# QD3SET-1: A Database with Quantum Dissipative Dynamics Data Sets

Arif Ullah  State Key Laboratory of Physical Chemistry of Solid Surfaces,
Fujian Provincial Key Laboratory of Theoretical and Computational Chemistry,
Department of Chemistry, and College of Chemistry and Chemical Engineering,
Xiamen University, Xiamen 361005, China  Luis E. Herrera Rodríguez  Department
of Physics and Astronomy, University of Delaware, Newark, Delaware 19716,
United States  Pavlo O. Dral  State Key Laboratory of Physical Chemistry of
Solid Surfaces, Fujian Provincial Key Laboratory of Theoretical and
Computational Chemistry, Department of Chemistry, and College of Chemistry and
Chemical Engineering, Xiamen University, Xiamen 361005, China  corresponding
author(s): Pavlo O. Dral<EMAIL_ADDRESS>and Alexei A. Kananenka
<EMAIL_ADDRESS> Alexei A. Kananenka  Department of Physics and Astronomy,
University of Delaware, Newark, Delaware 19716, United States  corresponding
author(s): Pavlo O. Dral<EMAIL_ADDRESS>and Alexei A. Kananenka
<EMAIL_ADDRESS>
###### Abstract

Simulations of the dynamics of dissipative quantum systems utilize many
methods such as physics-based quantum, semiclassical, and quantum-classical as
well as machine learning-based approximations, development and testing of
which requires diverse data sets. Here we present a new database QD3SET-1
containing eight data sets of quantum dynamical data for two systems of broad
interest, spin-boson (SB) model and the Fenna–Matthews–Olson (FMO) complex,
generated with two different methods solving the dynamics, approximate local
thermalizing Lindblad master equation (LTLME) and highly accurate hierarchy
equations of motion (HEOM). One data set was generated with the SB model which
is a two-level quantum system coupled to a harmonic environment using HEOM for
1,000 model parameters. Seven data sets were collected for the FMO complex of
different sizes (7- and 8-site monomer and 24-site trimer with LTLME and
8-site monomer with HEOM) for 500–879 model parameters. Our QD3SET-1 database
contains both population and coherence dynamics data and part of it has been
already used for machine learning-based quantum dynamics studies.

## Background & Summary

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 for 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, 2, 3, 4, 5, 6,
7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26,
27, 28] Data-driven machine learning (ML) methods for quantum dynamics emerged
as attractive alternative to the physics-based approximations due to their low
computational cost and high accuracy. [29, 30, 31, 32, 33, 34, 35, 36, 37, 38,
39, 40, 41, 42, 43, 44, 45, 46] 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 data sets of time-
evolved population dynamics of the two systems: spin-boson (SB) model and the
Fenna–Matthews–Olson (FMO) light-harvesting complex. The data sets are
summarized in Table 1. The SB model describes a (truncated or intrinsic) two-
level quantum system linearly coupled to a harmonic bath.[47, 48] The physics
of both the ground state and the dynamics of the SB model is very rich. It has
been a continuous object of study during the 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.[33, 40, 30]

The FMO system has become one of the most extensively studied natural light-
harvesting complexes.[49, 50, 51, 52, 53, 54, 55] 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).[56] 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. [57, 58, 59] These
observations triggered intense debates about the role this coherence might
play in the highly efficient excitation energy transfer (EET).

Early studies of the FMO complex considered only seven-site FMO models
comprising of 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 the sunlight.[60, 61, 62] 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.[63]

Ever since the discovery of the eighth BChla, the role of this pigment in the
EET has been extensively investigated.[64, 65, 66, 67, 53, 68, 69, 63, 70] In
particular, it was shown that while the population dynamics of the eight-site
FMO model is markedly different from a seven-site configuration, the EET
efficiencies in both models were predicted to remain comparable and very
high.[63] BChla 8 has also been suggested as 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 on the efficiency of EET (see e.g., Refs. 71, 72, 73, 74).
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 sites.[56,
75, 49, 76, 77, 78, 79] Simulations of the full FMO trimer containing 24 BChl
have also been performed.[80, 81]

Reported in this Data Descriptor a QD3SET-1 database contains seven data sets
of time-evolved population dynamics of FMO models with different system
Hamiltonians and initial excitations for several hundreds of bath and system-
bath parameters. Hierarchy of equations of motion (HEOM) approach[7, 5] was
used to simulate the population dynamics of SB and FMO models, in one of the
seven FMO data sets. 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 other
six FMO data sets, an approximate method—local thermalizing Lindblad master
equation (LTLME)[82, 83] was used.

