Charge transport limited by nonlocal electron-phonon interaction. II. Numerically exact quantum dynamics in the slow-phonon regime

Veljko Jankovi\'c

REVIEW 2 major objections 5 minor 2 cited by

Charge transport limited by nonlocal electron-phonon interaction. II. Numerically exact quantum dynamics in the slow-phonon regime

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Pith's one-line read In the 1D Peierls model at room-temperature rubrene parameters, the carrier's brief subdiffusive slowdown is transient, and diffusive transport returns from the superdiffusive side within about one phonon period.

desk verdict A careful, honest HEOM study that challenges TLS and QMC transport pictures for the slow-phonon Peierls model, but the load-bearing long-time features rest on a closing approximation, so the verdict is conditional, not numerically exact. read the letter →

arxiv 2501.05055 v3 pith:U6BVUHV3 submitted 2025-01-09 cond-mat.str-el cond-mat.mes-hallphysics.chem-phquant-ph

Veljko Janković This is my paper · ORCID

classification cond-mat.str-elcond-mat.mes-hallphysics.chem-phquant-ph

keywords chargetransportPeierlsmodelnonlocalelectron-phononinteractionhierarchicalequationsofmotiontransientlocalizationscenariodynamicalmobilityorganicsemiconductorsrubrene

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 asks how a single charge carrier moves when slow, thermally abundant phonons modulate its hopping amplitude, the setting believed to control room-temperature transport in soft molecular semiconductors. Using the hierarchical equations of motion (HEOM), a numerically exact real-time method, on the one-dimensional Peierls model near the adiabatic limit, it finds that the transient localization scenario (TLS) works well at high temperatures and strong coupling, where phonon-assisted current dominates and diffusion sets in before one phonon period. At the moderate parameters representative of rubrene, the situation changes: the super-to-subdiffusive crossover is transient, and diffusive transport is approached from the superdiffusive side on timescales of about one phonon period. The computed dynamical mobility then shows a displaced Drude peak, a dip near the phonon frequency, and a zero-frequency peak rather than the TLS's zero-frequency minimum. This matters because the shape of the low-frequency spectrum is an observable fingerprint of whether the phonon dynamics is treated as frozen or fully quantum.

What carries the argument

The load-bearing objects are the finite-temperature current-current correlation function $C_{jj}(t)$, the time-dependent diffusion constant $D(t)=\int_0^t ds\,\mathrm{Re}\,C_{jj}(s)$, the diffusion exponent $\alpha(t)=2tD(t)/\Delta x^2(t)$, and the dynamical mobility $\mathrm{Re}\,\mu(\omega)$. The numerical engine is the HEOM approach from the companion paper, which expresses the hierarchy auxiliaries in phonon creation and annihilation operators, decomposes the current into band, phonon-assisted, and cross parts, and closes the hierarchy with a Markovian-adiabatic scheme that stabilizes long-time propagation. The contrasting object is the TLS ansatz $C_{\mathrm{TLS}}^{jj}(t)=C_{\mathrm{dis}}^{jj}(t)e^{-|t|/\tau_d}$ with $\tau_d^{-1}=\alpha_d\,\omega_0$ and $\alpha_d=2.2$, which replaces the phonon field by frozen Gaussian disorder and restores dynamics by a relaxation-time cutoff. The comparison of these two objects identifies the regime in which frozen-phonon reasoning is adequate and the regime in which phonon-period-scale recovery changes the spectrum.

What would settle it

Use an independent real-time method that needs no hierarchy-closing approximation to propagate the same one-dimensional Peierls model at $\omega_0/J=0.044$, $\lambda=0.336$, and $T/J=0.175$, and check two predictions: the diffusion constant rises after $t\approx 1/\omega_0$ and the dynamical mobility has a local maximum (not a minimum) at zero frequency.

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Extended reading notes

Core claim

The central claim is that in the one-dimensional Peierls model with a single undamped optical phonon per site and $\omega_0/J=0.044$, $\lambda=0.336$, $T/J=0.175$---parameters standing in for room-temperature rubrene---the subdiffusive slowdown of the carrier is temporary. The diffusion constant $D(t)=\int_0^t ds\,\mathrm{Re}\,C_{jj}(s)$ falls to a minimum near $t\approx 1/\omega_0$, then rises and saturates by about $2\pi/\omega_0$, so the long-time diffusive limit is reached from the superdiffusive side. Correspondingly, the real-part dynamical mobility $\mathrm{Re}\,\mu(\omega)$ contains a displaced Drude peak at $\omega\approx 0.2J$, a dip near $\omega_0$, and a local maximum at zero frequency. The paper also establishes that the TLS with relaxation-time parameter $\alpha_d=2.2$ reproduces HEOM mobilities across much of the phase diagram, but misses these low-frequency features because its exponential cutoff erases the coupled carrier-phonon memory that drives the recovery. The Boltzmann bubble approximation fails for moderate coupling because the cross contribution between band and phonon-assisted currents is non-negligible.

Load-bearing premise

The paper's long-time results assume that numerically stabilizing its truncated set of coupled quantum equations does not distort the carrier's true motion once the phonons start moving; if the stabilization itself creates the observed recovery of the diffusion constant, the central conclusion would not survive.

Editorial extensions

If this is right

At room-temperature rubrene parameters, the diffusion constant should bottom out near $t\approx 1/\omega_0$ and saturate by about one phonon period, so mobilities extracted from simulations shorter than that will come out too low.
The dynamical mobility should show a displaced Drude peak, a dip near $\omega_0$, and a zero-frequency maximum, making terahertz absorption on rubrene the direct place to look for the three-feature structure.
The TLS's good mobility predictions at high temperature and strong coupling mean the frozen-phonon picture survives there, but its spectrum at moderate parameters lacks the zero-frequency peak, so mobility agreement alone is not spectral agreement.
The cross contribution between band and phonon-assisted currents is non-negligible at moderate coupling, so approximations that drop it will misestimate both the dc mobility and the line shape.

Reading between the lines

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

Because the HEOM and TLS mobilities agree within roughly ten percent while their low-frequency spectra differ qualitatively, the mobility itself is a weak discriminator; the zero-frequency shape of $\mathrm{Re}\,\mu(\omega)$ is the sharper experimental test.
The paper's interpretation of the TLS exponential cutoff as effective extra scattering suggests a testable hierarchy: adding static disorder or additional phonon branches to the model should suppress the post-minimum rise in $D(t)$ and push the spectrum back toward TLS, and this could be checked with the same HEOM machinery on structured spectral densities.
Read as a statement about the idealized single-mode model, the result implies the single undamped phonon is the hardest case for TLS; a realistic broadened phonon density of states should interpolate between the HEOM and TLS predictions, which is the author's stated expectation.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

First thing to know: this is a serious numerical study, and the central claim is not a trivial artifact, but it is not as ironclad as the title's 'numerically exact' suggests. The paper reports the first HEOM-based real-time results for the 1D Peierls model with slow phonons, and finds that at room-temperature-like parameters the transient subdiffusion reverses: D(t) turns up around t≈1/ω0 and diffusion is approached from the superdiffusive side, producing a dip at ω0 and a zero-frequency peak in Re μ(ω). That contradicts both the transient localization scenario and the best imaginary-axis QMC results, and supports the recent quantum-classical mapping-approach simulations.

