Pith. sign in

REVIEW 3 major objections 4 minor 73 references

Nonadiabatic ImF instanton rate theory

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Subtracting the zero-hop contribution from a mean-field ring-polymer potential gives a semiclassical nonadiabatic rate theory that reproduces both the Born–Oppenheimer instanton limit and the golden-rule Δ² scaling, with accurate…

desk verdict Genuinely new fix for MFRPI's golden-rule breakdown, solid deep-tunnelling benchmarks, but the prefactor transfer is unproven and the high-temperature extension fails as acknowledged. read the letter →

arxiv 2506.20852 v1 pith:XJWL2FPT submitted 2025-06-25 quant-ph physics.chem-ph

Rhiannon A. Zarotiadis , Jeremy O. Richardson This is my paper
classification quant-phphysics.chem-ph
keywords nonadiabaticratetheoryinstantonring-polymerdeeptunnellinggolden-rulelimitdiabaticcouplingspin-bosonmodelmean-fieldringpolymer
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

Reactions where nuclear tunnelling is accompanied by transitions between two electronic states are usually treated either in the strong-coupling Born–Oppenheimer limit or in the weak-coupling golden-rule limit. This paper builds a semiclassical (instanton) rate theory that claims to cover the full range between those limits. The new n-ImF theory keeps the mean-field ring-polymer potential of earlier nonadiabatic instanton theories but subtracts the 'zero-hop' contribution, the part of the path integral in which the electronic state never changes; because the exact rate is an even power series in the diabatic coupling, the leading term must come from paths that hop at least twice. On one-dimensional and multidimensional test models, the resulting deep-tunnelling rates agree well with numerically exact results across coupling strengths, while an attempted high-temperature extension fails outside the strong-coupling limit and is left for further study.

What carries the argument

The central object is the zero-hop-subtracted mean-field ring-polymer potential $$$U^{{\mathrm{n\text{-}}$ImF}}(x) = -\frac{1}{\beta_N}\ln\!\left(\mathrm{Tr}\!\left[\prod_i M_i\right] - \mathrm{Tr}\!\left[\prod_i $M_i^{{(0)}}$\right]\right),$$ where $M_i = e^{-\beta_N V(x_i)}$ is the matrix exponential of the $2\times2$ diabatic potential matrix and $M_i^{(0)}$ is its diagonal (zero-hop) part. This object carries the argument by removing all paths in which the ring polymer never changes diabatic state, so that the leading contribution to the rate is forced to be second order in the coupling $\Delta$; a second hop enters implicitly through the trace, and the permutational symmetry of the ring polymer is preserved. When the coupling is strong, the zero-hop term is negligible and the potential reduces to the Born–Oppenheimer ring-polymer potential, which is why the correct BO instanton limit is inherited. The paper also uses the equivalent bead-summed form of Eq. (19), which forces one hop explicitly at each bead and avoids numerically subtracting two large traces.

What would settle it

For the asymmetric linear-crossing model with slopes $\kappa_0=1$ and $\kappa_1=-10$ at inverse temperature $\beta=11$, compute the n-ImF rate at couplings below $\Delta=0.1$ and compare with the exact quantum rate: the theory is falsified if the coefficient of $\Delta^2$ in the n-ImF rate deviates from the exact golden-rule coefficient by more than the claimed accuracy, or if the ratio of the two rates diverges as $\Delta \to 0$.

Watch

Extended reading notes

Core claim

Within the ImF (imaginary free energy) approximation, the paper defines a new ring-polymer potential $$$U^{{\mathrm{n\text{-}}$ImF}}(x) = -\frac{1}{\beta_N}\ln\!\left(\mathrm{Tr}\!\left[\prod_i M_i\right] - \mathrm{Tr}\!\left[\prod_i $M_i^{{(0)}}$\right]\right),$$ where $M_i = e^{-\beta_N V(x_i)}$ is the matrix exponential of the $2\times2$ diabatic potential matrix and $M_i^{(0)}$ is its diagonal (zero-hop) part. Applying the standard ImF steepest-descent formula to this potential yields an instanton on which at least one electronic hop is forced, so the rate has the required $\Delta^2$ leading behaviour in the golden-rule limit; in the strong-coupling limit the zero-hop term is negligible and the theory reduces to Born–Oppenheimer instanton theory. On symmetric and asymmetric one-dimensional linear-crossing models and on spin-boson models in the low-friction regime, the deep-tunnelling rates match numerically exact results over couplings from $\Delta=0.1$ to $\Delta=10$. The paper also proposes a high-temperature extension based on a collapsed ring polymer, but reports that this extension is only reliable on the strong-coupling side and requires further improvement.

