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REVIEW 2 major objections 6 minor 83 references

Time-dependent Density Matrix Renormalization Group Quantum Dynamics for Realistic Chemical Systems

T0 review · 2 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read tDMRG's 2TDVP method matches MCTDH at lower cost for realistic chemical systems, and the authors argue it can scale to systems beyond the reach of exponential-cost methods.

desk verdict Solid, honest tDMRG benchmark with real new results; the transferred truncation thresholds and purely visual comparisons are the soft spots. read the letter →

arxiv 1908.10588 v3 pith:6KGMR37L submitted 2019-08-28 cond-mat.str-el

Xiaoyu Xie , Yuyang Liu , Yao Yao , Ulrich Schollwock , Chungen Liu , Haibo Ma This is my paper · ORCID
classification cond-mat.str-el
keywords tDMRGtime-dependentvariationalprinciplematrixproductstatequantumdynamicselectron-vibrationcouplingpyrazinesingletfissionMCTDHbenchmark
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper benchmarks four time-dependent density matrix renormalization group (tDMRG) algorithms against established multiconfiguration time-dependent Hartree (MCTDH) references and experimental data. Its central claim is that the two-site time-dependent variational principle (2TDVP) propagator describes the full quantum dynamics of molecules with dozens to hundreds of coupled vibrational modes accurately and efficiently. The authors show agreement on pyrazine internal-conversion spectra and singlet-fission populations, with the 24-mode pyrazine run taking 10.3 CPU-hours versus 13.7 for MCTDH. If the claim holds, tDMRG offers a practical, scalable route to quantum-dynamics simulations of realistic chemical systems.

What carries the argument

The central mechanism is the matrix product state (MPS) representation of the vibronic wavefunction, combined with a matrix product operator (MPO) form of the electron-vibration Hamiltonian. On top of that, the two-site time-dependent variational principle (2TDVP) sweeps a two-site effective Hamiltonian across the chain, adaptively growing the bond dimension, and truncates via singular value decomposition with a fixed threshold (DBSS). Site ordering that places strongly coupled and highly entangled small-basis sites near the center reduces the largest two-site tensor size and gives roughly a tenfold speedup, which together makes the 24- and 183-mode simulations feasible.

What would settle it

Rerun the 24-mode pyrazine 2TDVP dynamics with a truncation threshold of 1e-10 and a time step of 5 a.u., then compare the electronic populations and autocorrelation function with the paper's 1e-8 and 20 a.u. results; if the differences exceed the small-scale convergence pattern or the MCTDH comparison drifts at the tighter settings, the claimed accuracy is not converged.

Watch

Extended reading notes

Core claim

The paper's central claim is that the 2TDVP method reproduces MCTDH, ML-MCTDH, and experimental reference dynamics for the S1/S2 internal conversion in pyrazine (24 modes) and for singlet fission in a molecular dimer (up to 183 phonon modes), while keeping the wavefunction compressed and the computational cost comparable to or lower than the reference methods. It also finds that the one-site TDVP variant can fail quantitatively for larger systems unless the initial bond dimension is already large, that a second-order Taylor propagator crashes on realistic vibronic systems, and that fixed-threshold truncation (DBSS) with a threshold near $10^{-8}$ and a time step up to 20 a.u. gives converged results for the tested models.

Load-bearing premise

The fixed truncation threshold and time step are chosen on a four-mode test model and then carried over to the 24-mode and 183-mode systems without direct re-convergence checks, so the whole accuracy claim rests on the assumption that truncation error does not grow faster than that small-model test suggests.

Editorial extensions

If this is right

  • 2TDVP is a viable default propagator for vibronic dynamics with up to at least 183 modes, matching established quantum-dynamics benchmarks.
  • Because tDMRG costs scale polynomially with system size rather than exponentially like MCTDH, the approach should extend to molecular systems beyond current MCTDH reach.
  • 1TDVP is only reliable when started from a large-bond-dimension MPS; the paper shows it fails on the 24-mode and 183-mode systems otherwise.
  • Site ordering is a major efficiency lever: placing strongly coupled and entangled small-basis sites in the center accelerates the calculation by an order of magnitude.
  • Fixed-truncation (DBSS) control with a threshold near 1e-8 and a time step no larger than 20 a.u. appears sufficient for the tested classes of vibronic systems.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the MPS compression stays controlled in more entangled or high-temperature baths, the same protocol should scale to still larger vibronic systems; the paper gives indirect evidence but no direct convergence check at that scale.
  • The singlet-fission bath analysis identifies energy windows R4, R5, and R7 as the dynamically important ones, suggesting that future tDMRG studies could build reduced-bath models from those windows alone to cut cost further without losing the essential physics.
  • The parameter-transfer guidelines (truncation threshold and time step tuned on a four-mode test) may not transfer automatically to systems with different entanglement growth, so new system classes should re-run the same threshold-convergence tests.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 6 minor