Some of our data was already used in previous studies developing and
benchmarking ML models for quantum-dynamics simulations. [33, 32, 30, 31] Here
we regenerate one of the data sets to augment with more data and provide many
new data sets generated from scratch (Table 1). To facilitate their use, we
organize the data sets in a coherently formatted database and provided
metadata and extraction scripts. We expect our Database that accompany this
Data Descriptor will serve as valuable resources in the development of new
quantum-dynamics methods.

## Methods

### SB data set

This data set is re-generated with the same settings and the same parameters
as in on our previous SB data set[33] in order to include all the elements
(populations and coherences) of the system’s reduced density matrix (RDM). The
populations and population differences were published and used before.[33, 30]
Below we provide a brief summary for self-containing presentation of the data
set.

#### 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: $\hat{H}=\hat{H}_{s}+\hat{H}_{b}+\hat{H}_{sb}$. The
Hamiltonian of the TLS in the local (or site) basis
$\\{|+\rangle,|-\rangle\\}$ is given by ($\hslash$=1)

$\hat{H}_{s}=\epsilon\left(|+\rangle\langle+|-|-\rangle\langle-|\right)\,+\,\Delta\left(|+\rangle\langle-|+|-\rangle\langle+|\right),$
(1)

where $\epsilon$ is the so-called energetic bias and $\Delta$ is the tunneling
splitting. The harmonic bath is an ensemble of independent harmonic
oscillators

$\hat{H}_{b}=\sum_{j=1}^{N_{b}}\left(\frac{\hat{p}_{j}^{2}}{2m_{j}}+\frac{1}{2}m_{j}\omega_{j}^{2}\hat{x}_{j}^{2}\right),$
(2)

where $\\{\hat{x}_{j}\\}$ and $\\{\hat{p}_{j}\\}$ are the coordinates and
momenta, respectively, of $N_{b}$ independent harmonic bath modes with masses
$\\{m_{j}\\}$ and frequencies $\\{\omega_{j}\\}$. The TLS and bath are coupled
through the additional term

$\hat{H}_{sb}=-\sum_{j=1}^{N_{b}}c_{j}\hat{x}_{j}\left(|+\rangle\langle+|-|-\rangle\langle-|\right),$
(3)

where $\\{c_{j}\\}$ are the coupling coefficients.

The effects of the bath on the dynamics of TLS are collectively determined by
the spectral density function[84]

$J(\omega)=\frac{\pi}{2}\sum_{j=1}^{N_{b}}\frac{c_{j}^{2}}{m_{j}\omega_{j}}\delta(\omega-\omega_{j}).$
(4)

In this work we choose to employ the Debye form of the spectral density (Ohmic
spectral density with the Drude–Lorentz cut-off) [85]

$J(\omega)=2\lambda\frac{\omega\gamma}{\omega^{2}+\gamma^{2}},$ (5)

where $\lambda$ is the bath reorganization energy, which controls the strength
of system-bath coupling, and the cutoff frequency $\gamma=1/\tau_{c}$
($\tau_{c}$ is the bath relaxation time).

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

$\tilde{\rho}_{\alpha\beta}(t)=\text{Tr}_{b}\langle\alpha|e^{-i\hat{H}t/\hslash}\hat{\rho}(0)e^{i\hat{H}t/\hslash}|\beta\rangle,$
(6)

where $\alpha,\beta\in\\{|+\rangle,|-\rangle\\}$, $\hat{\rho}$ is the total
density operator, and the trace is taken over bath degrees of freedom. For
example, the commonly used in benchmark studies population difference is
obtained from the RDM as follows:
$p_{+}(t)-p_{-}(t)=\tilde{\rho}_{++}(t)-\tilde{\rho}_{--}(t)$.