Load-bearing premise

The load-bearing premise is that the standard imaginary-free-energy steepest-descent rate formula remains quantitatively accurate when evaluated on the modified mean-field potential of Eq. (17), a potential justified by the even-power structure of the exact rate rather than derived from a physical partition function.

Editorial extensions

If this is right

  • The deep-tunnelling n-ImF theory predicts accurate rate constants across the entire diabatic-coupling range, from the golden-rule limit to the Born–Oppenheimer limit, for one- and multidimensional systems below the crossover temperature.
  • Users of the method get an optimal tunnelling pathway, so they can identify the reaction mechanism rather than only a scalar rate, at the cost of one instanton search.
  • The high-temperature collapsed-ring-polymer extension, as constructed here, should be used only near the strong-coupling/Born–Oppenheimer side; in the weak-coupling or high-friction regimes it is unreliable.
  • At low temperature the n-ImF instanton continues to exist for all diabatic couplings, removing the premature collapse that afflicts the mean-field ring-polymer instanton at weak coupling.
  • The method is cheaper than the rigorous multiple-instanton nonadiabatic theory while providing mechanistic insight, making it suitable for preliminary studies of large-scale nonadiabatic reactions.

Reading between the lines

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

  • The zero-hop-subtraction principle is not tied to the ImF scheme; the same removal of no-hop paths could be enforced in other mean-field path-integral rate methods to impose the even-in-$\Delta$ expansion while keeping ring-polymer symmetry.
  • Because n-ImF averages over all two-hop configurations rather than selecting the optimal pair of hops, the high-temperature collapse point can drift off the diabatic crossing point; a variant that treats the two hops by steepest descent while preserving symmetry might cure the high-temperature failure.
  • The $\Delta$-dependence of the n-ImF crossover temperature could serve as a diagnostic: where it deviates from the golden-rule instanton's persistence to high temperature, one should compare against a rigorous golden-rule rate before trusting the mean-field result.
  • For molecular applications, comparing n-ImF deep-tunnelling rates against the rigorous nonadiabatic ring-polymer instanton theory on a proton-coupled electron transfer model would quantify the accuracy cost of the ad hoc zero-hop subtraction in realistic dimensions.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper examines the mean-field ring-polymer instanton (MFRPI) theory based on the ImF premise, shows that it fails to capture the golden-rule (Δ²) scaling of the rate in the weak-coupling limit, and proposes a new nonadiabatic ImF (n-ImF) theory. The n-ImF rate is defined through a modified ring-polymer potential (Eq. 17) obtained by subtracting the zero-hop contribution from the mean-field trace, with the standard ImF steepest-descent prefactor of Eq. (8) applied to this potential. The authors test the deep-tunnelling n-ImF theory on symmetric and asymmetric one-dimensional linear-crossing models and on multidimensional spin-boson models, reporting good agreement with numerically exact and HEOM benchmarks across a wide range of diabatic couplings. They also propose a high-temperature extension with an ad hoc parameter η = -2, which they show fails for asymmetric and high-friction systems in the golden-rule limit; the paper explicitly acknowledges this limitation and separates the high-temperature extension from the deep-tunnelling theory.

