning, quantum dissipative dynamics, hierarchical equations of motion, Fenna–Matthews–Olson complex, Lindblad master equation, spin-boson model

Citation: Ullah A, Herrera Rodríguez LE, Dral PO and Kananenka AA (2023) QD3SET-1: a database with quantum dissipative dynamics datasets. *Front. Phys.* 11:1223973. doi: 10.3389/fphy.2023.1223973

Received: 18 May 2023; Accepted: 03 July 2023;

Published: 07 August 2023.

Edited by:

Nguyen Hoang Lam, Memorial University of Newfoundland, CanadaReviewed by:

Arkajit Mandal, Columbia University, United StatesSaravana Prakash Thirumuruganandham, Universidad Technologica de Indoamerica, Ecuador

Copyright © 2023 Ullah, Herrera Rodríguez, Dral and Kananenka. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Alexei A. Kananenka, akanane@udel.edu; Pavlo O. Dral, dral@xmu.edu.cn