Summary. The paper benchmarks four time-dependent DMRG (tDMRG) time-evolution methods on two realistic electron-vibration models: the S1/S2 internal conversion of pyrazine (4- and 24-mode models) and singlet fission in a molecular dimer coupled to a continuous phonon bath (up to 183 modes). The methods compared are global Taylor, global Krylov, one-site TDVP, and two-site TDVP implemented through the SyTen package. The authors test the dependence of the dynamics on maximal phonon occupation numbers, time step, truncation threshold, site ordering, and bath-mode discretization, and they compare their spectra and populations against MCTDH, ML-MCTDH, and experimental results. Their central claim is that tDMRG, particularly 2TDVP, can describe the full quantum dynamics of large chemical systems accurately and efficiently.

Significance. If the claims hold, this is a useful benchmark paper for practical tDMRG applications in chemical dynamics. Its strengths are the use of external benchmarks (MCTDH, ML-MCTDH, and experimental spectra), the systematic parameter-convergence tests for the small pyrazine model, the direct CPU-time comparison for the 24-mode pyrazine model (10.3 h for 2TDVP versus 13.7 h for MCTDH), and the practical guidance on site ordering. The use of literature Hamiltonians and reference data makes the validation largely non-circular. The main issue is whether the error-control parameters calibrated on the 4-mode model remain adequate for the 24- and 183-mode systems, a point that needs explicit verification.

major comments (2)
  1. [Section III A, Figs. 1-2; Section III B, Figs. 5-6] The truncation threshold ε=1e-8 and time step δt=20 a.u. selected from the 4-mode pyrazine tests are applied to the 24-mode pyrazine run, and ε=1e-9 is used for the 183-mode singlet-fission run, but no re-convergence test at a tighter threshold or smaller time step is reported for these large systems. Because 2TDVP has both the SVD truncation controlled by ε and the tangent-space projection error from solving Eqs. (30)-(31), and the latter is nonzero for the 24-mode Hamiltonian with mMPO=14, convergence at 4 modes does not by itself establish that the error remains negligible at 24 or 183 modes. This missing checkpoint is load-bearing for the abstract's 'accurately' claim; the agreement with MCTDH/ML-MCTDH in Figs. 3 and 6 is good indirect evidence, but a direct tighter-epsilon run for each large system would close the gap.
  2. [Section III A, Fig. 3; Section III B, Figs. 5-6] The accuracy claims are supported only by visual comparison, with no quantitative error metrics reported. For example, the statement that 1TDVP 'has larger quantitative deviations' than 2TDVP in the 183-mode case (Section III B, Fig. 6(b)) is not backed by a number, and the population differences in Figs. 7(c,f) and 8(c,f,i) are plotted but not summarized. Since the MCTDH/ML-MCTDH reference data are available, reporting a time-integrated absolute population difference or a spectral peak-position error would make the benchmark conclusions falsifiable and is well within the manuscript's scope.
minor comments (6)
  1. [Section II A] The second-order Taylor propagator used for the 'Taylor' curves in Fig. 1 is not defined; please include its explicit expression, e.g., U(δt) ≈ 1 - iHδt/ħ - (Hδt)²/(2ħ²).
  2. [Fig. 1 caption] The caption states '(a, c, d) show results of population evolution of S2' but panel (e) is the TDVP population panel; this should read '(a, c, e)'.
  3. [Section III A, Fig. 4(c)] The two subpanels in Fig. 4(c) are not clearly labeled in the figure itself; the caption should state explicitly which panel corresponds to the optimized ordering and which to the default ordering.
  4. [Section III A] The quantity mlimit is used without definition in the bond-dimension discussion (e.g., 'mlimit = 410' and 'mlimit = 209'); please define it at first use, presumably as the maximal possible bond dimension from the local basis sizes.
  5. [Fig. 5 caption] The caption lists the last panel as '(f) 0.35-0.4 eV (R8)' but it should be '(h) 0.35-0.4 eV (R8)'.
  6. [Section III A, Fig. 3 caption] The captions of Fig. 3(a,b) and (c,d) use reference numbers 56 and 57 without clarifying which reference contains the linear 24-mode model and which contains the second-order model; please state this explicitly to avoid confusion.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: validation rests on external MCTDH/ML-MCTDH and experimental benchmarks; self-references are background only.