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

$\hat{\rho}(0)=\hat{\rho}_{\mathrm{s}}(0)\hat{\rho}_{\mathrm{b}}(0).$ (7)

In Eq. (7) the bath density operator is an equilibrium canonical density
operator
$\hat{\rho}_{b}(0)=e^{-\beta\hat{H}_{\mathrm{b}}}/\mathrm{Tr}_{\mathrm{b}}\left[e^{-\beta\hat{H}_{\mathrm{b}}}\right]$,
where $\beta=(k_{\mathrm{B}}T)^{-1}$ is the inverse temperature and
$k_{\mathrm{B}}$ is the Boltzmann constant. The initial density operator of
the system is chosen to be $\hat{\rho}_{\mathrm{s}}(0)=|+\rangle\langle+|$.
These conditions corresponds to instantaneous photoexcitation of the
subsystem.

#### Data generation with spin-boson model and the hierarchy equations of
motion approach

The data set for the spin-boson model was generated as described
previously.[33] We also summarize it below. The following system and bath
parameters were chosen: $\tilde{\epsilon}=\epsilon/\Delta=\\{0,1\\}$,
$\tilde{\lambda}=\lambda/\Delta=\\{0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1.0\\}$,
$\tilde{\gamma}=\gamma/\Delta=\\{1,2,3,4,5,6,7,8,9,10\\}$, and
$\tilde{\beta}=\beta\Delta=\\{0.1,0.25,0.5,0.75,1\\}$, where the tunneling
matrix element $\Delta$ is set as an energy unit. For all combinations of
these parameters the system’s RDM was propagated using HEOM approach
implemented in QuTiP software package. [86] The total propagation time was
$t_{\mathrm{max}}\Delta=20$ and the HEOM integration time-step was set to
$dt\Delta=0.05$. In total, 1,000 of HEOM calculations, 500 for symmetric
($\epsilon/\Delta=0$) and 500 for asymmetric ($\epsilon/\Delta=1$) spin-boson
Hamiltonian were performed. The data set contains a set of RDMs from
$t\Delta=0$ to $t_{\mathrm{max}}\Delta=20$, saved every $dt\Delta=0.05$, for
every combination of the parameters
$(\tilde{\epsilon},\tilde{\lambda},\tilde{\gamma},\tilde{\beta})$ described
above.

### Fenna–Matthews–Olson complex data sets

In this section we first describe the general theory behind the FMO model
Hamiltonian and later for each data set we provide specific technical details.
See also Table 1 for an overview of each data set.

#### FMO model Hamiltonian

The FMO complex in this work is described by the system-bath Hamiltonian with
the renormalization term
$\hat{H}=\hat{H}_{s}+\hat{H}_{b}+\hat{H}_{sb}+\hat{H}_{ren}$. The electronic
system is described by the Frenkel exciton Hamiltonian

$\hat{H}_{s}=\sum_{n=1}^{N_{e}}E_{n}|n\rangle\langle n|+\sum_{n,m=1,n\neq
m}^{N_{e}}V_{nm}|n\rangle\langle m|,$ (8)

where $|n\rangle$ denotes that only the $n$th site is in its electronically
excited state and all other sites are in their electronically ground states,
$E_{n}$ are 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). Note that the overall electronic ground state of the
pigment protein complex $|0\rangle$ is assumed to be only radiatively coupled
to the single-excitation manifold and as such it is not included in the
dynamics calculations. In analogy 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 $|n\rangle$ through the system-bath interaction term

$\hat{H}_{sb}=\sum_{n=1}^{N_{e}}\sum_{j=1}^{N_{b}}c_{nj}\hat{x}_{j}|n\rangle\langle
n|,$ (9)

where each subsystem’s state is independently coupled to its own harmonic
environment and $c_{nj}$ are the pigment-phonon coupling constants of
environmental phonons local to the $n$th BChla.

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

$\hat{H}_{ren}=\sum_{n=1}^{N_{e}}\lambda_{n}|n\rangle\langle n|,$ (10)

where $\lambda_{n}=\sum_{j}c_{nj}^{2}/(2m_{j}\omega_{j}^{2})$ is the bath
reorganization energy. The bath spectral density associated with each
electronic state is assumed to be given by the Lorentz–Drude spectral density
(Eq. 5).