Significance. If the central claim holds, the n-ImF theory is a valuable contribution: it is a single-instanton, mean-field method that correctly bridges the Born-Oppenheimer and golden-rule limits in the deep-tunnelling regime while providing a mechanistic optimal tunnelling pathway at a cost comparable to standard instanton theory. The numerical validation against exact one-dimensional rates and HEOM spin-boson results is a strength, and the authors are commendably transparent about the high-temperature extension's limitations. However, the theory is explicitly ad hoc: the subtraction of the zero-hop term is justified by the even-power expansion of the exact rate, and the steepest-descent prefactor of Eq. (8) is transferred to a modified potential that is not derived from a physical partition function. The central numerical agreement is encouraging, but the lack of a theoretical justification for the saddle-point structure and prefactor leaves the method's reliability in untested regimes uncertain. A direct comparison with the authors' rigorous NRPI theory would substantially strengthen the case.

major comments (3)
  1. [Sec. IV, Eq. (17) and Eq. (8)] The central rate expression applies the standard ImF steepest-descent prefactor of Eq. (8) to the modified potential U_n-ImF defined in Eq. (17), but this potential is not derived from a physical partition function and the required saddle-point structure is not established. The argument of the logarithm in Eq. (17) is a difference of two positive traces; the authors do not analyze its sign or the analytic continuation needed if it becomes non-positive, and they do not show that the Hessian at the n-ImF instanton has exactly one negative eigenvalue and one zero eigenvalue with all other modes positive. The zero-mode integration leading to the √B_N factor in Eq. (8) is likewise transferred without re-derivation for the modified potential. Because the quantitative deep-tunnelling claim depends on the prefactor as well as the exponential action, this unproven transfer is load-bearing and should be either justified analytically or supported by explicit numerical diagnostics of the saddle-point spectrum and the positivity of the traced difference.
  2. [Sec. IV A and Appendix A] The high-temperature extension introduces an ad hoc parameter η and sets η = -2 by requiring that the collapsed-ring-polymer expression match the classical golden-rule rate for a linear-crossing model. The derivation assumes the collapsed ring polymer sits at the diabatic crossing point, an assumption the authors themselves show to fail for asymmetric systems in Sec. V D. Consequently, the high-temperature n-ImF rates deviate from exact HEOM results by one to two orders of magnitude in the golden-rule limit for the spin-boson models in Tables II and III. The authors are transparent about this failure, but the abstract and introduction present the high-temperature analysis as part of the proposed theory. The paper would be strengthened by either developing a more robust high-temperature continuation or explicitly reframing the high-temperature formula as a separate, preliminary attempt outside the paper's core claims.
  3. [Sec. V] The paper does not benchmark n-ImF against the authors' own rigorous nonadiabatic ring-polymer instanton (NRPI) theory of Ref. 23, despite the fact that NRPI is claimed to bridge the same BO and GR limits from first principles. A direct comparison on the one-dimensional and spin-boson models would test the n-ImF approximation against a closely related semiclassical method and would clarify in which parameter regimes the ad hoc zero-hop subtraction and the transferred prefactor introduce errors. This is a natural and expected validation for a new approximation, and its absence leaves the central claim supported only by comparisons to exact/HEOM benchmarks, which are not always available for more complex systems.
minor comments (4)
  1. [Sec. V A and Fig. 3] The text states that the high-temperature limit is shown in Fig. 3(b), but the caption identifies Fig. 3(b) as the crossover-temperature region (β = 7) and Fig. 3(c) as the high-temperature regime (β = 3). Please correct the in-text reference or the figure caption.
  2. [Sec. IV, Eq. (19)] Equation (19) appears to have a typographical issue: the displayed expression includes an extraneous exclamation mark after the sum, and the trace symbol is missing inside the sum. Please check the typesetting.
  3. [Sec. III A, Eq. (16)] In the derivation of the BO limit, Eq. (16) reads U_MF(x) ≈ ln( exp( Σ V_BO(x_i) ) ), which is dimensionally inconsistent with the definition in Eq. (14); the argument of the exponential should include a factor -β_N, and the overall prefactor -1/β_N is missing. Please correct this expression.
  4. [Abstract] The sentence 'a number of methods have been proposed; almost all based on the less rigorous ImF premise' uses a semicolon where a comma (or 'almost all of them based') would be grammatically appropriate.