What the paper does well: the methodology is transparent, the data are open, and the author runs serious convergence tests - chain length, hierarchy depth, closing variants, optical sum rules at the 10^-4 level, cross-correlation symmetry. The decomposition into band, phonon-assisted, and cross contributions is useful, and the TLS comparison is honest, including a regime where TLS works well. The paper also flags its own limitations, noting that a realistic phonon density of states would likely reduce the qualitative differences.

Where it gets soft: the long-time dynamics, which is exactly what generates the zero-frequency peak and the dip, is stabilized by the Markovian-adiabatic closing scheme. Without closing, the hierarchy becomes unstable after about one phonon period. All three closing schemes tested share the same long-time Markovian-adiabatic approximation, so their mutual agreement does not rule out a common bias. The paper argues the D(t) upturn begins in the window where even the unclosed dynamics is still reasonable, which is a fair point, but the magnitude and shape of the low-frequency features at longer times are still closure-dependent. Also, the superdiffusive approach is clearly seen for D=4, while D=3 shows a weaker upturn and D=5 doesn't saturate; only D=4 supports the clean conclusion. That's a narrow basis for a headline claim.

The comparison to QMC is reasonable, and the author's interpretation that the difference comes from analytic continuation uncertainty is plausible. The 'numerically exact' label is stronger than the closing-scheme approximation warrants; the abstract's 'numerically exact' is fine if understood as the HEOM framework, but for this specific model the long-time results are not exact.

Who should read it: anyone working on organic-semiconductor transport, transient localization, or real-time quantum dynamics for electron-phonon models. It deserves a serious referee. My recommendation: send it to peer review, but the referee should push for a more guarded title and for additional evidence on the closing scheme - for example, a check against a method that does not rely on this closure, even if only at shorter times or for a smaller system.

Referee Report

2 major / 5 minor

Summary. The paper studies charge transport in the one-dimensional Peierls model with slow, dispersionless phonons, using a hierarchical-equations-of-motion (HEOM) approach developed in a companion paper. The central claims are: (i) in the moderate-interaction, moderate-temperature regime relevant to rubrene, the transient-localization scenario (TLS) reproduces HEOM mobilities but not the detailed dynamics; (ii) the super-to-subdiffusive crossover is transient, with subdiffusion lasting roughly half a phonon period and diffusion subsequently being approached from the superdiffusive side on timescales of about one phonon period; and (iii) the dynamical mobility in this regime contains a displaced Drude peak, a dip near the phonon frequency, and a zero-frequency peak, in contrast to TLS and imaginary-axis QMC results. The paper presents systematic checks of chain length and hierarchy depth, optical sum rules, cross-correlation symmetry, and comparisons with TLS, Boltzmann theory, QMC, and quantum-classical simulations.

Significance. If the central dynamical claims are correct, the paper provides the first real-axis numerically reliable solution of the slow-phonon Peierls model in a regime where approximate methods disagree qualitatively, and it sharpens the physical picture of charge transport in molecular semiconductors. The manuscript is unusually transparent: numerical parameters are tabulated, data are openly deposited, the closing approximation is discussed explicitly, and several independent error checks are reported. The comparison with quantum-classical surface-hopping results and the honest discussion of the limitations of TLS are valuable. The main uncertainty is whether the long-time dynamics that produces the zero-frequency peak and the superdiffusive approach is a true property of the Hamiltonian or an artifact of the hierarchy-closing approximation.

major comments (2)

[Sec. II.B and Appendix B] The load-bearing long-time dynamics (t ≳ 1/ω0) is stabilized by the Markovian-adiabatic closing, which is an approximation rather than a controlled truncation. The three closing schemes compared in Fig. 5(a,b,d) share the same Markovian-adiabatic long-time structure, so their mutual agreement does not exclude a common-mode bias. The comparison with the time-nonlocal scheme in Fig. 5(c) shows that the upturn of D(t) begins before the closing becomes active, but it does not certify that the subsequent saturation of D(t) and the associated zero-frequency peak of Re μ(ω) are exact model features. The optical sum rule (Table I) constrains only total spectral weight, and the cross-correlation symmetry error is of order 10^-2, comparable to the small-amplitude features in D(t) and α(t). I ask for an independent validation of the long-time region, for example a different hierarchy termination, an alternative real-time method on a smaller system, or a continuous-spectral-density calculation, or, failing that, a clear statement that the central conclusion is conditional on the closing approximation.
[Appendix A, Fig. 4 and Table I] The hierarchy-depth convergence check does not currently support the central conclusion. For D=3 the upturn is weaker, for D=5 the long-time dynamics does not saturate, and only D=4 exhibits the claimed superdiffusive approach to a plateau. The optical sum rule is satisfied to high accuracy also by the unreliable D=5 run, showing that δOSR is not a sufficient diagnostic for the low-frequency lineshape. Because the central physical claim depends on the intermediate-to-long-time behavior, the manuscript should explain why the failure of D=5 is a known limitation of the closing scheme rather than evidence that D=4 is not converged, and should quantify how tmin, D(tmin), and the zero-frequency peak height vary with D in the range where the method is usable.

minor comments (5)

[Title] The title contains a spacing artifact: “inte raction” should read “interaction”.
[Sec. V] In the conclusion, “may poor at treating” should read “may be poor at treating”.
[Throughout] Several mathematical comparison symbols appear as unrendered LaTeX commands such as “/greaterorsimilar”; these should be replaced by the intended notation.
[Fig. 4 and Fig. 5] The insets in Figs. 4 and 5 are essential for judging the claims but are very small; enlarging them or plotting the intermediate-to-long-time window as a separate panel would improve readability.
[Sec. III.B] The statement that “the results of the most recent quantum–classical approaches bear qualitative similarity” is appropriately cautious, but the later sentence in Sec. IV that a more realistic phonon density of states would reduce the differences is presented without direct numerical support; it would be helpful to label this explicitly as a conjecture.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: HEOM transport results are parameter-free solutions of the stated model, with the only fitted TLS constant imported from prior QMC calibration and external benchmarks used throughout.