Analogously to the SB data set 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 $\hat{\rho}_{s}(0)$ was varied as described below.
The bath density operator is taken to be the equilibrium canonical density
operator.

#### FMO-Ia, FMO-Ib, and FMO-II data sets: 7-site FMO models with the local
thermalizing Lindblad master equation approach

We generated data sets for the two 7-site system ($N_{e}=7$) Hamiltonians.
FMO-I data set was generated for the system Hamiltonian parameterized by
Adolphs and Renger[49] and given by (in cm-1)

$H_{s}=\begin{pmatrix}200&-87.7&5.5&-5.9&6.7&-13.7&-9.9\\\
-87.7&320&30.8&8.2&0.7&11.8&4.3\\\ 5.5&30.8&0&-53.5&-2.2&-9.6&6.0\\\
-5.9&8.2&-53.5&110&-70.7&-17.0&-63.6\\\ 6.7&0.7&-2.2&-70.7&270&81.1&-1.3\\\
-13.7&11.8&-9.6&-17.0&81.1&420&39.7\\\
-9.9&4.3&6.0&-63.3&-1.3&39.7&230\end{pmatrix},$ (11)

FMO-Ia data set comes directly from our previous studies[32, 31] and FMO-Ib
data set was generated here for a broader parameter space as described below.

FMO-II data set was generated for the Hamiltonian parameterized by Cho et
al.[76] which takes the following form (in cm-1)

$H_{s}=\begin{pmatrix}280&-106&8&-5&6&-8&-4\\\ -106&420&28&6&2&13&1\\\
8&28&0&-62&-1&-9&17\\\ -5&6&-62&175&-70&-19&-57\\\ 6&2&-1&-70&320&40&-2\\\
-8&13&-9&-19&40&360&32\\\ -4&1&17&-57&-2&32&260\end{pmatrix}.$ (12)

The diagonal offset of 12210 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 FMO-Ia data set, the following spectral density parameters and
temperatures were employed: $\lambda$ = $\\{$10, 40, 70, …, 310$\\}$ cm-1,
$\gamma$ = $\\{$25, 50, 75, …, 300$\\}$ fs rad-1, and T = $\\{$30, 50, 70, …,
310$\\}$ K. For FMO-Ib and FMO-II data sets, the spectral density parameters
and temperatures were: $\lambda$ = $\\{$10, 40, 70, …, 520$\\}$ cm-1, $\gamma$
= $\\{$25, 50, 75, …, 500$\\}$ cm-1, and T = $\\{$30, 50, 70, …, 510$\\}$ K.

For FMO-Ia, FMO-Ib, and FMO-II data sets, the farthest-point sampling[87] was
employed to select the most distant points in the Euclidean space[32] of
parameters which typically more efficiently covers relevant space compared to
random sampling.[87] We choose the top 500 (most distant) combinations of
($\lambda$ , $\gamma$, $T$) based on farthest-point sampling. For each
selected set of parameters the system RDM was calculated using the local
thermalizing Lindblad master equation (LTLME) approach[82, 83] implemented in
the quantum_HEOM package.[88, 83] Two subsets of the data set were generated,
one for the initial electronic density operator
$\hat{\rho}_{s}(0)=|1\rangle\langle 1|$ corresponding to the initial
excitation of site-1 and the other one for the initial density operator
$\hat{\rho}_{s}(0)=|6\rangle\langle 6|$ which corresponds to the initial
excitation of site-6. In each case, 500 RDM trajectories were generated. The
data set contains both diagonal (populations) and off-diagonal (coherences)
elements of the RDM on a time grid from 0 to 1 ns (in the case of FMO-Ia) and
0 to 50 ps (in the case of FMO-Ib and FMO-II) with the 5 fs time step.

