Although recent advances in simulating open quantum systems have led to significant progress, the applicability of numerically exact methods is still restricted to rather small systems. Hence, more approximate methods remain relevant due to their computational efficiency, enabling simulations of larger systems over extended timescales. In this study, we present advances for one such method, namely, the numerical integration of Schrödinger equation (NISE). First, we introduce a modified ensemble-averaging procedure that improves the long-time behavior of the thermalized variant of the NISE scheme, termed thermalized NISE. Second, we demonstrate how to use the NISE in conjunction with (highly) structured spectral densities by utilizing a noise generating algorithm for arbitrary structured noise. This algorithm also serves as a tool for establishing best practices in determining spectral densities from excited state calculations along molecular dynamics or quantum mechanics/molecular mechanics trajectories. Finally, we assess the ability of the NISE approach to calculate absorption spectra and demonstrate the utility of the proposed modifications by determining population dynamics.
REFERENCES
1.
C. P.
Koch
, “Controlling open quantum systems: Tools, achievements, and limitations
,” J. Phys.: Condens. Matter
28
, 213001
(2016
).2.
Z.
Hu
, R.
Xia
, and S.
Kais
, “A quantum algorithm for evolving open quantum dynamics on quantum computing devices
,” Sci. Rep.
10
, 3301
(2020
).3.
J.
Cao
, R. J.
Cogdell
, D. F.
Coker
, H.-G.
Duan
, J.
Hauer
, U.
Kleinekathöfer
, T. L. C.
Jansen
, T.
Mančal
, R. J. D.
Miller
, J. P.
Ogilvie
, V. I.
Prokhorenko
, T.
Renger
, H.-S.
Tan
, R.
Tempelaar
, M.
Thorwart
, E.
Thyrhaug
, S.
Westenhoff
, and D.
Zigmantas
, “Quantum biology revisited
,” Sci. Adv.
6
, eaaz4888
(2020
).4.
S.
Maity
and U.
Kleinekathöfer
, “Recent progress in atomistic modeling of light-harvesting complexes: A mini review
,” Photosynth. Res.
156
, 147
(2023
).5.
G. A.
Worth
, H.-D.
Meyer
, H.
Köppel
, L.
Cederbaum
, and I.
Burghardt
, “Using the MCTDH wavepacket propagation method to describe multimode non-adiabatic dynamics
,” Int. Rev. Phys. Chem.
27
, 569
(2008
).6.
H.-D.
Meyer
, “Studying molecular quantum dynamics with the multiconfiguration time-dependent Hartree method
,” Wiley Interdiscip. Rev.: Comput. Mol. Sci.
2
, 351
(2012
).7.
O.
Vendrell
and H.-D.
Meyer
, “Multilayer multiconfiguration time-dependent Hartree method: Implementation and applications to a Henon–Heiles Hamiltonian and to pyrazine
,” J. Chem. Phys.
134
, 044135
(2011
).8.
H.
Wang
, “Multilayer multiconfiguration time-dependent Hartree theory
,” J. Phys. Chem. A
119
, 7951
(2015
).9.
J.
Schulze
, M. F.
Shibl
, M. J.
Al-Marri
, and O.
Kühn
, “Multi-layer multi-configuration time-dependent Hartree (ML-MCTDH) approach to the correlated exciton-vibrational dynamics in the FMO complex
,” J. Chem. Phys.
144
, 185101
(2016
).10.
L.
Chen
, M.
Gelin
, and Y.
Zhao
, “Dynamics of the spin-boson model: A comparison of the multiple Davydov D1, D1.5, D2 Ansätze
,” Chem. Phys.
515
, 108
(2018
), part of the Special Issue: Ultrafast Photoinduced Processes in Polyatomic Molecules: Electronic Structure, Dynamics and Spectroscopy.11.
Y.
Zhao
, “The hierarchy of Davydov’s Ansätze: From guesswork to numerically ‘exact’ many-body wave functions
,” J. Chem. Phys.
158
, 080901
(2023
).12.
M.
Cygorek
, M.
Cosacchi
, A.
Vagov
, V. M.