full rationale

No circular step was found. The paper's central claim is that tDMRG methods, especially 2TDVP, can describe the quantum dynamics of large chemical systems accurately and efficiently. This claim is supported by comparison with external benchmarks: MCTDH results for pyrazine from Refs. 56 and 57, ML-MCTDH results for singlet fission from Ref. 83, and experimental spectra from Ref. 63. These benchmarks are independent of the present implementation and parameters; no quantity that is labeled a prediction is defined in terms of a fitted tDMRG parameter. The convergence parameters (e.g., dt = 20 a.u. and ε = 1e-8 for pyrazine, ε = 1e-9 for singlet fission) are numerical accuracy controls established on smaller models, not fitted outputs, and the large-system results are still compared with independent ML-MCTDH and experimental data. The only self-references, such as Ref. 43 and Ref. 46, and the use of the SyTen package (Refs. 52-53), are methodological or implementation details and are not load-bearing for the validation. The skeptical concern about transferring the truncation threshold from the 4-mode model to larger systems is a correctness/convergence risk, not a circularity, because the paper does not define its claimed accuracy in terms of that threshold and instead checks against external results.

Assumptions & free parameters 4 free parameters · 3 assumptions · 0 invented entities

No invented physical entities are introduced. The central claim rests on standard tensor-network truncation and on literature Hamiltonians; the only hand-chosen numerical parameters are convergence and broadening parameters.

free parameters (4)
  • Spectral damping time tau = 30 fs for most pyrazine spectra; 50 fs for the 24-mode second-order model
    Chosen by hand to broaden the Fourier-transformed autocorrelation function for spectral comparison. Affects linewidth and visual agreement with experiment.
  • Maximal phonon occupation numbers nmax = 24, 18, 10, 18 for modes v6a, v1, v9a, v10 in pyrazine; 10, 8, 6 for LE, CT, TT bath modes in singlet fission
    Convergence parameters for the bosonic local basis, chosen by testing dynamics at different nmax. They are representation parameters, not physically fitted constants.
  • Truncation threshold epsilon (DBSS) = 1e-8 for 2TDVP pyrazine; 1e-9 for 2TDVP singlet fission
    Chosen by convergence tests on small models and assumed to transfer to larger systems. This is the load-bearing assumption identified in the report.
  • Number of discretized bath modes = 90 modes for the reduced R4+R5+R7 bath; 183 modes for the full 0-0.4 eV bath
    Chosen by increasing the mode count until population dynamics converged. Represents the continuous Debye spectral density in the singlet-fission model.
assumptions (3)
  • domain assumption The vibronic Hamiltonian truncated at second order in the electron-vibration coupling (Eqs. 36-39) adequately describes pyrazine and singlet-fission dynamics.
    Taken from earlier literature (refs 56, 57, 83). The benchmark validates method and model jointly, so errors in the model are not separated from method errors.
  • domain assumption The MPS/MPO compression with the DBSS threshold epsilon gives a controlled approximation to the exact time-evolved state.
    Standard tensor-network truncation. Convergence with epsilon is demonstrated on the 4-mode pyrazine model but not exhaustively on the 24-mode or 183-mode systems.
  • domain assumption Discretization of the Debye spectral density via Eq. 44 with the chosen mode counts faithfully represents the continuous bath.
    Convergence with mode number is tested per energy region, but the full-bath result is only compared to ML-MCTDH using the same discretization scheme.

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Cite this review

Pith. "Pith review of Time-dependent Density Matrix Renormalization Group Quantum Dynamics for Realistic Chemical Systems." pith.science (2026). https://pith.science/paper/6KGMR37L

@misc{pith2026190810588,
  author       = {Pith},
  title        = {Pith review of: Time-dependent Density Matrix Renormalization Group Quantum Dynamics for Realistic Chemical Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6KGMR37L}},
  note         = {Machine review of arXiv:1908.10588}
}
read the original abstract

Electronic and/or vibronic coherence has been found by recent ultrafast spectroscopy experiments in many chemical, biological and material systems. This indicates that there are strong and complicated interactions between electronic states and vibration modes in realistic chemical systems. Therefore, simulations of quantum dynamics with a large number of electronic and vibrational degrees of freedom are highly desirable. Due to the efficient compression and localized representation of quantum states in the matrix-product state (MPS) formulation, time-evolution methods based on the MPS framework, which we summarily refer to as tDMRG (time-dependent density-matrix renormalization group) methods, are considered to be promising candidates to study the quantum dynamics of realistic chemical systems. In this work, we benchmark the performances of four different tDMRG methods, including global Taylor, global Krylov, local one-site and two-site time-dependent variational principle (1TDVP and 2TDVP), with a comparison to multi-configuration time-dependent Hartree (MCTDH) and experimental results. Two typical chemical systems of internal conversion and singlet fission are investigated, one containing strong and high-order local and non-local electron-vibration couplings, the other exhibiting a continuous phonon bath. The comparison shows that the tDMRG methods (particularly, the 2TDVP method) can describe the full quantum dynamics in large chemical systems accurately and efficiently. Several key parameters in the tDMRG calculation including the truncation error threshold, time interval and ordering of local sites were also investigated to strike the balance between efficiency and accuracy of results.