Circularity Check

0 steps flagged · score 1.0 of 10

No load-bearing circularity: the n-ImF construction is an openly stated ad hoc design, validated against external exact benchmarks, with only minor non-load-bearing self-citation.

full rationale

The central n-ImF rate is not circular. Equation (17) defines a new ring-polymer potential by subtracting the zero-hop term from the mean-field potential, and the rate is then computed with the standard ImF steepest-descent expression, Eq. (8), and compared to numerically exact external benchmarks (Greens-function grids and HEOM results taken from Lawrence et al., Ref. 52). No parameter is fitted to those benchmark rates. The Δ² scaling in the golden-rule limit is intentionally built in by the zero-hop subtraction, but the paper states this explicitly as an ad hoc design choice ('This argument is the justification for the ad hoc correction to a mean-field instanton approach presented here'), so it is not a hidden prediction made to look like an independent result. The high-temperature parameter η=-2 is derived analytically in Appendix A to match the known classical golden-rule rate for a linear-crossing model; this is a calibration to a known classical limit rather than a fit to the tested quantum rates, and the resulting analytic prefactor 6√6/π^(5/2) ≈ 1.19 is quoted honestly as a deviation from the classical rate. Self-citations (Refs. 23 and 47) are used for context and for the exact even-power expansion k = Δ²k2 + Δ⁴k4 + …, but that expansion is a published, externally accessible result and is an input motivation, not the target claim. The main unresolved issue — transferring the steepest-descent prefactor of Eq. (8) to the ad hoc subtracted potential of Eq. (17) without re-deriving its analytic-continuation structure — is a correctness/validity concern rather than circularity, because the potential is not defined in terms of the rates being predicted and the numerical tests provide independent evidence. Overall, no load-bearing circular step is identified; the score of 1 reflects only minor non-load-bearing self-citation.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The theory introduces no new physical entities and no fitted parameters in its deep-tunnelling form. Its epistemic burden is concentrated in three ad hoc assumptions: the exclusion of the zero-hop term, the transfer of the ImF steepest-descent formula to the modified potential, and the reduction to the BO limit. A single hand-chosen exponent η enters only the unsuccessful high-temperature extension.

free parameters (1)
  • eta (η) in high-temperature n-ImF rate formula = -2
    Hand-chosen exponent in Eq. (22) so that the classical limit of the n-ImF rate matches the classical golden-rule rate for a linear-crossing model; this parameter is not used in the deep-tunnelling formulation and belongs to the extension that the paper itself shows fails in the GR limit.
assumptions (4)
  • domain assumption The ImF premise, k ≈ −(2/ℏ) Im F (Eq. 7), relates the rate to the imaginary part of a free energy.
    Standard in instanton theory but not rigorously derived; the paper notes it is less rigorous than flux-correlation derivations and agrees with them in the BO limit.
  • ad hoc to paper The exact nonadiabatic rate has an even-power expansion in the diabatic coupling Δ, so the zero-hop contribution must be excluded from the mean-field potential.
    Used to justify subtracting Tr[∏ M_i^(0)] in Eq. (17); the subtraction is motivated by the expansion rather than derived from the path integral.
  • ad hoc to paper The steepest-descent instanton prefactor of Eq. (8) remains quantitatively accurate when applied to the modified n-ImF potential.
    The ImF apparatus is transferred to a potential that is not a physical partition function; validity is supported only by numerical tests, not by derivation.
  • ad hoc to paper At strong coupling, the zero-hop term is negligible, so U_n-ImF reduces to the BO ring-polymer potential.
    The paper argues this from the dominance of hop-containing terms and verifies it numerically for symmetric and asymmetric systems.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Nonadiabatic ImF instanton rate theory." pith.science (2026). https://pith.science/paper/XJWL2FPT

@misc{pith2026250620852,
  author       = {Pith},
  title        = {Pith review of: Nonadiabatic ImF instanton rate theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XJWL2FPT}},
  note         = {Machine review of arXiv:2506.20852}
}
read the original abstract

Semiclassical instanton theory captures nuclear quantum effects such as tunnelling in chemical reactions. It was originally derived from two different starting points, the flux correlation function and the ImF premise. In pursuit of a nonadiabatic rate theory, a number of methods have been proposed; almost all based on the less rigorous ImF premise. Only recently, we introduced a rigorous nonadiabatic ring-polymer instanton rate theory in the flux-correlation function framework which successfully bridges from the Born-Oppenheimer to the golden-rule limit. Here, we examine the previous ImF-based attempts and conclude that they do not capture the two limits correctly. In particular, we will highlight how the last in a series of developments, called mean-field ring-polymer instanton theory, breaks down in the golden-rule limit. We develop a new nonadiabatic ImF rate theory to remedy the failings of previous attempts while taking inspiration from them. We also consider the crossover from deep tunnelling to a high-temperature rate theory. We test our new nonadiabatic ImF theory on a range of models including asymmetric and multidimensional systems and we show reliable results for the deep-tunnelling regime but limitations for the related high-temperature rate theory.