Axt
, B. W.
Lovett
, J.
Keeling
, and E. M.
Gauger
, “Simulation of open quantum systems by automated compression of arbitrary environments
,” Nat. Phys.
18
, 662
(2022
).13.
A.
Nitzan
, Chemical Dynamics in Condensed Phases: Relaxation, Transfer, and Reactions in Condensed Molecular Systems
(Oxford University Press
, 2006
).14.
V.
May
and O.
Kühn
, Charge and Energy Transfer in Molecular Systems
, 3rd ed. (Wiley-VCH
, 2011
).15.
N.
Makri
and D. E.
Makarov
, “Tensor propagator for iterative quantum time evolution of reduced density matrices. I. Theory
,” J. Chem. Phys.
102
, 4600
(1995
).16.
M.
Thorwart
, P.
Reimann
, and P.
Hänggi
, “Iterative algorithm versus analytic solutions of the parametrically driven dissipative quantum harmonic oscillator
,” Phys. Rev. E
62
, 5808
(2000
).17.
Y.
Tanimura
, “Numerically ‘exact’ approach to open quantum dynamics: The hierarchical equations of motion (HEOM)
,” J. Chem. Phys.
153
, 020901
(2020
).18.
T.
Ikeda
and G. D.
Scholes
, “Generalization of the hierarchical equations of motion theory for efficient calculations with arbitrary correlation functions
,” J. Chem. Phys.
152
, 204101
(2020
).19.
D.
Suess
, A.
Eisfeld
, and W. T.
Strunz
, “Hierarchy of stochastic pure states for open quantum system dynamics
,” Phys. Rev. Lett.
113
, 150403
(2014
).20.
R.
Hartmann
and W. T.
Strunz
, “Exact open quantum system dynamics using the hierarchy of pure states (HOPS)
,” J. Chem. Theory Comput.
13
, 5834
(2017
).21.
L.
Varvelo
, J. K.
Lynd
, and D. I. G.
Bennett
, “Formally exact simulations of mesoscale exciton dynamics in molecular materials
,” Chem. Sci.
12
, 9704
(2021
).22.
T.
Gera
, L.
Chen
, A.
Eisfeld
, J. R.
Reimers
, E. J.
Taffet
, and D. I. G. B.
Raccah
, “Simulating optical linear absorption for mesoscale molecular aggregates: An adaptive hierarchy of pure states approach
,” J. Chem. Phys.
158
, 174103
(2023
).23.
Y.
Tanimura
and R.
Kubo
, “Time evolution of a quantum system in contact with a nearly Gaussian-Markoffian noise bath
,” J. Phys. Soc. Jpn.
58
, 101
(1989
).24.
C.
Kreisbeck
and T.
Kramer
(2017
). “Exciton dynamics lab for light-harvesting complexes (GPU-HEOM)
,” nanoHUB
. https://nanohub.org/resources/gpuheompop25.
Q.
Shi
, Y.
Xu
, Y.
Yan
, and M.
Xu
, “Efficient propagation of the hierarchical equations of motion using the matrix product state method
,” J. Chem. Phys.
148
, 174102
(2018
).26.
R.
Borrelli
and S.
Dolgov
, “Expanding the range of hierarchical equations of motion by tensor-train implementation
,” J. Phys. Chem. B
125
, 5397
(2021
).27.
E.
Mangaud
, A.
Jaouadi
, A.
Chin
, and M.
Desouter-Lecomte
, “Survey of the hierarchical equations of motion in tensor-train format for non-Markovian quantum dynamics
,” Eur. Phys. J.: Spec. Top.
232
, 1847
(2023
).28.
Z.
Tang
, X.
Ouyang
, Z.
Gong
, H.
Wang
, and J.
Wu
, “Extended hierarchy equation of motion for the spin-boson model
,” J. Chem. Phys.
143
, 224112
(2015
).29.
X.
Chen
and I.
Franco
, “Bexcitonics: Quasiparticle approach to open quantum dynamics
,” J. Chem. Phys.
160
, 204116
(2024
).30.
Z.-H.
Chen
, Y.