full rationale

The paper's central claims are extracted from numerical HEOM solutions of the one-dimensional Peierls model (Eqs. 1-3), and no target observable is used as an input to obtain them. The only fitted parameter in the TLS comparison, αd=2.2, is adopted from Ref. 57, where it was calibrated against QMC and Holstein-model reference data, not against the HEOM results presented here; the paper explicitly states that 'the TLS can reproduce HEOM mobilities very well once the free-parameter αd ∼ 1 is appropriately tuned,' which is an honest description of tuning rather than a hidden fit. Self-citations to the companion paper (arXiv:2501.05054) supply the HEOM machinery and the Markovian-adiabatic closing scheme, but the long-time dynamics that carries the main conclusion is not justified only by that citation: Appendix B shows that the increase in D(t) after t_min≈1/ω0 appears even with zero closing terms up to about half a phonon period, and that MA, MA-avg, and DR closing schemes give essentially the same D(t), α(t), and Re μ(ω), with reported optical-sum-rule and cross-correlation-symmetry accuracies. The acknowledged need for closing at t≳1/ω0, and the possibility that a common-mode closure bias could mimic the recovery, is a numerical robustness concern, not a logical circularity: the paper states this limitation explicitly ('the hierarchy closing is vital to computing long-time transport dynamics') and attempts to bound it by the closure comparisons and by agreement with quantum–classical simulations and QMC mobilities. Thus no step in the derivation reduces by construction to its inputs.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The paper's own calculation is a parameter-free numerical solution of a standard model Hamiltonian; the only fitted constant (α_d=2.2) sits inside the TLS benchmark, not the HEOM result. The HEOM closure is an ad hoc numerical approximation that the central claim depends on. No invented entities are introduced, and the model's single-mode phonon spectrum is a deliberate simplification the authors flag as limiting the real-world reach of their findings.

free parameters (1)

TLS proportionality constant α_d = 2.2
Adopted from Fratini and Ciuchi (Phys. Rev. Res. 2, 013001, 2020), calibrated against quantum Monte Carlo results for the Holstein/Peierls models; used in all TLS computations in this paper. Reasonable variations around unity affect quantitative mobilities but not the qualitative TLS picture, according to the paper.

assumptions (5)

domain assumption The one-dimensional Peierls model with a single undamped dispersionless optical phonon mode per site (Eqs. 1-3) captures the physics of room-temperature charge transport in rubrene.
Standard model for nonlocal carrier-phonon coupling in organic semiconductors; parameters J=143 meV, ω0=6.2 meV, a=7.2 Å are taken from rubrene literature.
ad hoc to paper The Markovian-adiabatic hierarchy closing scheme for HEOM truncation yields long-time transport dynamics that are numerically reliable.
Introduced in the companion paper and used in Sec. II.B; without closing, long-time oscillations prevent mobility extraction; validated in Appendix B across closing variants.
domain assumption The TLS ansatz (C_jj^TLS(t) = C_jj^dis(t) e^{-|t|/τ_d}, Eq. 12) is the appropriate approximate benchmark for the slow-phonon regime.
Standard transient-localization ansatz from Refs. 55-58; the decay rate τ_d^{-1}=α_d ω0 with α_d=2.2 is adopted from prior calibration to QMC.
domain assumption Phonons in the TLS benchmark can be replaced by classical Gaussian disorder with variance σ²=2λJT on timescales short compared to 1/ω0.
Standard TLS assumption, not used in HEOM; used only to construct C_jj^dis in Sec. II.C.
standard math The Kubo formula and Einstein relation connect the current autocorrelation function to the mobility.
Standard linear-response framework used throughout (Eqs. 5-10).

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Pith. "Pith review of Charge transport limited by nonlocal electron-phonon interaction. II. Numerically exact quantum dynamics in the slow-phonon regime." pith.science (2026). https://pith.science/paper/U6BVUHV3

@misc{pith2026250105055,
  author       = {Pith},
  title        = {Pith review of: Charge transport limited by nonlocal electron-phonon interaction. II. Numerically exact quantum dynamics in the slow-phonon regime},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/U6BVUHV3}},
  note         = {Machine review of arXiv:2501.05055}
}

read the original abstract

Transport of charge carriers in mechanically soft semiconductors is mainly limited by their interaction with slow intermolecular phonons. Carrier motion exhibits a crossover from superdiffusive to subdiffusive, producing a distinct low-frequency peak in the dynamical-mobility profile. These features can be understood within approaches relying on the timescale separation between carrier and phonon dynamics, such as the transient localization scenario (TLS). However, recovering them from fully quantum dynamics has proved elusive. Using the hierarchical equations of motion (HEOM)-based approach exposed in a companion paper (arXiv:2501.05054), we study carrier transport in the one-dimensional Peierls model near the adiabatic limit. We find that the TLS approximates HEOM dynamics very well at higher temperatures and for stronger interactions. Then, the transport is predominantly phonon-assisted, and turns diffusive from the subdiffusive side well before one phonon period. In contrast, the band current dominates at moderate temperatures and interactions, relevant for transport in realistic materials. We then conclude that the super-to-subdiffusive crossover is transient, so that the diffusive motion sets in from the superdiffusive side on timescales comparable to the phonon period. The low-frequency dynamical mobility then additionally exhibits a dip at approximately one phonon frequency, and the zero-frequency peak. Our findings in this moderate regime show limitations of the TLS, and support the results of the most advanced quantum--classical simulations. We expect that the qualitative differences between HEOM and TLS dynamics would diminish for a more realistic phonon density of states.

Figures

Figures reproduced from arXiv: 2501.05055 by the authors.

**Figure 2.** FIG. 2. Dynamics of the diffusion constant [(a1) and (b1)] [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** FIG. 3. (a1)–(d1): Dynamical-mobility profiles computed us [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: FIG. 4. Effects of finite maximum hierarchy depth [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]

**Figure 5.** Figure 5: FIG. 5. (a)–(c): Dynamics of (a) the diffusion constant, (b) t [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Charge transport limited by nonlocal electron-phonon interaction. I. Hierarchical equations of motion approach
cond-mat.str-el 2025-01 conditional novelty 6.0 of 10

HEOM auxiliaries are expressed through phonon operators, the generalized Wick theorem is proved, and exact mobility is computed for the 1D Peierls model.
Dynamical quantum typicality: Simple method for investigating transport properties applied to the Holstein model
cond-mat.str-el 2024-12 conditional novelty 6.0 of 10

Quantum typicality produces well-converged frequency-dependent electron mobility for the Holstein model at strong coupling, confirmed by quantum Monte Carlo and used to quantify vertex corrections.