Figures

Figures reproduced from arXiv: 2506.20852 by the authors.

Figure 1
Figure 1. FIG. 1: Asymmetric linear-crossing model with [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Cartoon of the contributions to the “n-ImF” [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Log-log plot of rate coefficients [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: FIG. 4: Rate constants as a function of the inverse [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: Asymmetric linear-crossing model with [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

73 extracted references · 59 canonical work pages

  1. [1]

    Hammes-Schiffer ,\ title title Proton-coupled electron transfer: Moving together and charging forward , \ 10.1021/jacs.5b04087 journal journal J

    author author S. Hammes-Schiffer ,\ title title Proton-coupled electron transfer: Moving together and charging forward , \ 10.1021/jacs.5b04087 journal journal J. Am. Chem. Soc. \ volume 137 ,\ pages 8860--8871 ( year 2015 ) NoStop

    work page doi:10.1021/jacs.5b04087 2015

  2. [2]

    author author W. H. \ Miller ,\ title title Semiclassical quantization of nonseparable systems: A new look at periodic orbit theory , \ @noop journal journal J. Chem. Phys. \ volume 63 ,\ pages 996--999 ( year 1975 a ) NoStop

    work page 1975

  3. [3]

    author author R. P. \ Feynman \ and\ author A. R. \ Hibbs ,\ @noop title Quantum Mechanics and Path Integrals \ ( publisher McGraw-Hill ,\ address New York ,\ year 1965 ) NoStop

    work page 1965

  4. [4]

    author author J. O. \ Richardson ,\ title title Perspective: Ring-polymer instanton theory , \ 10.1063/1.5028352 journal journal J. Chem. Phys. \ volume 148 ,\ pages 200901 ( year 2018 a ) NoStop

    doi:10.1063/1.5028352 2018

  5. [5]

    author author J. B. \ Rommel , author Y. Liu , author H.-J. \ Werner , \ and\ author J. K\"astner ,\ title title Role of tunneling in the enzyme glutamate mutase , \ 10.1021/jp308526t journal journal J. Phys. Chem. B \ volume 116 ,\ pages 13682--13689 ( year 2012 ) NoStop

    work page doi:10.1021/jp308526t

  6. [6]

    \'A sgeirsson , author A

    author author V. \'A sgeirsson , author A. Arnaldsson , \ and\ author H. J \'o nsson ,\ title title Efficient evaluation of atom tunneling combined with electronic structure calculations , \ 10.1063/1.5007180 journal journal J. Chem. Phys. \ volume 148 ,\ pages 102334 ( year 2018 ) NoStop

    work page doi:10.1063/1.5007180

  7. [7]

    Simultaneous Deep Tunneling and Classical Hopping for Hydrogen Diffusion on Metals

    author author W. Fang , author J. O. \ Richardson , author J. Chen , author X.-Z. \ Li , \ and\ author A. Michaelides ,\ title title Simultaneous deep tunneling and classical hopping for hydrogen diffusion on metals , \ 10.1103/PhysRevLett.119.126001 journal journal Phys. Rev. Lett. \ volume 119 ,\ pages 126001 ( year 2017 ) ,\ http://arxiv.org/abs/1709.0...

    work page Pith review arXiv

  8. [8]

    author author E. R. \ Heller \ and\ author J. O. \ Richardson ,\ title title Spin crossover of thiophosgene via multidimensional heavy-atom quantum tunneling , \ 10.1021/jacs.1c10088 journal journal J. Am. Chem. Soc. \ volume 143 ,\ pages 20952--20961 ( year 2021 ) ,\ note pMID: 34846871 NoStop

    work page doi:10.1021/jacs.1c10088

Show all 73 references
  1. [9]

    author author E. R. \ Heller \ and\ author J. O. \ Richardson ,\ title title Heavy-atom quantum tunnelling in spin crossovers of nitrenes , \ @noop journal journal Angew. Chem. Int. Ed. \ volume 61 ( year 2022 ) NoStop