#### FMO-III and FMO-IV data sets: 8-site FMO models with the local
thermalizing Lindblad master equation approach

Using the same LTLME-based approach, we generated a data set for two different
Hamiltonians for the 8-site FMO model. The first Hamiltonian (FMO-III data
set) was parameterized by Jia et al.[89] The electronic system Hamiltonian is
given by (in cm-1)

$H_{s}=\begin{pmatrix}218&-91.0&4.1&-6.3&6.3&-8.8&-7.8&32.4\\\
-91.0&81&28.7&8.2&1.0&8.8&3.4&6.3\\\ 4.1&28.7&0&-46.6&-4.4&-9.3&1.3&1.3\\\
-6.3&8.2&-46.6&105&-73.9&-17.7&-59.1&-1.9\\\
6.3&1.0&-4.4&-73.9&105&76.0&-3.1&4.2\\\
-8.8&8.8&-9.3&-17.7&76.0&186&25.9&-11.6\\\
-7.8&3.4&1.3&-59.1&-3.1&25.9&169&-11.9\\\
32.4&6.3&1.3&-1.9&4.2&-11.6&-11.9&154\end{pmatrix},$ (13)

with the diagonal offset of 11332 cm-1.

The FMO-IV data set was generated for the Hamiltonian parameterized by Busch
et al.[64] (site energies) and Olbrich et al. [67] (excitonic couplings) and
takes the following form (in cm-1)

$H_{s}=\begin{pmatrix}310&-80.3&3.5&-4.0&4.5&-10.2&-4.9&21.0\\\
-80.3&230&23.5&6.7&0.5&7.5&1.5&3.3\\\ 3.5&23.5&0&-49.8&-1.5&-6.5&1.2&0.7\\\
-4.0&6.7&-49.8&180&63.4&-13.3&-42.2&-1.2\\\
4.5&0.5&-1.5&63.4&450&55.8&4.7&2.8\\\
-10.2&7.5&-6.5&-13.3&55.8&320&33.0&-7.3\\\
-4.9&1.5&1.2&-42.2&4.7&33.0&270&-8.7\\\
21.0&3.3&0.7&-1.2&2.8&-7.3&-8.7&505\end{pmatrix},$ (14)

with the diagonal offset of 12195 cm-1.

The same set of spectral density parameters and temperatures that was used in
generation of the FMO-Ib and FMO-II data sets was used here. LTLME method was
used to propagate system’s RDM from 0 to 50 ps with 5 fs time-step and three
initial states of the electronic system were considered: sites-1, 6 and 8. The
data set contains both diagonal (populations) and off-diagonal (coherences)
elements of the RDM. The calculations was performed with the quantum_HEOM
package[88] with some local modifications to make it compatible for the
Hamiltonians with larger dimension. We will refer to this as modified-
quantum_HEOM implementation.

#### FMO-V data set: FMO trimer with local thermalizing Lindblad master
equation approach

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

$H_{s}=\begin{pmatrix}H_{A}&H_{B}&H_{B}^{T}\\\ H_{B}^{T}&H_{A}&H_{B}\\\
H_{B}&H_{B}^{T}&H_{A}\end{pmatrix}$ (15)

where $H_{A}$ is the subunit Hamiltonian for which we used the same
Hamiltonian as in FMO-IV data set (Eq. 14), while $H_{B}$ is the intra-subunit
Hamiltonian which is taken from the work of Olbrich et al.[67] and is given by
(in cm-1)

$H_{B}=\begin{pmatrix}1.0&0.3&-0.6&0.7&2.3&1.5&0.9&0.1\\\
1.5&-0.4&-2.5&-1.5&7.4&5.2&1.5&0.7\\\ 1.4&0.1&-2.7&5.7&4.6&2.3&4.0&0.8\\\
0.3&0.5&0.7&1.9&-0.6&-0.4&1.9&-0.8\\\ 0.7&0.9&1.1&-0.1&1.8&0.1&-0.7&1.3\\\
0.1&0.7&0.8&1.4&-1.4&-1.5&1.6&-1.0\\\ 0.3&0.2&-0.7&4.8&-1.6&0.1&5.7&-2.3\\\
0.1&0.6&1.5&-1.1&4.0&-3.1&-5.2&3.6\end{pmatrix}.$ (16)

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

#### FMO-VI data set: 8-site FMO model with the hierarchy of equations of
motion approach

The LTLME approach provides only approximate description of quantum dynamics
of the FMO complex. Therefore, the FMO-I—FMO-V data sets are useful merely for
the 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 data sets (e.g., via transfer
learning). However, LTLME dynamics cannot be used to benchmark other quantum
dynamics methods. In the latter case the high-quality reference data is
needed.