Reference graph

Works this paper leans on

118 extracted references · 55 canonical work pages · cited by 2 Pith papers

[1]

and abundantly present ( T /greaterorsimilarω0) phonons. Then, charge dynamics on timescales short compared to ω−1 0 , when phonons can be considered as frozen, is essentially the same as the dynamics in the presence of Gaussian static disorder in the hopping amplitude, whose strength is σ2 = 2 2g2 βω0 = 2λJ T. Formally, one replaces the phonon operator g...
[2]

( 5) by the average over diﬀerent disorder realizations

and ( 6) by a Gaussian random variable Xn,n+1 of zero mean and variance σ2, and the trace in Eq. ( 5) by the average over diﬀerent disorder realizations. The current–current cor- relation function computed upon introducing these as- sumptions in Eq. (
[3]

M. J. Schilcher, P. J. Robinson, D. J. Abramovitch, L. Z. Tan, A. M. Rappe, D. R. Reichman, and D. A. Egger, The signiﬁcance of polarons and dynamic disorder in halide perovskites, ACS Energy Lett. 6, 2162 (2021)

2021
[4]

is denoted as Cdis jj (t). Charge diﬀu- sion, which is inhibited in the static-disorder setup [54], 4 is ultimately established through the coupled charge– phonon dynamics, whose eﬀects become appreciable on timescale τd. The TLS eﬀectively restores phonon dy- namics by virtue of the RTA, in which CTLS jj (t) = Cdis jj (t)e−|t|/τd. (12) Physically, τ −1 d...
[5]

denote averaging over the equilibrium state ρeq = e−βH /Tre−βH of the interacting 3 carrier–phonon system at temperature T = β−1. The current operator j = je + je−ph = −i ∑ n [ −J + g ( bn + b† n − bn+1 − b† n+1 )] × (|n⟩⟨n + 1| − |n + 1⟩⟨n|) (6) has the so-called phonon-assisted contribution je−ph [the term proportional to g in Eq. ( 6)] in addition to t...
[6]

( 1)–(3)]

eﬀectively takes into account inter- actions of the carrier with additional phonon modes not considered in the model [Eqs. ( 1)–(3)]. Such a possibility is supported by the most recent quantum–classical sim- ulations [67]. There, the authors conclude that consid- ering a continuous phonon spectrum centered around ω0 instead of the delta-like spectrum used...
[7]

At realistic tem- peratures and interactions, we argue that Eq

is suf- ﬁciently good to describe this predominantly phonon- 9 assisted transport, even when the timescales of carrier and phonon dynamics are comparable. At realistic tem- peratures and interactions, we argue that Eq. (
[8]

can be considered to eﬀectively take into account other scatter- ing mechanisms not included in the present model. Apart from providing numerically exact results for carrier dy- namics, this piece of research can be regarded as a formal justiﬁcation of the already well-established practical ap- plicability of the TLS to realistic systems, for which the mo...

Show all 118 references

[9]

M. Z. Mayers, L. Z. Tan, D. A. Egger, A. M. Rappe, and D. R. Reichman, How lattice and charge ﬂuctu- ations control carrier dynamics in halide perovskites, Nano Lett. 18, 8041 (2018)

2018
[10]

D. A. Egger, A. Bera, D. Cahen, G. Hodes, T. Kir- chartz, L. Kronik, R. Lovrincic, A. M. Rappe, D. R. Reichman, and O. Yaﬀe, What remains unex- plained about the properties of halide perovskites?, Adv. Mater. 30, 1800691 (2018)

2018
[11]

W. P. Su, J. R. Schrieﬀer, and A. J. Heeger, Solitons in polyacetylene, Phys. Rev. Lett. 42, 1698 (1979)

1979
[12]

( 1)–(3)] considered

is es- tablished on the basis of physical arguments [55, 58], we note that similar ans¨ atze are often used to eﬀectively take into account the inﬂuence of other scattering mechanisms not included in the model [Eqs. ( 1)–(3)] considered. For example, the polaron transformation...
[13]

Mladenovi´ c and N

M. Mladenovi´ c and N. Vukmirovi´ c, Charge carrier localization and transport in organic semiconduc- tors: Insights from atomistic multiscale simulations, Adv. Funct. Mater. 25, 1915 (2015)

2015
[14]

Ostroverkhova, Organic optoelectronic materials: Mechanisms and applications, Chem

O. Ostroverkhova, Organic optoelectronic materials: Mechanisms and applications, Chem. Rev. 116, 13279 (2016)

2016
[15]

Fratini, M

S. Fratini, M. Nikolka, A. Salleo, G. Schweicher, and H. Sirringhaus, Charge transport in high-mobility conjugated polymers and molecular semiconductors, Nat. Mater. 19, 491 (2020)

2020
[16]

Nematiaram and A

T. Nematiaram and A. Troisi, Modeling charge transport in high-mobility molecular semiconduc- tors: Balancing electronic structure and quantum dynamics methods with the help of experiments, J. Chem. Phys. 152, 190902 (2020)

2020
[17]

Giannini and J

S. Giannini and J. Blumberger, Charge trans- port in organic semiconductors: The perspec- tive from nonadiabatic molecular dynamics, Acc. Chem. Res. 55, 819 (2022)

2022
[18]

Troisi and G

A. Troisi and G. Orlandi, Charge-transport regime of crystalline organic semiconductors: Diﬀusion lim- ited by thermal oﬀ-diagonal electronic disorder, Phys. Rev. Lett. 96, 086601 (2006)

2006
[19]

Troisi, Prediction of the absolute charge mobil- ity of molecular semiconductors: the case of rubrene, Adv

A. Troisi, Prediction of the absolute charge mobil- ity of molecular semiconductors: the case of rubrene, Adv. Mater. 19, 2000 (2007)

2007
[20]

Capone, W

M. Capone, W. Stephan, and M. Grilli, Small- polaron formation and optical absorption in Su-Schrieﬀer-Heeger and Holstein models, Phys. Rev. B 56, 4484 (1997)

1997
[21]

Fratini and S

S. Fratini and S. Ciuchi, Bandlike motion and mo- bility saturation in organic molecular semiconductors, Phys. Rev. Lett. 103, 266601 (2009)

2009
[22]

R. Kubo, M. Toda, and N. Hashitsume, Statisti- cal Physics II: Nonequilibrium Statistical Mechanics (Springer-Verlag Berlin Heidelberg, 1985)

1985
[23]

Mahan, Many-Particle Physics (Kluwer Academic, New York, 2000)

G. Mahan, Many-Particle Physics (Kluwer Academic, New York, 2000)

2000
[24]

Vukmirovi´ c, C

N. Vukmirovi´ c, C. Bruder, and V. M. Stojanovi´ c, Electron-phonon coupling in crystalline organic semi- conductors: Microscopic evidence for nonpolaronic charge carriers, Phys. Rev. Lett. 109, 126407 (2012)

2012
[25]

R. W. Munn and R. Silbey, Theory of electronic trans- port in molecular crystals. III. Diﬀusion coeﬃcient in- corporating nonlocal linear electron–phonon coupling, J. Chem. Phys. 83, 1854 (1985)