Wang
, X.
Zheng
, R.-X.
Xu
, and Y.
Yan
, “Universal time-domain Prony fitting decomposition for optimized hierarchical quantum master equations
,” J. Chem. Phys.
156
, 221102
(2022
).31.
M.
Xu
, Y.
Yan
, Q.
Shi
, J.
Ankerhold
, and J. T.
Stockburger
, “Taming quantum noise for efficient low temperature simulations of open quantum systems
,” Phys. Rev. Lett.
129
, 230601
(2022
).32.
H.
Takahashi
, S.
Rudge
, C.
Kaspar
, M.
Thoss
, and R.
Borrelli
, “High accuracy exponential decomposition of bath correlation functions for arbitrary and structured spectral densities: Emerging methodologies and new approaches
,” J. Chem. Phys.
160
, 204105
(2024
).33.
Y.
Yan
, M.
Xu
, T.
Li
, and Q.
Shi
, “Efficient propagation of the hierarchical equations of motion using the Tucker and hierarchical Tucker tensors
,” J. Chem. Phys.
154
, 194104
(2021
).34.
G.
Pleasance
, B. M.
Garraway
, and F.
Petruccione
, “Generalized theory of pseudomodes for exact descriptions of non-Markovian quantum processes
,” Phys. Rev. Res.
2
, 043058
(2020
).35.
A.
Bose
and P. L.
Walters
, “Tensor network path integral study of dynamics in B850 LH2 ring with atomistically derived vibrations
,” J. Chem. Theory Comput.
18
, 4095
(2022
).36.
N.
Makri
, “Modular path integral methodology for real-time quantum dynamics
,” J. Chem. Phys.
149
, 214108
(2018
).37.
S.
Kundu
and N.
Makri
, “Intramolecular vibrations in excitation energy transfer: Insights from real-time path integral calculations
,” Annu. Rev. Phys. Chem.
73
, 349
(2022
).38.
A.
Ishizaki
and G. R.
Fleming
, “On the adequacy of the Redfield equation and related approaches to the study of quantum dynamics in electronic energy transfer
,” J. Chem. Phys.
130
, 234110
(2009
).39.
T. L. C.
Jansen
, W.
Zhuang
, and S.
Mukamel
, “Stochastic Liouville equation simulation of multidimensional vibrational line shapes of trialanine
,” J. Chem. Phys.
121
, 10577
(2004
).40.
H.
Zhu
, V.
May
, and B.
Röder
, “Mixed quantum classical simulations of electronic excitation energy transfer: The pheophorbide-a DAB dendrimer P4 in solution
,” Chem. Phys.
351
, 117
(2008
).41.
J.
Megow
, B.
Röder
, A.
Kulesza
, V.
Bonacic Koutecky
, and V.
May
, “A mixed quantum-classical description of excitation energy transfer in supramolecular complexes: Förster theory and beyond
,” ChemPhysChem
12
, 645
(2011
).42.
T. L. C.
Jansen
and J.
Knoester
, “Nonadiabatic effects in the two-dimensional infrared spectra of peptides: Application to alanine dipeptide
,” J. Phys. Chem. B
110
, 22910
(2006
).43.
T. L. C.
Jansen
, “Simple quantum dynamics with thermalization
,” J. Phys. Chem. A
122
, 172
(2018
).44.
A. S.
Bondarenko
, J.
Knoester
, and T. L. C.
Jansen
, “Comparison of methods to study excitation energy transfer in molecular multichromophoric systems
,” Chem. Phys.
529
, 110478
(2020
).45.
Y.
Holtkamp
, M.
Kowalewski
, J.
Jasche
, and U.
Kleinekathöfer
, “Machine-learned correction to ensemble-averaged wave packet dynamics
,” J. Chem. Phys.
159
, 094107
(2023
).46.
M.
Aghtar
, J.
Liebers
, J.
Strümpfer
, K.
Schulten
, and U.
Kleinekathöfer
, “Juxtaposing density matrix and classical path-based wave packet dynamics
,” J. Chem. Phys.
136
, 214101
(2012
).47.
V.
Erić
, J. L.