    2022

  2. [10]

    Ansari , author E

    author author I. Ansari , author E. R. \ Heller , author G. Trenins , \ and\ author J. O. \ Richardson ,\ title title Heavy-atom tunnelling in singlet oxygen deactivation using instanton theory with branch-point singularities , \ 10.1038/s41467-024-48463-2 journal journal Nat....

    doi

  3. [11]

    author author M. A. \ Manae \ and\ author J. O. \ Richardson ,\ title title Temperature-Dependent Mechanistic Control of Nonadiabatic Tunnelling in Triplet Carbenes , \ 10.1002/anie.202503066 journal journal Angew. Chem. Int. Ed. \ ( year 2025 ),\ 10.1002/anie.202503066 NoStop

    doi

  4. [12]

    Litman , author J

    author author Y. Litman , author J. O. \ Richardson , author T. Kumagai , \ and\ author M. Rossi ,\ title title Elucidating the quantum dynamics of intramolecular double hydrogen transfer in porphycene , \ 10.1021/jacs.8b12471 journal journal J. Am. Chem. Soc. \ volume 141 ,\ ...

    arXiv

  5. [13]

    Han , author W

    author author E. Han , author W. Fang , author M. Stamatakis , author J. O. \ Richardson , \ and\ author J. Chen ,\ title title Quantum Tunnelling Driven H _2 Formation on Graphene , \ 10.1021/acs.jpclett.2c00520 journal journal J. Phys. Chem. Lett. \ volume 13 ,\ pages 3173--...

    doi

  6. [14]

    author author E. E. \ Claveau , author E. R. \ Heller , author J. O. \ Richardson , \ and\ author E. Miliordos ,\ title title Methane against methanol: The tortoise and the hare of the oxidation race , \ 10.1021/acs.jpclett.3c02274 journal journal J. Phys. Chem. Lett. \ volume...

    doi

  7. [15]

    \ Li , author W

    author author S.-J. \ Li , author W. Fang , author J. O. \ Richardson , \ and\ author D.-C. \ Fang ,\ title title Tunnelling assisted hydrogen elimination mechanisms of FeCl _3 /TEMPO , \ 10.1039/d1cc06035j journal journal Chem. Commun. \ volume 58 ,\ pages 565--568 ( year 202...

    doi

  8. [16]

    Fang , author E

    author author W. Fang , author E. R. \ Heller , \ and\ author J. O. \ Richardson ,\ title title Competing quantum effects in heavy-atom tunnelling through conical intersections , \ 10.1039/D3SC03706A journal journal Chem. Sci. \ volume 14 ,\ pages 10777--10785 ( year 2023 ) NoStop

    doi

  9. [17]

    author author J. O. \ Richardson ,\ title title Derivation of instanton rate theory from first principles , \ 10.1063/1.4943866 journal journal J. Chem. Phys. \ volume 144 ,\ pages 114106 ( year 2016 ) ,\ http://arxiv.org/abs/1512.04292 arXiv:1512.04292 [physics.chem-ph] NoStop

    2016 arXiv

  10. [18]

    author author J. O. \ Richardson ,\ title title Ring-polymer instanton theory , \ 10.1080/0144235X.2018.1472353 journal journal Int. Rev. Phys. Chem. \ volume 37 ,\ pages 171--216 ( year 2018 b ) NoStop

    2018

  11. [19]

    author author J. O. \ Richardson , author R. Bauer , \ and\ author M. Thoss ,\ title title Semiclassical G reen's functions and an instanton formulation of electron-transfer rates in the nonadiabatic limit , \ 10.1063/1.4932361 journal journal J. Chem. Phys. \ volume 143 ,\ pa...

    arXiv

  12. [20]

    author author J. O. \ Richardson ,\ title title Ring-polymer instanton theory of electron transfer in the nonadiabatic limit , \ 10.1063/1.4932362 journal journal J. Chem. Phys. \ volume 143 ,\ pages 134116 ( year 2015 ) ,\ http://arxiv.org/abs/1508.05195 arXiv:1508.05195 [phy...