To generate a data set with accurate FMO dynamics we performed HEOM
calculations for the 8-site FMO model with the Hamiltonian given by Eq. 14.
HEOM calculations were performed using the parallel hierarchy integrator (PHI)
code.[90] The initial data set was chosen on the basis of farthest-point
sampling similar to how it was done in the FMO-Ib—FMO-V data sets with the
only difference being that instead of 500 most distant sets of parameters that
were chosen in the preparation of FMO-Ib—FMO-V data sets, 1100 most distant
set of parameters were used to prepare the initial FMO-VI data set. 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
data set. Excluded parameters correspond to low temperatures, high
reorganization energies, and low cut-off frequency. Such strong non-Markovian
regimes pose significant challenges in the computational studies of open
quantum systems. Approximately 20% of the initial data set was removed because
of prohibitive memory requirements. We note that even though graphics
processing units (GPU) implementations of HEOM (e.g., Ref. 91) 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 data set HEOM calculations were performed for 2.0
ps. To speed up calculations, an adaptive integration Runge–Kutta–Fehlberg
4/5[92] (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.[90] In
this work the PHI default accuracy threshold of $1\cdot 10^{-6}$ 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 compared to the convergence thresholds discussed below in Technical
Validation. The final FMO-VI data set contains 879 entries each comprising all
the populations and coherences for the RDM from 0 to 2 ps with the time-step
of 0.1 fs.

## Data Records

All data sets can be accessed at https://figshare.com/s/ed24594205ab87404238.
The data sets are stored in standard NumPy[93] 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 energetic bias ($\tilde{\epsilon}$), Y is the reorganization energy
$\tilde{\lambda}$, Z is the cut-off frequency $\tilde{\gamma}$, and XX is the
inverse temperature $\tilde{\beta}$. 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.

## Technical Validation

Central to the HEOM approach is the assumption that the bath correlation
function $C_{a}(t)$ for site $a$ can be represented by an infinite sum of
exponentially decaying terms
$C_{a}(t)=\sum_{k}^{\infty}c_{ak}\exp(-\nu_{ak}t)$, where $\nu_{ak}=2\pi
k/\beta\hslash$ are Matsubara frequencies. Further, each exponential term
leads to a set of auxiliary density matrices which take into account the non-
Markovian evolution of the system’s RDM under the influence of bath. In
practice, the summation must be truncated at a finite level, $K$, which is
called Matsubara cut-off and the set of auxiliary density matrices needs to be
truncated at a finite number $M$. In the truncated set of auxiliary matrices
are indexed by $\mathbf{n}=(n_{10},\ldots,n_{1K},n_{M0},\ldots,n_{MK})$. The
hierarchy truncation level is given by $L=\sum_{a=1}^{M}\sum_{k=0}^{K}n_{ak}$,
where $n_{ak}$ is the index of an auxiliary density matrix. The computational
cost of the HEOM method rises steeply with the hierarchy level $L$.[90]

The hierarchy truncation level $L$ depends on how non-Markovian the system is.
Although, there is some guidance on how to choose the Matsubara cut-off and
hierarchy truncation level based on bath and spectral density parameters,[5,
50] in practice, the values of $M$ and $K$ have to be chosen by requiring the
convergence of the RDM to acceptable accuracy level. In this work HEOM
calculations for the SB model were performed by setting $L=30$ for all
temperatures. The Matsubara cut-off was chosen depending on the temperature as
follows: for $\beta$ = 0.1 $K$ was set to 2; for $\beta=0.25,K=3$, for
$\beta=0.5,K=3$, for $\beta=0.75,K=4$, and for $\beta=1.0,K=5$. These values
are chosen sufficiently high to ensure the convergence of the populations with
respect to $K$ and $L$. Choosing high truncation levels in the HEOM
calculations of a TLS does not present a problem given the presently available
computational resources.