Castro
, X.
Li
, L.
Dsouza
, S. K.
Frehan
, A.
Huijser
, A. R.
Holzwarth
, F.
Buda
, G. J. A.
Sevink
, H. J. M.
de Groot
, and T. L. C.
Jansen
, “Ultrafast anisotropy decay reveals structure and energy transfer in supramolecular aggregates
,” J. Phys. Chem. B
127
, 7487
(2023
).48.
V.
Erić
, X.
Li
, L.
Dsouza
, S. K.
Frehan
, A.
Huijser
, A. R.
Holzwarth
, F.
Buda
, G. J. A.
Sevink
, H. J. M.
de Groot
, and T. L. C.
Jansen
, “Manifestation of hydrogen bonding and exciton delocalization on the absorption and two-dimensional electronic spectra of chlorosomes
,” J. Phys. Chem. B
127
, 1097
(2023
).49.
V.
Erić
, X.
Li
, L.
Dsouza
, A.
Huijser
, A. R.
Holzwarth
, F.
Buda
, G. J. A.
Sevink
, H. J. M.
de Groot
, and T. L. C.
Jansen
, “Observation of dark states in two-dimensional electronic spectra of chlorosomes
,” J. Phys. Chem. B
128
, 3575
(2024
).50.
S.
Roy
, M. S.
Pshenichnikov
, and T. L. C.
Jansen
, “Analysis of 2D CS spectra for systems with non-Gaussian dynamics
,” J. Phys. Chem. B
115
, 5431
(2011
).51.
F.
van Mourik
, M.
Chergui
, and G.
van der Zwan
, “The occurrence of non-Gaussian spectral line shapes of molecules in electrostatically disordered media
,” J. Phys. Chem. B
105
, 9715
(2001
).52.
A.
Singharoy
, C.
Maffeo
, K. H.
Delgado-Magnero
, D. J. K.
Swainsbury
, M.
Şener
, U.
Kleinekathöfer
, J. W.
Vant
, J.
Nguyen
, A.
Hitchcock
, B.
Isralewitz
, I.
Teo
, D. E.
Chandler
, J. E.
Stone
, J. C.
Phillips
, T. V.
Pogorelov
, M. I.
Mallus
, C.
Chipot
, Z.
Luthey-Schulten
, D. P.
Tieleman
, C. N.
Hunter
, E.
Tajkhorshid
, A.
Aksimentiev
, and K.
Schulten
, “Atoms to phenotypes: Molecular design principles of cellular energy metabolism
,” Cell
179
, 1098
(2019
).53.
M. H.
Farag
, B. J.
Hoenders
, J.
Knoester
, and T. L. C.
Jansen
, “Spectral line shapes in linear absorption and two-dimensional spectroscopy with skewed frequency distributions
,” J. Chem. Phys.
146
, 234201
(2017
).54.
J.
Seibt
and T.
Mančal
, “Treatment of Herzberg-Teller and non-Condon effects in optical spectra with hierarchical equations of motion
,” Chem. Phys.
515
, 129
(2018
).55.
A.
Damjanović
, I.
Kosztin
, U.
Kleinekathöfer
, and K.
Schulten
, “Excitons in a photosynthetic light-harvesting system: A combined molecular dynamics, quantum chemistry, and polaron model study
,” Phys. Rev. E
65
, 031919
(2002
).56.
S.
Maity
, B. M.
Bold
, J. D.
Prajapati
, M.
Sokolov
, T.
Kubař
, M.
Elstner
, and U.
Kleinekathöfer
, “DFTB/MM molecular dynamics simulations of the FMO light-harvesting complex
,” J. Phys. Chem. Lett.
11
, 8660
(2020
).57.
E.
Cignoni
, V.
Slama
, L.
Cupellini
, and B.
Mennucci
, “The atomistic modeling of light-harvesting complexes from the physical models to the computational protocol
,” J. Chem. Phys.
156
, 120901
(2022
).58.
P.
Sarngadharan
, Y.
Holtkamp
, and U.