Figures

Figures reproduced from arXiv: 1908.10588 by the authors.

Figure 1
Figure 1. FIG. 1. The dynamics simulation tests of 4-modes pyrazine S [PITH_FULL_IMAGE:figures/full_fig_p015_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. The parameters tests of TDVP methods with the dynamics simulation of 4-mode pyrazine S [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Results for the spectra of the molecule pyrazine. (a) Spectrum (black line) of the 4-mode model [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Dynamics behaviour of bond dimension for the systems in Fig. 1-3. (a). Different methods testing [PITH_FULL_IMAGE:figures/full_fig_p019_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. The time-evolution for the populations of three electronic states (LE state with black lines, CT state [PITH_FULL_IMAGE:figures/full_fig_p022_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. The time-evolution for the populations of three electronic states of the SF dynamics coupled to the [PITH_FULL_IMAGE:figures/full_fig_p023_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Dynamics results of bond dimension and population of electronic states for SF systems with differ [PITH_FULL_IMAGE:figures/full_fig_p023_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Dynamics results of bond dimension and population of electronic states for SF systems with differ [PITH_FULL_IMAGE:figures/full_fig_p024_8.png]

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Reference graph

Works this paper leans on

83 extracted references · 69 canonical work pages

  1. [1]

    author author T. G. \ Kolda \ and\ author B. W. \ Bader ,\ @noop journal journal SIAM Rev. \ volume 51 ,\ pages 455 ( year 2009 ) NoStop

    2009

  2. [2]

    Dolgov ,\ title Tensor product methods in numerical simulation of high-dimensional dynamical problems ,\ @noop Ph.D

    author author S. Dolgov ,\ title Tensor product methods in numerical simulation of high-dimensional dynamical problems ,\ @noop Ph.D. thesis ,\ school University of Leipzig ( year 2014 ) NoStop

    2014

  3. [3]

    author author L. R. \ Tucker ,\ @noop journal journal Psychometrika \ volume 31 ,\ pages 279 ( year 1966 ) NoStop

    1966

  4. [4]

    \ Meyer , author U

    author author H.-D. \ Meyer , author U. Manthe , \ and\ author L. S. \ Cederbaum ,\ @noop journal journal Chem. Phys. Lett. \ volume 165 ,\ pages 73 ( year 1990 ) NoStop

    work page 1990

  5. [5]

    author author M. H. \ Beck , author A. J \"a ckle , author G. A. \ Worth , \ and\ author H.-D. \ Meyer ,\ @noop journal journal Phys. Rep. \ volume 324 ,\ pages 1 ( year 2000 ) NoStop

    2000

  6. [6]

    De , author B

    author author L. De , author B. De Moor , \ and\ author J. Vandewalle ,\ @noop journal journal SIAM J. Matrix Anal. Appl. \ volume 21 ,\ pages 1324 ( year 2000 ) NoStop

    work page 2000

  7. [7]

    De Lathauwer , author B

    author author L. De Lathauwer , author B. De Moor , \ and\ author J. Vandewalle ,\ @noop journal journal SIAM J. Matrix Anal. Appl. \ volume 21 ,\ pages 1253 ( year 2000 ) NoStop

    work page 2000

  8. [8]

    Wang \ and\ author M

    author author H. Wang \ and\ author M. Thoss ,\ @noop journal journal J. Chem. Phys. \ volume 119 ,\ pages 1289 ( year 2003 ) NoStop

    2003

Show all 83 references
  1. [9]

    author author I. V. \ Oseledets ,\ @noop journal journal SIAM J. Sci. Comput. \ volume 33 ,\ pages 2295 ( year 2011 ) NoStop

    2011

  2. [10]

    Verstraete , author V

    author author F. Verstraete , author V. Murg , \ and\ author J. I. \ Cirac ,\ @noop journal journal Adv. Phys. \ volume 57 ,\ pages 143 ( year 2008 ) NoStop

    2008

  3. [11]

    Schollw \"o ck ,\ @noop journal journal Ann

    author author U. Schollw \"o ck ,\ @noop journal journal Ann. Phys. \ volume 326 ,\ pages 96 ( year 2011 ) NoStop

    2011

  4. [12]

    author author S. R. \ White ,\ @noop journal journal Phys. Rev. Lett. \ volume 69 ,\ pages 2863 ( year 1992 ) NoStop

    1992

  5. [13]

    Schollw \"o ck ,\ @noop journal journal Rev

    author author U. Schollw \"o ck ,\ @noop journal journal Rev. Mod. Phys. \ volume 77 ,\ pages 259 ( year 2005 ) NoStop

    2005

  6. [14]

    Verstraete \ and\ author J

    author author F. Verstraete \ and\ author J. I. \ Cirac ,\ @noop journal journal arXiv preprint cond-mat/0407066 \ ( year 2004 ) NoStop