1985
[26]

R. W. Munn and R. Silbey, Theory of electronic trans- port in molecular crystals. II. Zeroth order states in- corporating nonlocal linear electron–phonon coupling, J. Chem. Phys. 83, 1843 (1985)

1985
[27]

Hannewald, V

K. Hannewald, V. M. Stojanovi´ c, J. M. T. Schellekens, P. A. Bobbert, G. Kresse, and J. Hafner, Theory of po- laron bandwidth narrowing in organic molecular crys- tals, Phys. Rev. B 69, 075211 (2004)

2004
[28]

Hannewald and P

K. Hannewald and P. A. Bobbert, Anisotropy eﬀects in phonon-assisted charge-carrier transport in organic molecular crystals, Phys. Rev. B 69, 075212 (2004)

2004
[29]

V. M. Stojanovi´ c, P. A. Bobbert, and M. A. J. Michels, Nonlocal electron-phonon coupling: Consequences for the nature of polaron states, Phys. Rev. B 69, 144302 (2004)

2004
[30]

Hannewald and P

K. Hannewald and P. A. Bobbert, Ab initio theory of charge-carrier conduction in ultrapure organic crystals, Appl. Phys. Lett. 85, 1535 (2004)

2004
[31]

Y. Zhao, D. W. Brown, and K. Lindenberg, On the Munn–Silbey approach to nonlocal exciton–phonon cou- pling, J. Chem. Phys. 100, 2335 (1994)

1994
[32]

D. Chen, J. Ye, H. Zhang, and Y. Zhao, On the Munn-Silbey Approach to Polaron Transport with Oﬀ-Diagonal Coupling and Temperature-Dependent Canonical Transformations, J. Phys. Chem. B 115, 5312 (2011)

2011
[33]

Cataudella, G

V. Cataudella, G. De Filippis, and C. A. Per- roni, Transport properties and optical conductivity of the adiabatic su-schrieﬀer-heeger model: A show- case study for rubrene-based ﬁeld eﬀect transistors, Phys. Rev. B 83, 165203 (2011)

2011
[34]

Gargiulo, C

F. Gargiulo, C. A. Perroni, V. M. Ramaglia, and V. Cataudella, Electronic transport within a quasi-two- dimensional model for rubrene single-crystal ﬁeld eﬀect transistors, Phys. Rev. B 84, 245204 (2011)

2011
[35]

De Filippis, V

G. De Filippis, V. Cataudella, A. S. Mishchenko, N. Nagaosa, A. Fierro, and A. de Candia, Crossover from super- to subdiﬀusive motion and mem- ory eﬀects in crystalline organic semiconductors, Phys. Rev. Lett. 114, 086601 (2015)

2015
[36]

de Candia, G

A. de Candia, G. D. Filippis, L. M. Cangemi, A. S. Mishchenko, N. Nagaosa, and V. Cataudella, Two- channel model for optical conductivity of high-mobility organic crystals, EPL 125, 47002 (2019)

2019
[37]

Wang and Y

Y.-C. Wang and Y. Zhao, Diagrammatic quan- tum Monte Carlo toward the calculation of trans- port properties in disordered semiconductors, J. Chem. Phys. 156, 204116 (2022)

2022
[38]

Ostmeyer, T

J. Ostmeyer, T. Nematiaram, A. Troisi, and P. Buivi- dovich, First-principles quantum Monte Carlo study of charge-carrier mobility in organic molecular semicon- ductors, Phys. Rev. Appl. 22, L031004 (2024)

2024
[39]

J. Ren, W. Li, T. Jiang, Y. Wang, and Z. Shuai, Time-dependent density matrix renormalization group method for quantum dynamics in complex systems, WIREs Comput. Mol. Sci. 12, e1614 (2022)

2022
[40]

W. Li, J. Ren, and Z. Shuai, A general charge transport picture for organic semiconduc- tors with nonlocal electron-phonon couplings, Nat. Commun. 12, 4260 (2021) . 13

2021
[41]

Y. Ge, W. Li, J. Ren, and Z. Shuai, Roles of nonlocal electron-phonon coupling on the electrical conductivity and seebeck coeﬃcient: A time-dependent dmrg study, Phys. Rev. B 110, 035201 (2024)

2024
[42]

D. Wang, L. Chen, R. Zheng, L. Wang, and Q. Shi, Communications: A nonperturbative quantum master equation approach to charge carrier transport in organic molecular crystals, J. Chem. Phys. 132, 081101 (2010)

2010
[43]

Chen and Y

L. Chen and Y. Zhao, Finite temperature dynamics of a Holstein polaron: The thermo-ﬁeld dynamics approach, J. Chem. Phys. 147, 214102 (2017)

2017
[44]

Lian, Y.-C

M. Lian, Y.-C. Wang, Y. Ke, and Y. Zhao, Non- Markovian stochastic Schr¨ odinger equation in k-space toward the calculation of carrier dynamics in organic semiconductors, J. Chem. Phys. 151, 044115 (2019)

2019
[45]

L. Zhou, X. Gao, and Z. Shuai, A stochastic Schr¨ odinger equation and matrix product state ap- proach to carrier transport in organic semicon- ductors with nonlocal electron–phonon interaction, J. Chem. Phys. 161, 084118 (2024)

2024
[46]

Jankovi´ c, Holstein polaron transport from numeri- cally “exact” real-time quantum dynamics simulations, J

V. Jankovi´ c, Holstein polaron transport from numeri- cally “exact” real-time quantum dynamics simulations, J. Chem. Phys. 159, 094113 (2023)

2023
[47]

Jankovi´ c, P

V. Jankovi´ c, P. Mitri´ c, D. Tanaskovi´ c, and N. Vuk- mirovi´ c, Vertex corrections to conductivity in the Holstein model: A numerical-analytical study, Phys. Rev. B 109, 214312 (2024)

2024
[48]

Tanimura, Numerically “exact” approach to open quantum dynamics: The hierarchical equations of mo- tion (HEOM), J

Y. Tanimura, Numerically “exact” approach to open quantum dynamics: The hierarchical equations of mo- tion (HEOM), J. Chem. Phys. 153, 020901 (2020)

2020
[49]

Ishizaki and G

A. Ishizaki and G. R. Fleming, Uniﬁed treatment of quantum coherent and incoherent hopping dynamics in electronic energy transfer: Reduced hierarchy equation approach, J. Chem. Phys. 130, 234111 (2009)

2009
[50]

Xu and Y

R.-X. Xu and Y. J. Yan, Dynamics of quantum dissipation systems interacting with bosonic canoni- cal bath: Hierarchical equations of motion approach, Phys. Rev. E 75, 031107 (2007)

2007
[51]