Kleinekathöfer
, “Protein effects on the excitation energies and exciton dynamics of the CP24 antenna complex
,” J. Phys. Chem. B
128
, 5201
(2024
).59.
T.
Fujita
, J. C.
Brookes
, S. K.
Saikin
, and A.
Aspuru Guzik
, “Memory-assisted exciton diffusion in the chlorosome light-harvesting antenna of green sulfur bacteria
,” J. Phys. Chem. Lett.
3
, 2357
(2012
).60.
C.
Olbrich
and U.
Kleinekathöfer
, “Time-dependent atomistic view on the electronic relaxation in light-harvesting system II
,” J. Phys. Chem. B
114
, 12427
(2010
).61.
S.
Valleau
, A.
Eisfeld
, and A.
Aspuru Guzik
, “On the alternatives for bath correlators and spectral densities from mixed quantum-classical simulations
,” J. Chem. Phys.
137
, 224103
(2012
).62.
M. K.
Lee
and D. F.
Coker
, “Modeling electronic-nuclear interactions for excitation energy transfer processes in light-harvesting complexes
,” J. Phys. Chem. Lett.
7
, 3171
(2016
).63.
C. W.
Kim
, B.
Choi
, and Y. M.
Rhee
, “Excited state energy fluctuations in the Fenna–Matthews–Olson complex from molecular dynamics simulations with interpolated chromophore potentials
,” Phys. Chem. Chem. Phys.
20
, 3310
(2018
).64.
D.
Loco
and L.
Cupellini
, “Modeling the absorption lineshape of embedded systems from molecular dynamics: A tutorial review
,” Int. J. Quantum Chem.
119
, e25726
(2019
).65.
P. V.
Parandekar
and J. C.
Tully
, “Detailed balance in Ehrenfest mixed quantum-classical dynamics
,” J. Chem. Theory Comput.
2
, 229
(2006
).66.
A.
Bastida
, C.
Cruz
, J.
Zuniga
, A.
Requena
, and B.
Miguel
, “A modified Ehrenfest method that achieves Boltzmann quantum state populations
,” Chem. Phys. Lett.
417
, 53
(2006
).67.
V. I.
Novoderezhkin
and R.
van Grondelle
, “Physical origins and models of energy transfer in photosynthetic light-harvesting
,” Phys. Chem. Chem. Phys.
12
, 7352
(2010
).68.
T.
Renger
and F.
Müh
, “Understanding photosynthetic light-harvesting: A bottom up theoretical approach
,” Phys. Chem. Chem. Phys.
15
, 3348
(2013
).69.
S.
Jurinovich
, L.
Viani
, I. G.
Prandi
, T.
Renger
, and B.
Mennucci
, “Towards an ab initio description of the optical spectra of light-harvesting antennae: Application to the CP29 complex of photosystem II
,” Phys. Chem. Chem. Phys.
17
, 14405
(2015
).70.
L.
Cupellini
, F.
Lipparini
, and J.
Cao
, “Absorption and circular dichroism spectra of molecular aggregates with the full cumulant expansion
,” J. Phys. Chem. B
124
, 8610
(2020
).71.
R.
Gutiérrez
, R.
Caetano
, P. B.
Woiczikowski
, T.
Kubař
, M.
Elstner
, and G.
Cuniberti
, “Structural fluctuations and quantum transport through DNA molecular wires: A combined molecular dynamics and model Hamiltonian approach
,” New J. Phys.
12
, 023022
(2010
).72.
A.
Caldeira
and A.
Leggett
, “Path integral approach to quantum Brownian motion
,” Physica A
121
, 587
(1983
).73.
E.
Geva
, E.
Rosenman
, and D. J.
Tannor
, “On the second-order corrections to the quantum canonical equilibrium density matrix
,” J. Chem. Phys.
113
, 1380
(2000
).74.
A.
Gelzinis
and L.
Valkunas
, “Analytical derivation of equilibrium state for open quantum system
,” J. Chem. Phys.
152
, 051103
(2020
).75.
V.
Mannalatha
, S.
Mishra
, and A.
Pathak
, “A comprehensive review of quantum random number generators: Concepts, classification and the origin of randomness
,” Quantum Inf. Process.