    2004 arXiv

  7. [15]

    \ Shi , author L.-M

    author author Y.-Y. \ Shi , author L.-M. \ Duan , \ and\ author G. Vidal ,\ @noop journal journal Phys. Rev. A \ volume 74 ,\ pages 022320 ( year 2006 ) NoStop

    2006

  8. [16]

    Nakatani \ and\ author G

    author author N. Nakatani \ and\ author G. K.-L. \ Chan ,\ @noop journal journal J. Chem. Phys. \ volume 138 ,\ pages 134113 ( year 2013 ) NoStop

    2013

  9. [17]

    Gunst , author F

    author author K. Gunst , author F. Verstraete , author S. Wouters , author O. Legeza , \ and\ author D. Van Neck ,\ @noop journal journal J. Chem. Theory Comput. \ volume 14 ,\ pages 2026 ( year 2018 ) NoStop

    2026

  10. [18]

    author author F. A. \ Schr \"o der , author D. H. \ Turban , author A. J. \ Musser , author N. D. \ Hine , \ and\ author A. W. \ Chin ,\ @noop journal journal Nat. Commun. \ volume 10 ,\ pages 1062 ( year 2019 ) NoStop

    2019

  11. [19]

    author author S. R. \ White \ and\ author A. E. \ Feiguin ,\ @noop journal journal Phys. Rev. Lett. \ volume 93 ,\ pages 076401 ( year 2004 ) NoStop

    2004

  12. [20]

    author author A. J. \ Daley , author C. Kollath , author U. Schollw \"o ck , \ and\ author G. Vidal ,\ @noop journal journal J. Stat. Mech.: Theory Exp. \ volume 2004 ,\ pages P04005 ( year 2004 ) NoStop

    2004

  13. [21]

    Verstraete , author J

    author author F. Verstraete , author J. J. \ Garcia-Ripoll , \ and\ author J. I. \ Cirac ,\ @noop journal journal Phys. Rev. Lett. \ volume 93 ,\ pages 207204 ( year 2004 ) NoStop

    2004

  14. [22]

    Vidal ,\ @noop journal journal Phys

    author author G. Vidal ,\ @noop journal journal Phys. Rev. Lett. \ volume 91 ,\ pages 147902 ( year 2003 ) NoStop

    2003

  15. [23]

    Vidal ,\ @noop journal journal Phys

    author author G. Vidal ,\ @noop journal journal Phys. Rev. Lett. \ volume 93 ,\ pages 040502 ( year 2004 ) NoStop

    2004

  16. [24]

    Suzuki ,\ @noop journal journal Commun

    author author M. Suzuki ,\ @noop journal journal Commun. Math. Phys. \ volume 51 ,\ pages 183 ( year 1976 ) NoStop

    1976

  17. [25]

    Ronca , author Z

    author author E. Ronca , author Z. Li , author C. A. \ Jimenez-Hoyos , \ and\ author G. K.-L. \ Chan ,\ @noop journal journal J. Chem. Theory Comput. \ volume 13 ,\ pages 5560 ( year 2017 ) NoStop

    2017

  18. [26]

    Ren , author Z

    author author J. Ren , author Z. Shuai , \ and\ author G. Kin-Lic Chan ,\ @noop journal journal J. Chem. Theory Comput. \ volume 14 ,\ pages 5027 ( year 2018 ) NoStop

    2018

  19. [27]

    \ Frahm \ and\ author D

    author author L.-H. \ Frahm \ and\ author D. Pfannkuche ,\ @noop journal journal J. Chem. Theory Comput. \ volume 15 ,\ pages 2154 ( year 2019 ) NoStop

    2019

  20. [28]

    o hler , author A. Swoboda , author S. R. \ Manmana , author U. Schollw \

    author author S. Paeckel , author T. K \"o hler , author A. Swoboda , author S. R. \ Manmana , author U. Schollw \"o ck , \ and\ author C. Hubig ,\ @noop journal journal Ann. Phys. \ volume 411 ,\ pages 167998 ( year 2019 ) NoStop

    2019

  21. [29]

    author author J. J. \ Garc \' a-Ripoll ,\ @noop journal journal New J. Phys. \ volume 8 ,\ pages 305 ( year 2006 ) NoStop

    2006

  22. [30]

    Haegeman , author J

    author author J. Haegeman , author J. I. \ Cirac , author T. J. \ Osborne , author I. Pi z orn , author H. Verschelde , \ and\ author F. Verstraete ,\ @noop journal journal Phys. Rev. Lett. \ volume 107 ,\ pages 070601 ( year 2011 ) NoStop

    2011

  23. [31]