Gosar and S.-i

P. Gosar and S.-i. Choi, Linear-response theory of the electron mobility in molecular crystals, Phys. Rev. 150, 529 (1966)

1966
[52]

Gosar and I

P. Gosar and I. Vilfan, Phonon-assisted current in or- ganic molecular crystals, Mol. Phys. 18, 49 (1970)

1970
[53]

Song and Q

L. Song and Q. Shi, A new approach to calculate charge carrier transport mobility in organic molecular crystals from imaginary time path integral simulations, J. Chem. Phys. 142, 174103 (2015)

2015
[54]

L. Wang, O. V. Prezhdo, and D. Beljonne, Mixed quantum-classical dynamics for charge transport in or- ganics, Phys. Chem. Chem. Phys. 17, 12395 (2015)

2015
[55]

ten Brink, S

M. ten Brink, S. Gr¨ aber, M. Hopjan, D. Jansen, J. Stolpp, F. Heidrich-Meisner, and P. E. Bl¨ ochl, Real- time non-adiabatic dynamics in the one-dimensional Holstein model: Trajectory-based vs exact methods, J. Chem. Phys. 156, 234109 (2022)

2022
[56]

Hultell and S

M. Hultell and S. Stafstr¨ om, Polaron dy- namics in highly ordered molecular crystals, Chem. Phys. Lett. 428, 446 (2006)

2006
[57]

J. C. Tully, Molecular dynamics with electronic transi- tions, J. Chem. Phys. 93, 1061 (1990)

1990
[58]

Carof, S

A. Carof, S. Giannini, and J. Blumberger, How to calculate charge mobility in molecular materi- als from surface hopping non-adiabatic molecular dynamics – beyond the hopping/band paradigm, Phys. Chem. Chem. Phys. 21, 26368 (2019)

2019
[59]

Ishii, K

H. Ishii, K. Honma, N. Kobayashi, and K. Hi- rose, Wave-packet approach to transport proper- ties of carrier coupled with intermolecular and in- tramolecular vibrations of organic semiconductors, Phys. Rev. B 85, 245206 (2012)

2012
[60]

Ishii, J

H. Ishii, J. Inoue, N. Kobayashi, and K. Hirose, Quan- titative mobility evaluation of organic semiconductors using quantum dynamics based on density functional theory, Phys. Rev. B 98, 235422 (2018)

2018
[61]

Ishii, N

H. Ishii, N. Kobayashi, and K. Hirose, Car- rier transport calculations of organic semi- conductors with static and dynamic disorder, Jpn. J. Appl. Phys. 58, 110501 (2019)

2019
[62]

P. W. Anderson, Absence of diﬀusion in certain random lattices, Phys. Rev. 109, 1492 (1958)

1958
[63]

Ciuchi, S

S. Ciuchi, S. Fratini, and D. Mayou, Transient localization in crystalline organic semiconductors, Phys. Rev. B 83, 081202 (2011)

2011
[64]

Ciuchi and S

S. Ciuchi and S. Fratini, Electronic transport and quantum localization eﬀects in organic semiconductors, Phys. Rev. B 86, 245201 (2012)

2012
[65]

Fratini and S

S. Fratini and S. Ciuchi, Dynamical lo- calization corrections to band transport, Phys. Rev. Res. 2, 013001 (2020)

2020
[66]

Fratini, D

S. Fratini, D. Mayou, and S. Ciuchi, The transient lo- calization scenario for charge transport in crystalline or - ganic materials, Adv. Funct. Mater. 26, 2292 (2016)

2016
[67]

Fratini, S

S. Fratini, S. Ciuchi, D. Mayou, G. T. de Laissardi` ere, and A. Troisi, A map of high-mobility molecular semi- conductors, Nat. Mater. 16, 998 (2017)

2017
[68]

Nematiaram, S

T. Nematiaram, S. Ciuchi, X. Xie, S. Fratini, and A. Troisi, Practical computation of the charge mobility in molecular semiconductors using transient localization theory, J. Phys. Chem. C 123, 6989 (2019)

2019
[69]

Fratini, S

S. Fratini, S. Ciuchi, and D. Mayou, Phenomenological model for charge dynamics and optical response of disor- dered systems: Application to organic semiconductors, Phys. Rev. B 89, 235201 (2014)

2014
[70]

H. Yada, R. Uchida, H. Sekine, T. Terashige, S. Tao, Y. Matsui, N. Kida, S. Fratini, S. Ciuchi, Y. Okada, T. Uemura, J. Takeya, and H. Okamoto, Car- rier dynamics of rubrene single-crystals revealed by transient broadband terahertz spectroscopy, Appl. Phys. Lett. 105, 143302 (2014)

2014
[71]

Y. Han, T. Miyamoto, T. Otaki, N. Takamura, N. Kida, N. Osakabe, J. Tsurumi, S. Watanabe, T. Okamoto, J. Takeya, and H. Okamoto, Scatter- ing mechanism of hole carriers in organic molecu- lar semiconductors deduced from analyses of tera- hertz absorption spectra using Drude–An...

2022
[72]

Giannini, L

S. Giannini, L. Di Virgilio, M. Bardini, J. Hausch, J. J. Geuchies, W. Zheng, M. Volpi, J. Elsner, K. Broch, Y. H. Geerts, F. Schreiber, G. Schwe- icher, H. I. Wang, J. Blumberger, M. Bonn, and D. Beljonne, Transiently delocalized states enhance hole mobility in organic molecu...

2023
[73]

Lacroix, G

A. Lacroix, G. T. de Laissardi` ere, P. Qu´ emerais, J.-P. Julien, and D. Mayou, Modeling of electronic mobilities in halide perovskites: Adiabatic quantum localization scenario, Phys. Rev. Lett. 124, 196601 (2020)

2020
[74]

P. V. Parandekar and J. C. Tully, 14 Mixed quantum-classical equilibrium, J. Chem. Phys. 122, 094102 (2005)

2005
[75]

J. E. Runeson, T. J. G. Drayton, and D. E. Manolopoulos, Charge transport in organic semiconduc- tors from the mapping approach to surface hopping, J. Chem. Phys. 161, 144102 (2024)

2024
[76]

J. R. Mannouch and J. O. Richardson, A mapping approach to surface hopping, J. Chem. Phys. 158, 104111 (2023)

2023
[77]

J. E. Runeson and D. E. Manolopoulos, A multi- state mapping approach to surface hopping, J. Chem. Phys. 159, 094115 (2023)

2023
[78]

J. E. Runeson, T. P. Fay, and D. E. Manolopoulos, Ex- citon dynamics from the mapping approach to surface hopping: Comparison with F¨ orster and Redﬁeld theo- ries, Phys. Chem. Chem. Phys. 26, 4929 (2024)

2024
[79]