22
, 439
(2023
).76.
R. F.
Fox
, I. R.
Gatland
, R.
Roy
, and G.
Vemuri
, “Fast, accurate algorithm for numerical simulation of exponentially correlated colored noise
,” Phys. Rev. A
38
, 5938
(1988
).77.
G. A.
Mastin
, P. A.
Watterberg
, and J. F.
Mareda
, “Fourier synthesis of ocean scenes
,” IEEE Comput. Graphics Appl.
7
, 16
(1987
).78.
M. L.
Ravalec
, B.
Noetinger
, and L. Y.
Hu
, “The FFT moving average (FFT-MA) generator: An efficient numerical method for generating and conditioning Gaussian simulations
,” Math. Geol.
32
, 701
(2000
).79.
A. G.
Dijkstra
, “Models for nonlinear optical spectra of coupled oscillators
,” Ph.D. thesis, University of Groningen
, Groningen
, 2009
.80.
R.
Hartmann
(2016
). “StocProc: Time continuous stochastic process generator for Python
,” GitHub
. https://github.com/cimatosa/stocproc81.
L.
Fontanella
and L.
Ippoliti
, “Karhunen–Loéve expansion of temporal and spatio-temporal processes
,” in Handbook of Statistics: Time Series Analysis: Methods and Applications
, edited by T.
Subba Rao
, S.
Subba Rao
, and C.
Rao
(Elsevier
, 2012
), Vol. 30
, pp. 497
–520
.82.
J. W.
Cooley
and J. W.
Tukey
, “An algorithm for the machine calculation of complex Fourier series
,” Math. Comput.
19
, 297
(1965
).83.
P.
Bloomfield
, Fourier Analysis of Time Series: An Introduction
, 2nd ed. (Wiley
, 2000
).84.
C. R.
Dietrich
and G. N.
Newsam
, “Fast and exact simulation of stationary Gaussian processes through circulant embedding of the covariance matrix
,” SIAM J. Sci. Comput.
18
, 1088
(1997
).85.
M.
Rätsep
and A.
Freiberg
, “Electron-phonon and vibronic couplings in the FMO bacteriochlorophyll a antenna complex studied by difference fluorescence line narrowing
,” J. Lumin.
127
, 251
(2007
).86.
H.
Nyquist
, “Certain topics in telegraph transmission theory
,” Trans. Am. Inst. Electr. Eng.
47
, 617
(1928
).87.
C.
Shannon
, “Communication in the presence of noise
,” Proc. IRE
37
, 10
(1949
).88.
M.
Unser
, “Sampling-50 years after Shannon
,” Proc. IEEE
88
, 569
(2000
).89.
S.
Maity
, V.
Daskalakis
, M.
Elstner
, and U.
Kleinekathöfer
, “Multiscale QM/MM molecular dynamics simulations of the trimeric major light-harvesting complex II
,” Phys. Chem. Chem. Phys.
23
, 7407
(2021
).90.
S.
Wei
, D. F.
Jiang
, and M.
Fattouche
, “A review of super-resolution methods for nonorthogonal analysis of highly correlated signals in noise
,” Appl. Mech. Mater.
543–547
, 2201
–2204
(2014
).91.
T.
Markovich
, S. M.
Blau
, J.
Parkhill
, C.
Kreisbeck
, J. N.
Sanders
, X.
Andrade
, and A.
Aspuru-Guzik
, “Accelerating the computation of bath spectral densities with super-resolution
,” Theor. Chem. Acc.
135
, 215
(2016
).92.
T. J.
Zuehlsdorff
, A.
Montoya-Castillo
, J. A.
Napoli
, T. E.
Markland
, and C. M.
Isborn
, “Optical spectra in the condensed phase: Capturing anharmonic and vibronic features using dynamic and static approaches
,” J. Chem. Phys.
151
, 074111
(2019
).93.
L.
Cupellini
and F.
Lipparini
(2020
). “FCE program to compute optical spectra with the full cumulant expansion
,” OpenAIRE Zenodo. https://doi.org/10.5281/zenodo.390019994.