    Lubich , author I

    author author C. Lubich , author I. V. \ Oseledets , \ and\ author B. Vandereycken ,\ @noop journal journal SIAM J. Numer. Anal. \ volume 53 ,\ pages 917 ( year 2015 ) NoStop

    2015

  24. [32]

    Haegeman , author C

    author author J. Haegeman , author C. Lubich , author I. Oseledets , author B. Vandereycken , \ and\ author F. Verstraete ,\ @noop journal journal Phys. Rev. B \ volume 94 ,\ pages 165116 ( year 2016 ) NoStop

    2016

  25. [33]

    Kurashige ,\ @noop journal journal J

    author author Y. Kurashige ,\ @noop journal journal J. Chem. Phys. \ volume 149 ,\ pages 194114 ( year 2018 ) NoStop

    2018

  26. [34]

    Kloss , author D

    author author B. Kloss , author D. R. \ Reichman , \ and\ author R. Tempelaar ,\ 10.1103/PhysRevLett.123.126601 journal journal Phys. Rev. Lett. \ volume 123 ,\ pages 126601 ( year 2019 ) NoStop

    2019 doi

  27. [35]

    Shi , author Y

    author author Q. Shi , author Y. Xu , author Y. Yan , \ and\ author M. Xu ,\ @noop journal journal J. Chem. Phys. \ volume 148 ,\ pages 174102 ( year 2018 ) NoStop

    2018

  28. [36]

    Borrelli ,\ @noop journal journal J

    author author R. Borrelli ,\ @noop journal journal J. Chem. Phys. \ volume 150 ,\ pages 234102 ( year 2019 ) NoStop

    2019

  29. [37]

    author author S. M. \ Greene \ and\ author V. S. \ Batista ,\ @noop journal journal J. Chem. Theory Comput. \ volume 13 ,\ pages 4034 ( year 2017 ) NoStop

    2017

  30. [38]

    Borrelli \ and\ author M

    author author R. Borrelli \ and\ author M. F. \ Gelin ,\ @noop journal journal Sci. Rep. \ volume 7 ,\ pages 9127 ( year 2017 ) NoStop

    2017

  31. [39]

    Borrelli \ and\ author M

    author author R. Borrelli \ and\ author M. F. \ Gelin ,\ @noop journal journal J. Chem. Phys. \ volume 145 ,\ pages 224101 ( year 2016 ) NoStop

    2016

  32. [40]

    de Vega \ and\ author M.-C

    author author I. de Vega \ and\ author M.-C. \ Ba \ n uls ,\ @noop journal journal Phys. Rev. A \ volume 92 ,\ pages 052116 ( year 2015 ) NoStop

    2015

  33. [41]

    Borrelli ,\ @noop journal journal Chem

    author author R. Borrelli ,\ @noop journal journal Chem. Phys. \ volume 515 ,\ pages 236 ( year 2018 ) NoStop

    2018

  34. [42]

    author author M. F. \ Gelin , author R. Borrelli , \ and\ author W. Domcke ,\ @noop journal journal J. Phys. Chem. Lett. \ volume 10 ,\ pages 2806 ( year 2019 ) NoStop

    2019

  35. [43]

    Yao , author K.-W

    author author Y. Yao , author K.-W. \ Sun , author Z. Luo , \ and\ author H. Ma ,\ @noop journal journal J. Phys. Chem. Lett. \ volume 9 ,\ pages 413 ( year 2018 ) NoStop

    2018

  36. [44]

    Baiardi \ and\ author M

    author author A. Baiardi \ and\ author M. Reiher ,\ @noop journal journal J. Chem. Theory Comput. \ volume 15 ,\ pages 3481 ( year 2019 ) NoStop

    2019

  37. [45]

    Li , author J

    author author W. Li , author J. Ren , \ and\ author Z. Shuai ,\ @noop journal journal arXiv preprint arXiv:1907.12044 \ ( year 2019 ) NoStop

    1907 arXiv

  38. [46]

    Ma , author Z

    author author H. Ma , author Z. Luo , \ and\ author Y. Yao ,\ @noop journal journal Mol. Phys. \ volume 116 ,\ pages 854 ( year 2018 ) NoStop

    2018

  39. [47]

    O . Legeza , author J. R \

    author author \"O . Legeza , author J. R \"o der , \ and\ author B. Hess ,\ @noop journal journal Physical Review B \ volume 67 ,\ pages 125114 ( year 2003 ) NoStop

    2003

  40. [48]

    Tal-Ezer \ and\ author R

    author author H. Tal-Ezer \ and\ author R. Kosloff ,\ @noop journal journal J. Chem. Phys. \ volume 81 ,\ pages 3967 ( year 1984 ) NoStop

    1984

  41. [49]

    Nevanlinna ,\ @noop title Convergence of iterations for linear equations \ ( publisher Birkh \"a user ,\ year 2012 ) NoStop

    author author O. Nevanlinna ,\ @noop title Convergence of iterations for linear equations \ ( publisher Birkh \"a user ,\ year 2012 ) NoStop