J. H. Fetherolf, D. Goleˇ z, and T. C. Berkelbach, A uniﬁcation of the holstein polaron and dynamic dis- order pictures of charge transport in organic crystals, Phys. Rev. X 10, 021062 (2020)

2020
[80]

J. H. Fetherolf, P. Shih, and T. C. Berkelbach, Conduc- tivity of an electron coupled to anharmonic phonons: Quantum-classical simulations and comparison of ap- proximations, Phys. Rev. B 107, 064304 (2023)

2023
[81]

Schubert, G

G. Schubert, G. Wellein, A. Weisse, A. Alvermann, and H. Fehske, Optical absorption and activated transport in polaronic systems, Phys. Rev. B 72, 104304 (2005)

2005
[82]

Jansen, J

D. Jansen, J. Bonˇ ca, and F. Heidrich-Meisner, Finite-temperature optical conductivity with density- matrix renormalization group methods for the hol- stein polaron and bipolaron with dispersive phonons, Phys. Rev. B 106, 155129 (2022)

2022
[83]

Mitri´ c, Dynamical quantum typicality: A simple method for investigating transport prop- erties applied to the Holstein model (2024), arXiv:2412.17436 [cond-mat.str-el]

P. Mitri´ c, Dynamical quantum typicality: A simple method for investigating transport prop- erties applied to the Holstein model (2024), arXiv:2412.17436 [cond-mat.str-el]

2024 arXiv
[84]

Mitri´ c, V

P. Mitri´ c, V. Dobrosavljevi´ c, and D. Tanaskovi´ c, Precursors to Anderson localization in the Holstein model: Quantum and quantum-classical solutions, Phys. Rev. B 111, L161105 (2025)

2025
[85]

Rammal, A

H. Rammal, A. Ralko, S. Ciuchi, and S. Fratini, Tran- sient localization from the interaction with quantum bosons, Phys. Rev. Lett. 132, 266502 (2024)

2024
[87]

Jankovi´ c and N

V. Jankovi´ c and N. Vukmirovi´ c, Spectral and thermodynamic properties of the Holstein po- laron: Hierarchical equations of motion approach, Phys. Rev. B 105, 054311 (2022)

2022
[88]

Song and Q

L. Song and Q. Shi, Calculation of correlated ini- tial state in the hierarchical equations of motion method using an imaginary time path integral approach, J. Chem. Phys. 143, 194106 (2015)

2015
[89]

T. Xing, T. Li, Y. Yan, S. Bai, and Q. Shi, Applica- tion of the imaginary time hierarchical equations of mo- tion method to calculate real time correlation functions, J. Chem. Phys. 156, 244102 (2022)

2022
[90]

Z. H. Li, N. H. Tong, X. Zheng, D. Hou, J. H. Wei, J. Hu, and Y. J. Yan, Hierarchical Liouville-space approach for accurate and universal characterization of quantum im- purity systems, Phys. Rev. Lett. 109, 266403 (2012)

2012
[91]

Bhattacharyya, T

S. Bhattacharyya, T. Sayer, and A. Montoya-Castillo, Anomalous transport of small polarons arises from transient lattice relaxation or immovable boundaries, J. Phys. Chem. Lett. 15, 1382 (2024)

2024
[92]

Yan, Theory of open quantum systems with bath of electrons and phonons and spins: Many-dissipaton density matrixes approach, J

Y. Yan, Theory of open quantum systems with bath of electrons and phonons and spins: Many-dissipaton density matrixes approach, J. Chem. Phys. 140, 054105 (2014)

2014
[93]

Y. Yan, J. Jin, R.-X. Xu, and X. Zheng, Dissipation equation of motion approach to open quantum systems, Front. Phys. 11, 110306 (2016)

2016
[94]

Zhang, R.-X

H.-D. Zhang, R.-X. Xu, X. Zheng, and Y. Yan, Statis- tical quasi-particle theory for open quantum systems, Mol. Phys. 116, 780 (2018)

2018
[95]

Wang and Y

Y. Wang and Y. Yan, Quantum mechan- ics of open systems: Dissipaton theories, J. Chem. Phys. 157, 170901 (2022)

2022
[96]

I. S. Dunn, R. Tempelaar, and D. R. Reichman, Re- moving instabilities in the hierarchical equations of mo- tion: Exact and approximate projection approaches, J. Chem. Phys. 150, 184109 (2019)

2019
[97]

Y. Yan, T. Xing, and Q. Shi, A new method to im- prove the numerical stability of the hierarchical equa- tions of motion for discrete harmonic oscillator modes, J. Chem. Phys. 153, 204109 (2020)

2020
[98]

T. Li, Y. Yan, and Q. Shi, Is there a ﬁnite mobility for the one vibrational mode Holstein model? Implications from real time simulations, J. Chem. Phys. 160, 111102 (2024)

2024
[99]

See the Supplemental Material for (i) the details of HEOM computations, (ii) the details of the TLS compu- tations, (iii) the details of computations using the Boltz- mann transport equation, (iv) the detailed comparison between HEOM results and TLS predictions for ω0/J = 0.5,...
[101]

Sumi, Theory of electrical conduction in organic molecular crystals: Temperature-independent mobili- ties, J

H. Sumi, Theory of electrical conduction in organic molecular crystals: Temperature-independent mobili- ties, J. Chem. Phys. 70, 3775 (1979)

1979
[102]

27 use J = 93 meV, and to enable a direct comparison between our HEOM and their QMC mobilities in physical units, we rescale their results by 143/93, as was done in Ref

The authors of Ref. 27 use J = 93 meV, and to enable a direct comparison between our HEOM and their QMC mobilities in physical units, we rescale their results by 143/93, as was done in Ref. 58
[103]

Podzorov, E

V. Podzorov, E. Menard, A. Borissov, V. Kiryukhin, J. A. Rogers, and M. E. Gershenson, Intrinsic charge transport on the surface of organic semiconductors, Phys. Rev. Lett. 93, 086602 (2004)

2004
[104]

Podzorov, E

V. Podzorov, E. Menard, J. A. Rogers, and M. E. Gershenson, Hall eﬀect in the accumula- tion layers on the surface of organic semiconductors, Phys. Rev. Lett. 95, 226601 (2005)

2005
[105]

Takeya, J

J. Takeya, J. Kato, K. Hara, M. Yamagishi, R. Hira- hara, K. Yamada, Y. Nakazawa, S. Ikehata, K. Tsuk- agoshi, Y. Aoyagi, T. Takenobu, and Y. Iwasa, In- crystal and surface charge transport of electric-ﬁeld- induced carriers in organic single-crystal semiconduc- tors, Phys. Re...