Similar approach of taking excessively large values of $K$ and $L$, however,
is infeasible in the FMO calculations because the computational cost of HEOM
grows steeply with the size of the quantum system. Therefore, the following
approach was adopted for the HEOM calculations of the 8-site FMO model (FMO-VI
data set). Starting from $K=0$ and $L=1$, $K$ was increased until the maximum
difference in the populations between calculations with $K$ and $K+1$ falls
below a threshold $\Delta$, i.e.,

$\delta=\max_{\begin{subarray}{c}n=1,\ldots,N_{el}\\\
t=0,\ldots,t_{max}\end{subarray}}\bigg{|}\rho_{n,n}^{K,L}(t)-\rho_{n,n}^{K+1,L}(t)\bigg{|}<\Delta.$
(17)

When Eq.17 is satisfied for a given $\Delta$ the convergence with respect to
Matsubara cut-off is deemed to have been achieved. Then, for a fixed $K$ a
series of calculations were performed with increasing values of $L$ until the
maximum difference in populations between two consecutive calculations becomes
less than the same threshold value $\Delta$. When this condition is satisfied
the convergence with respect to hierarchy truncation level as well as the
overall convergence is declared. These steps were performed in the HEOM
calculations for each parameter set for an 8-site FMO model until either the
overall convergence is achieved or $K$ and/or $L$ become large enough so the
calculation becomes intractable exceeding RAM available on our machines (1
TB).

In this work we set the threshold $\Delta=0.01$. This threshold was chosen
such that the population errors would be almost imperceptible which is
illustrated in Figure 1. This data set is converged enough to be helpful in
benchmarks of approximate methods describing quantum dynamics because the
errors of these methods often exceed the threshold used in this work.
Additionally, Figures 2 and 3 show the number of Matsubara terms and the
hierarchy truncation level required for achieving the overall convergence
depending on spectral density parameters and temperature.

## Usage Notes

A Python package for extracting data is provided together with the data set
and can be accessed at https://github.com/Arif-PhyChem/QD3SET.

## Code availability

PHI code (version 1.0) used in HEOM calculations was downloaded from
http://www.ks.uiuc.edu/Research/phi/. QuTip software package (version 4.6)
used in HEOM calculations of the spin-boson model and was downloaded from
https://qutip.org/. LTLME calculations of FMO models were performed with the
basic quantum_HEOM package https://github.com/jwa7/quantum_HEOM and was
modified to enable compatibility with the Hamiltonian of larger than the
default dimension.

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## Acknowledgements

A.A.K. acknowledges the Ralph E. Powe Junior Faculty Enhancement Award from
Oak Ridge Associated Universities. This work was also 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. P.O.D. 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 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.

## Author contributions statement

P.O.D. and A.U. conceived the idea of creating a HEOM-based spin-boson
database. A.U. conceived the idea of creating LTLME-based database for FMO
complex. A.A.K conceived the idea of creating an FMO dataset with the HEOM
method. A.U. performed the HEOM calculations for spin-boson along with the
LTLME calculations for FMO complex. A.U. wrote the provided package for easy
extraction of the data. A.A.K and L.E.H.R. performed the calculations and
created database files for the FMO-VI data set. All authors analysed the
results. A.A.K. took the lead in writing the original draft of the manuscript.
All authors reviewed and approved the manuscript.

## Competing interests

The authors declare no competing interest.

## Figures & Tables

Figure 1: Example of the technical validation of the convergence of HEOM
calculations for the following parameters $T=30$ K, $\lambda=70$ cm-1,
$\gamma=500$ cm-1. The convergence within $\Delta=0.01$ threshold was achieved
for $K=7$ and $L=4$. Left plot shows the populations obtained with $K=7$ and
$L=4$ (solid lines) compared to populations obtained with $K=7$ and $L=5$
(dashed lines) for all 8 sites. The largest population difference is
$\delta=3.14\cdot 10^{-4}$. The right plot shows the populations obtained with
$K=7$ and $L=4$ (solid lines) compared to the populations obtained with $K=8$
and $L=4$. The largest population difference is $\delta=4.62\cdot 10^{-3}$. In
both cases the difference is very small illustrating the validity of the
chosen threshold. Figure 2: Number of Matsubara terms (Matsubara cut-off, $M$)
required for converging populations of an 8-site FMO model with the system
Hamiltonian given by Eq. 14 for three selected temperatures ($T$),
reorganization energies ($\lambda$), and bath cut-off frequencies ($\gamma$).
Figure 3: Hierarchy truncation level $L$ required for converging populations
of an 8-site FMO model with the system Hamiltonian given by Eq. 14 for three
selected temperatures ($T$), reorganization energies ($\lambda$), and bath
cut-off frequencies ($\gamma$).