    2012

  42. [50]

    Saad ,\ @noop title Iterative methods for sparse linear systems ,\ Vol

    author author Y. Saad ,\ @noop title Iterative methods for sparse linear systems ,\ Vol. volume 82 \ ( publisher siam ,\ year 2003 ) NoStop

    2003

  43. [51]

    Moler \ and\ author C

    author author C. Moler \ and\ author C. Van Loan ,\ @noop journal journal SIAM Rev. \ volume 45 ,\ pages 3 ( year 2003 ) NoStop

    2003

  44. [52]

    Hubig , author F

    author author C. Hubig , author F. Lachenmaier , author N.-O. \ Linden , author T. Reinhard , author L. Stenzel , author A. Swoboda , \ and\ author M. Grundner ,\ https://syten.eu title The SyTen toolkit , \ ( year 2019 ) NoStop

    2019

  45. [53]

    Hubig ,\ title Symmetry-Protected Tensor Networks ,\ https://edoc.ub.uni-muenchen.de/21348/ Ph.D

    author author C. Hubig ,\ title Symmetry-Protected Tensor Networks ,\ https://edoc.ub.uni-muenchen.de/21348/ Ph.D. thesis ,\ school LMU München ( year 2017 ) NoStop

    2017

  46. [54]

    Seidner , author G

    author author L. Seidner , author G. Stock , author A. Sobolewski , \ and\ author W. Domcke ,\ @noop journal journal J. Chem. Phys. \ volume 96 ,\ pages 5298 ( year 1992 ) NoStop

    1992

  47. [55]

    Woywod , author W

    author author C. Woywod , author W. Domcke , author A. L. \ Sobolewski , \ and\ author H.-J. \ Werner ,\ @noop journal journal J. Chem. Phys. \ volume 100 ,\ pages 1400 ( year 1994 ) NoStop

    1994

  48. [56]

    author author G. A. \ Worth , author H.-D. \ Meyer , \ and\ author L. Cederbaum ,\ @noop journal journal J. Chem. Phys. \ volume 105 ,\ pages 4412 ( year 1996 ) NoStop

    1996

  49. [57]

    Raab , author G

    author author A. Raab , author G. A. \ Worth , author H.-D. \ Meyer , \ and\ author L. Cederbaum ,\ @noop journal journal J. Chem. Phys. \ volume 110 ,\ pages 936 ( year 1999 ) NoStop

    1999

  50. [58]

    Burghardt , author K

    author author I. Burghardt , author K. Giri , \ and\ author G. Worth ,\ @noop journal journal J. Chem. Phys. \ volume 129 ,\ pages 174104 ( year 2008 ) NoStop

    2008

  51. [59]

    author author M. A. \ Saller \ and\ author S. Habershon ,\ @noop journal journal J. Chem. Theory Comput. \ volume 11 ,\ pages 8 ( year 2014 ) NoStop

    2014

  52. [60]

    Kloss , author I

    author author B. Kloss , author I. Burghardt , \ and\ author C. Lubich ,\ @noop journal journal J. Chem. Phys. \ volume 146 ,\ pages 174107 ( year 2017 ) NoStop

    2017

  53. [61]

    Xiang ,\ @noop journal journal Phys

    author author T. Xiang ,\ @noop journal journal Phys. Rev. B \ volume 53 ,\ pages R10445 ( year 1996 ) NoStop

    1996

  54. [62]

    Legeza \ and\ author J

    author author \"O . Legeza \ and\ author J. S \'o lyom ,\ @noop journal journal Physical Review B \ volume 68 ,\ pages 195116 ( year 2003 ) NoStop

    2003

  55. [63]

    Yamazaki , author T

    author author I. Yamazaki , author T. Murao , author T. Yamanaka , \ and\ author K. Yoshihara ,\ @noop journal journal Faraday Disc. Chem. Soc. \ volume 75 ,\ pages 395 ( year 1983 ) NoStop

    1983

  56. [64]

    Vendrell \ and\ author H.-D

    author author O. Vendrell \ and\ author H.-D. \ Meyer ,\ @noop journal journal J. Chem. Phys. \ volume 134 ,\ pages 044135 ( year 2011 ) NoStop

    2011

  57. [65]

    author author M. B. \ Smith \ and\ author J. Michl ,\ @noop journal journal Chem. Rev. \ volume 110 ,\ pages 6891 ( year 2010 ) NoStop

    2010

  58. [66]

    author author M. B. \ Smith \ and\ author J. Michl ,\ @noop journal journal Annu. Rev. Phys. Chem. \ volume 64 ,\ pages 361 ( year 2013 ) NoStop

    2013

  59. [67]