2007
[106]

Buividovich, J

P. Buividovich, J. Ostmeyer, and A. Troisi, High- precision Quantum Monte-Carlo study of charge trans- port in a lattice model of molecular organic semicon- ductors, PoS LATTICE2024, 067 (2024) . 15

2024
[107]

Vuˇ ciˇ cevi´ c, J

J. Vuˇ ciˇ cevi´ c, J. Kokalj, R. ˇZitko, N. Wentzell, D. Tanaskovi´ c, and J. Mravlje, Conductivity in the Square Lattice Hubbard Model at High Tem- peratures: Importance of Vertex Corrections, Phys. Rev. Lett. 123, 036601 (2019)

2019
[108]

Mukamel, Principles of Nonlinear Optical Spec- troscopy (Oxford University Press, Inc., New York, 1995)

S. Mukamel, Principles of Nonlinear Optical Spec- troscopy (Oxford University Press, Inc., New York, 1995)

1995
[109]

May and O

V. May and O. K¨ uhn,Charge and Energy Transfer Dy- namics in Molecular Systems , 3rd ed. (WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim, 2011)

2011
[110]

Valkunas, D

L. Valkunas, D. Abramavicius, and T. Manˇ cal, Molecu- lar Excitation Dynamics and Relaxation (WILEY-VCH Verlag GmbH & Co. KGaA, 2013)

2013
[111]

Troisi, D

A. Troisi, D. L. Cheung, and D. Andrienko, Charge transport in semiconductors with multiscale conforma- tional dynamics, Phys. Rev. Lett. 102, 116602 (2009)

2009
[112]

H. Liu, L. Zhu, S. Bai, and Q. Shi, Reduced quantum dynamics with arbitrary bath spectral densities: Hierarchical equations of motion based on several diﬀerent bath decomposition schemes, J. Chem. Phys. 140, 134106 (2014)

2014
[113]

Tanimura, Reduced hierarchical equations of motion in real and imaginary time: Corre- lated initial states and thermodynamic quantities, J

Y. Tanimura, Reduced hierarchical equations of motion in real and imaginary time: Corre- lated initial states and thermodynamic quantities, J. Chem. Phys. 141, 044114 (2014)

2014
[114]

Takahashi and R

H. Takahashi and R. Borrelli, Tensor-Train Format Hierarchical Equations of Motion For- malism: Charge Transfer in Organic Semi- conductors via Dissipative Holstein Models, J. Chem. Theory Comput. 20, 7052 (2024)

2024
[115]

Bhattacharyya, T

S. Bhattacharyya, T. Sayer, and A. Montoya-Castillo, Mori generalized master equations oﬀer an eﬃcient route to predict and interpret polaron transport, Chem. Sci. 15, 16715 (2024) . Supplemental Material for Charge transport limited by nonlocal electron–phonon inte raction. I...

2024
[116]

Jankovi´ c, Numerical investigation of transport pro perties of the one-dimensional Peierls model based on the hi erarchical equations of motion, 10.5281/zenodo.14637019 (2025)

V. Jankovi´ c, Numerical investigation of transport pro perties of the one-dimensional Peierls model based on the hi erarchical equations of motion, 10.5281/zenodo.14637019 (2025)

2025 doi
[117]

Nematiaram, S

T. Nematiaram, S. Ciuchi, X. Xie, S. Fratini, and A. Trois i, Practical computation of the charge mobility in molecula r semiconductors using transient localization theory, J. Ph ys. Chem. C 123, 6989 (2019)

2019
[118]

Jankovi´ c, Charge transport limited by nonlocal electron–phonon interaction

V. Jankovi´ c, Charge transport limited by nonlocal electron–phonon interaction. I. Hierarchical equations of mot ion approach (2025), arXiv:2501.05054 [cond-mat.str-el]

2025 arXiv
[119]

Fratini and S

S. Fratini and S. Ciuchi, Dynamical localization correc tions to band transport, Phys. Rev. Res. 2, 013001 (2020)

2020
[120]

J. E. Runeson, T. J. G. Drayton, and D. E. Manolopoulos, Ch arge transport in organic semiconductors from the mapping approach to surface hopping, J. Chem. Phys. 161, 144102 (2024)

2024

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

and abundantly present ( T /greaterorsimilarω0) phonons. Then, charge dynamics on timescales short compared to ω−1 0 , when phonons can be considered as frozen, is essentially the same as the dynamics in the presence of Gaussian static disorder in the hopping amplitude, whose strength is σ2 = 2 2g2 βω0 = 2λJ T. Formally, one replaces the phonon operator g...

[2] [2]

( 5) by the average over diﬀerent disorder realizations

and ( 6) by a Gaussian random variable Xn,n+1 of zero mean and variance σ2, and the trace in Eq. ( 5) by the average over diﬀerent disorder realizations. The current–current cor- relation function computed upon introducing these as- sumptions in Eq. (

[3] [3]

M. J. Schilcher, P. J. Robinson, D. J. Abramovitch, L. Z. Tan, A. M. Rappe, D. R. Reichman, and D. A. Egger, The signiﬁcance of polarons and dynamic disorder in halide perovskites, ACS Energy Lett. 6, 2162 (2021)

2021

[4] [4]

is denoted as Cdis jj (t). Charge diﬀu- sion, which is inhibited in the static-disorder setup [54], 4 is ultimately established through the coupled charge– phonon dynamics, whose eﬀects become appreciable on timescale τd. The TLS eﬀectively restores phonon dy- namics by virtue of the RTA, in which CTLS jj (t) = Cdis jj (t)e−|t|/τd. (12) Physically, τ −1 d...

[5] [5]

denote averaging over the equilibrium state ρeq = e−βH /Tre−βH of the interacting 3 carrier–phonon system at temperature T = β−1. The current operator j = je + je−ph = −i ∑ n [ −J + g ( bn + b† n − bn+1 − b† n+1 )] × (|n⟩⟨n + 1| − |n + 1⟩⟨n|) (6) has the so-called phonon-assisted contribution je−ph [the term proportional to g in Eq. ( 6)] in addition to t...

[6] [6]

( 1)–(3)]

eﬀectively takes into account inter- actions of the carrier with additional phonon modes not considered in the model [Eqs. ( 1)–(3)]. Such a possibility is supported by the most recent quantum–classical sim- ulations [67]. There, the authors conclude that consid- ering a continuous phonon spectrum centered around ω0 instead of the delta-like spectrum used...

[7] [7]

At realistic tem- peratures and interactions, we argue that Eq

is suf- ﬁciently good to describe this predominantly phonon- 9 assisted transport, even when the timescales of carrier and phonon dynamics are comparable. At realistic tem- peratures and interactions, we argue that Eq. (

[8] [8]

can be considered to eﬀectively take into account other scatter- ing mechanisms not included in the present model. Apart from providing numerically exact results for carrier dy- namics, this piece of research can be regarded as a formal justiﬁcation of the already well-established practical ap- plicability of the TLS to realistic systems, for which the mo...