    2015 arXiv

  13. [21]

    author author E. R. \ Heller \ and\ author J. O. \ Richardson ,\ title title Instanton formulation of F ermi's golden rule in the M arcus inverted regime , \ 10.1063/1.5137823 journal journal J. Chem. Phys. \ volume 152 ,\ pages 034106 ( year 2020 ) NoStop

    doi

  14. [22]

    author author J. O. \ Richardson ,\ title title Nonadiabatic tunneling in chemical reactions , \ 10.1021/acs.jpclett.4c01098 journal journal J. Phys. Chem. Lett. \ volume 15 ,\ pages 7387--7397 ( year 2024 ) NoStop

    2024 doi

  15. [23]

    author author R. A. \ Zarotiadis , author J. E. \ Lawrence , \ and\ author J. O. \ Richardson ,\ https://arxiv.org/abs/2505.04770 title Nonadiabatic ring-polymer instanton rate theory: a generalised dividing-surface approach , \ ( year 2025 ),\ http://arxiv.org/abs/2505.04770 ...

    2025 arXiv

  16. [24]

    author author W. H. \ Miller ,\ title title Quantum mechanical transition state theory and a new semiclassical model for reaction rate constants , \ 10.1063/1.1682181 journal journal J. Chem. Phys. \ volume 61 ,\ pages 1823 ( year 1974 ) NoStop

    doi

  17. [25]

    author author W. H. \ Miller , author S. D. \ Schwartz , \ and\ author J. W. \ Tromp ,\ title title Quantum mechanical rate constants for bimolecular reactions , \ 10.1063/1.445581 journal journal J. Chem. Phys. \ volume 79 ,\ pages 4889--4898 ( year 1983 ) NoStop

    doi

  18. [26]

    author author J. E. \ Lawrence \ and\ author D. E. \ Manolopoulos ,\ title title A general non-adiabatic quantum instanton approximation , \ 10.1063/5.0009109 journal journal J. Chem. Phys. \ volume 152 ,\ pages 204117 ( year 2020 ) NoStop

    doi

  19. [27]

    Cao , author C

    author author J. Cao , author C. Minichino , \ and\ author G. A. \ Voth ,\ title title The computation of electron transfer rates: T he nonadiabatic instanton solution. \ 10.1063/1.469762 journal journal J. Chem. Phys. \ volume 103 ,\ pages 1391--1399 ( year 1995 ) NoStop

    doi

  20. [28]

    Cao \ and\ author G

    author author J. Cao \ and\ author G. A. \ Voth ,\ title title A unified framework for quantum activated rate processes. II . The nonadiabatic limit , \ 10.1063/1.474123 journal journal J. Chem. Phys. \ volume 106 ,\ pages 1769--1779 ( year 1997 ) NoStop

    doi

  21. [29]

    author author C. D. \ Schwieters \ and\ author G. A. \ Voth ,\ title title The semiclassical calculation of nonadiabatic tunneling rates , \ 10.1063/1.475467 journal journal J. Chem. Phys. \ volume 108 ,\ pages 1055 ( year 1998 ) NoStop

    doi

  22. [30]

    Ranya \ and\ author N

    author author S. Ranya \ and\ author N. Ananth ,\ title title Multistate ring polymer instantons and nonadiabatic reaction rates , \ 10.1063/1.5132807 journal journal J. Chem. Phys. \ volume 152 ,\ pages 114112 ( year 2020 ) NoStop

    2020 doi

  23. [31]

    author author J. S. \ Langer ,\ title title Theory of the condensation point , \ 10.1016/0003-4916(67)90200-X journal journal Ann. Phys.--New York \ volume 41 ,\ pages 108--157 ( year 1967 ) NoStop

    1967 doi

  24. [32]

    author author J. S. \ Langer ,\ title title Statistical theory of the decay of metastable states , \ 10.1016/0003-4916(69)90153-5 journal journal Ann. Phys.--New York \ volume 54 ,\ pages 258--275 ( year 1969 ) NoStop