Data set | System | Hamiltonian(s) | Method |  Data set size | Cases |  Propagation time (time-step) | Package | Parameter space | References
---|---|---|---|---|---|---|---|---|---
SB | SB | SB | HEOM | 1000 |  Symmetric and Asymmetric | 20$/\Delta$ (0.05$/\Delta$) | QuTiP[86] | $\mathcal{E}^{b}$ |  Regenerated based on Ref. 33
FMO-Ia | 7-site FMO |  Adolphs and Renger[49] | LTLME | Sites 1 and 6 | 1 ns (5 fs) | quantum_HEOM[88, 83] | $\mathcal{F}^{c}$ |  Regenerated based on Ref. 32
FMO-Ib | 50 ps (5 fs) |  modified- quantum_HEOM a |  $\mathcal{G}^{d}$ | This work
FMO-II | Cho[76]
FMO-III | 8-site FMO | Jia[70] |  |
FMO-IV |  Busch[64] and Olbrich[67] | 1500 |  Sites 1, 6 and 8
FMO-V | FMO trimer |  |
FMO-VI | 8-site FMO | HEOM | 879 | Site 1 | 2 ps (0.1 fs) | PHI[90] | $\mathcal{H}^{e}$

Table 1:  Summary of all data sets. See more details in the main text. Here
“SB" stands for spin-boson model. amodified-quantum_HEOM is the quantum_HEOM
package with some local modifications to make it compatible for larger
Hamiltonians. bIn parameter space $\mathcal{E}$, we define
$\tilde{\epsilon}=\epsilon/\Delta=\\{0,1\\}$,
$\tilde{\lambda}=\lambda/\Delta=\\{0.1$, $0.2$, $0.3$, $0.4$, $0.5$, $0.6$,
$0.7$, $0.8$, $0.9$, $1.0\\}$, $\tilde{\gamma}=\gamma/\Delta=\\{1$, $2$, $3$,
$4$, $5$, $6$, $7$, $8$, $9$, $10\\}$, and $\tilde{\beta}=\beta\Delta=\\{0.1$,
$0.25$, $0.5$, $0.75$, $1\\}$, where the tunneling matrix element $\Delta$ is
set as an energy unit. For all combinations of these parameters the system’s
RDM was propagated. cIn parameter space $\mathcal{F}$, we choose the top 500
(most distant) combinations of ($\lambda$ , $\gamma$, $T$) based on farthest-
point sampling. Parameter range for each dimension is $\lambda$ = $\\{$10, 40,
70, …, 310$\\}$ cm-1, $\gamma$ = $\\{$25, 50, 75, …, 300$\\}$ fs rad-1, and T
= $\\{$30, 50, 70, …, 310$\\}$ K. dIn parameter space $\mathcal{G}$, we adopt
the same procedure as in parameter space $\mathcal{F}$ and choose the most
distant 500 points (based on farthest-point sampling) from 3D space ($\lambda$
, $\gamma$, $T$) where $\lambda$ = $\\{$10, 40, 70, …, 520$\\}$ cm-1, $\gamma$
= $\\{$25, 50, 75, …, 500$\\}$ cm-1, and T = $\\{$30, 50, 70, …, 510$\\}$ K.
eIn parameter space $\mathcal{H}$, parameters range remains the same as in
$\mathcal{G}$. In addition, the same farthest point sampling was adopted but
with the only difference being that instead of 500, 1100 most distant set of
parameters were chosen. Approximately 20% of the initial data set was removed
because of the prohibitive memory requirements. For the remaining 80% of the
data set, HEOM calculations were performed for 2.0 ps using 0.1 fs as a time
step.