S.
Maity
, P.
Sarngadharan
, V.
Daskalakis
, and U.
Kleinekathöfer
, “Time-dependent atomistic simulations of the CP29 light-harvesting complex
,” J. Chem. Phys.
155
, 055103
(2021
).95.
P.
Sarngadharan
, S.
Maity
, and U.
Kleinekathöfer
, “Spectral densities and absorption spectra of the core antenna complex CP43 from photosystem II
,” J. Chem. Phys.
156
, 215101
(2022
).96.
T. L. C.
Jansen
(2017
). “NISE 2017: A quantum classical package for calculating coherent multidimensional spectra
,” GitHub. https://github.com/GHlacour/NISE_201797.
T. L.
Jansen
, “Computational spectroscopy of complex systems
,” J. Chem. Phys.
155
, 170901
(2021
).98.
F.
Müh
, M.
Madjet
, and T.
Renger
, “Structure-based simulation of linear optical spectra of the CP43 core antenna of photosystem II
,” Photosynth. Res.
111
, 87
(2012
).99.
S.
Valleau
, S. K.
Saikin
, D.
Ansari-Oghol Beig
, M.
Rostami
, H.
Mossallaei
, and A.
Aspuru Guzik
, “Electromagnetic study of the chlorosome antenna complex of Chlorobium tepidum
,” ACS Nano
8
, 3884
(2014
).100.
M.
Frigo
and S.
Johnson
, “The design and implementation of FFTW3
,” Proc. IEEE
93
, 216
(2005
).101.
P.
Virtanen
, R.
Gommers
, T. E.
Oliphant
, M.
Haberland
, T.
Reddy
, D.
Cournapeau
, E.
Burovski
, P.
Peterson
, W.
Weckesser
, J.
Bright
, S. J.
van der Walt
, M.
Brett
, J.
Wilson
, K. J.
Millman
, N.
Mayorov
, A. R. J.
Nelson
, E.
Jones
, R.
Kern
, E.
Larson
, C. J.
Carey
, İ.
Polat
, Y.
Feng
, E. W.
Moore
, J.
VanderPlas
, D.
Laxalde
, J.
Perktold
, R.
Cimrman
, I.
Henriksen
, E. A.
Quintero
, C. R.
Harris
, A. M.
Archibald
, A. H.
Ribeiro
, F.
Pedregosa
, P.
van Mulbregt
, A.
Vijaykumar
et al, “SciPy 1.0: Fundamental algorithms for scientific computing in Python
,” Nat. Methods
17
, 261
(2020
).102.
A.
Tokmakoff
, “Time-dependent quantum mechanics and spectroscopy
,” lecture notes, The University of Chicago
, https://tdqms.uchicago.edu/ (2014).103.
J.
Bioucas-Dias
and M.
Figueiredo
, “A new twist: Two-step iterative shrinkage/thresholding algorithms for image restoration
,” IEEE Trans. Image Process.
16
, 2992
(2007
).104.
D. C.
Liu
and J.
Nocedal
, “On the limited memory BFGS method for large scale optimization
,” Math. Program.
45
, 503
(1989
).105.
A.
Paszke
, S.
Gross
, F.
Massa
, A.
Lerer
, J.
Bradbury
, G.
Chanan
, T.
Killeen
, Z.
Lin
, N.
Gimelshein
, L.
Antiga
, A.
Desmaison
, A.
Kopf
, E.
Yang
, Z.
DeVito
, M.
Raison
, A.
Tejani
, S.
Chilamkurthy
, B.
Steiner
, L.
Fang
, J.
Bai
, and S.
Chintala
, “PyTorch: An imperative style, high-performance deep learning library
,” in Advances in Neural Information Processing Systems
, edited by H.
Wallach
, H.
Larochelle
, A.
Beygelzimer
, F.
d’Alché-Buc
, E.
Fox
, and R.
Garnett
(Curran Associates, Inc.
, 2019
), Vol. 32
.© 2024 Author(s). Published under an exclusive license by AIP Publishing.
2024
Author(s)
You do not currently have access to this content.