    Casanova ,\ @noop journal journal Chem

    author author D. Casanova ,\ @noop journal journal Chem. Rev. \ volume 118 ,\ pages 7164 ( year 2018 ) NoStop

    2018

  60. [68]

    author author P. E. \ Teichen \ and\ author J. D. \ Eaves ,\ @noop journal journal J. Phys. Chem. B \ volume 116 ,\ pages 11473 ( year 2012 ) NoStop

    2012

  61. [69]

    author author T. C. \ Berkelbach , author M. S. \ Hybertsen , \ and\ author D. R. \ Reichman ,\ @noop journal journal J. Chem. Phys. \ volume 138 ,\ pages 510 ( year 2013 a ) NoStop

    2013

  62. [70]

    author author T. C. \ Berkelbach , author M. S. \ Hybertsen , \ and\ author D. R. \ Reichman ,\ @noop journal journal J. Chem. Phys. \ volume 138 ,\ pages 114102 ( year 2013 b ) NoStop

    2013

  63. [71]

    author author T. C. \ Berkelbach , author M. S. \ Hybertsen , \ and\ author D. R. \ Reichman ,\ @noop journal journal J. Chem. Phys. \ volume 141 ,\ pages 6891 ( year 2014 ) NoStop

    2014

  64. [72]

    author author A. A. \ Bakulin , author S. E. \ Morgan , author T. B. \ Kehoe , author M. W. \ Wilson , author A. W. \ Chin , author D. Zigmantas , author D. Egorova , \ and\ author A. Rao ,\ @noop journal journal Nat. Chem. \ volume 8 ,\ pages 16 ( year 2016 ) NoStop

    2016

  65. [73]

    Tamura , author M

    author author H. Tamura , author M. Huix-Rotllant , author I. Burghardt , author Y. Olivier , \ and\ author D. Beljonne ,\ @noop journal journal Phys. Rev. Lett. \ volume 115 ,\ pages 107401 ( year 2015 ) NoStop

    2015

  66. [74]

    Yao ,\ @noop journal journal Phys

    author author Y. Yao ,\ @noop journal journal Phys. Rev. B \ volume 93 ,\ pages 115426 ( year 2016 ) NoStop

    2016

  67. [75]

    Fujihashi , author L

    author author Y. Fujihashi , author L. Chen , author A. Ishizaki , author J. Wang , \ and\ author Y. Zhao ,\ @noop journal journal J. Chem. Phys. \ volume 146 ,\ pages 044101 ( year 2017 ) NoStop

    2017

  68. [76]

    author author A. F. \ Morrison \ and\ author J. M. \ Herbert ,\ @noop journal journal J. Phys. Chem. Lett. \ volume 8 ,\ pages 1442 ( year 2017 ) NoStop

    2017

  69. [77]

    Miyata , author Y

    author author K. Miyata , author Y. Kurashige , author K. Watanabe , author T. Sugimoto , author S. Takahashi , author S. Tanaka , author J. Takeya , author T. Yanai , \ and\ author Y. Matsumoto ,\ @noop journal journal Nat. Chem. \ volume 9 ,\ pages 983 ( year 2017 ) NoStop

    2017

  70. [78]

    Zang , author Y

    author author H. Zang , author Y. Zhao , \ and\ author W. Liang ,\ @noop journal journal J. Phys. Chem. Lett. \ volume 8 ,\ pages 5105 ( year 2017 ) NoStop

    2017

  71. [79]

    Tempelaar \ and\ author D

    author author R. Tempelaar \ and\ author D. R. \ Reichman ,\ @noop journal journal J. Chem. Phys. \ volume 146 ,\ pages 330 ( year 2017 a ) NoStop

    2017

  72. [80]

    Tempelaar \ and\ author D

    author author R. Tempelaar \ and\ author D. R. \ Reichman ,\ @noop journal journal J. Chem. Phys. \ volume 146 ,\ pages 510 ( year 2017 b ) NoStop

    2017

  73. [81]

    Tempelaar \ and\ author D

    author author R. Tempelaar \ and\ author D. R. \ Reichman ,\ @noop journal journal J. Chem. Phys. \ volume 148 ,\ pages 244701 ( year 2018 ) NoStop

    2018

  74. [82]

    Xie , author A

    author author X. Xie , author A. Santana-Bonilla , author W. Fang , author C. Liu , author A. Troisi , \ and\ author H. Ma ,\ @noop journal journal J. Chem. Theory Comput. \ volume 15 ,\ pages 3721 ( year 2019 ) NoStop

    2019

  75. [83]

    Zheng , author Y

    author author J. Zheng , author Y. Xie , author S. Jiang , \ and\ author Z. Lan ,\ @noop journal journal J. Phys. Chem. C \ volume 120 ,\ pages 1375 ( year 2016 ) NoStop

    2016

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.