    1969 doi

  25. [33]

    Coleman ,\ title title Fate of the false vacuum: S emiclassical theory , \ 10.1103/PhysRevD.15.2929 journal journal Phys

    author author S. Coleman ,\ title title Fate of the false vacuum: S emiclassical theory , \ 10.1103/PhysRevD.15.2929 journal journal Phys. Rev. D \ volume 15 ,\ pages 2929--2936 ( year 1977 a ) NoStop

    1977 doi

  26. [34]

    author author C. G. \ Callan , Jr \ and\ author S. Coleman ,\ title title Fate of the false vacuum. II .\ F irst quantum corrections , \ 10.1103/PhysRevD.16.1762 journal journal Phys. Rev. D \ volume 16 ,\ pages 1762--1768 ( year 1977 ) NoStop

    doi

  27. [35]

    Coleman ,\ title title The uses of instantons , \ in\ @noop booktitle Proc

    author author S. Coleman ,\ title title The uses of instantons , \ in\ @noop booktitle Proc. Int. School of Subnuclear Physics \ ( organization Erice ,\ year 1977 )\ note also in S. Coleman, Aspects of Symmetry, chapter 7, pp. 265--350 (Cambridge University Press, 1985) NoStop

    1977

  28. [36]

    Affleck ,\ title title Quantum-statistical metastability , \ 10.1103/PhysRevLett.46.388 journal journal Phys

    author author I. Affleck ,\ title title Quantum-statistical metastability , \ 10.1103/PhysRevLett.46.388 journal journal Phys. Rev. Lett. \ volume 46 ,\ pages 388--391 ( year 1981 ) NoStop

    1981 doi

  29. [37]

    author author V. A. \ Benderskii , author D. E. \ Makarov , \ and\ author C. A. \ Wight ,\ 10.1002/9780470141472 title Chemical Dynamics at Low Temperatures ,\ series Adv. Chem. Phys. , Vol. volume 88 \ ( publisher Wiley ,\ address New York ,\ year 1994 ) NoStop

    doi

  30. [38]

    Cao \ and\ author G

    author author J. Cao \ and\ author G. A. \ Voth ,\ title title A unified framework for quantum activated rate processes. I. General theory , \ 10.1063/1.471980 journal journal J. Chem. Phys. \ volume 105 ,\ pages 6856--6870 ( year 1996 ) NoStop

    doi

  31. [39]

    author author J. O. \ Richardson \ and\ author S. C. \ Althorpe ,\ title title Ring-polymer molecular dynamics rate-theory in the deep-tunneling regime: Connection with semiclassical instanton theory , \ 10.1063/1.3267318 journal journal J. Chem. Phys. \ volume 131 ,\ pages 21...

    doi

  32. [40]

    author author H. Kleinert ,\ @noop title Path Integrals in Quantum Mechanics, Statistics, Polymer Physics and Financial Markets ,\ edition 5th \ ed.\ ( publisher World Scientific ,\ address Singapore ,\ year 2009 ) NoStop

    2009

  33. [41]

    author author W. H. \ Miller ,\ title title Semiclassical limit of quantum mechanical transition state theory for nonseparable systems , \ 10.1063/1.430676 journal journal J. Chem. Phys. \ volume 62 ,\ pages 1899--1906 ( year 1975 b ) NoStop

    1906 doi

  34. [42]

    author author S. C. \ Althorpe ,\ title title On the equivalence of two commonly used forms of semiclassical instanton theory , \ 10.1063/1.3563045 journal journal J. Chem. Phys. \ volume 134 ,\ pages 114104 ( year 2011 ) NoStop

    2011 doi

  35. [43]

    Upadhyayula \ and\ author E

    author author S. Upadhyayula \ and\ author E. Pollak ,\ title title ^2 corrections to semiclassical transmission coefficients , \ 10.1021/acs.jpca.4c00452 journal journal J. Phys. Chem. A \ volume 128 ,\ pages 3434--3448 ( year 2024 ) NoStop

    doi

  36. [44]

    author author J. E. \ Lawrence ,\ title title Semiclassical instanton theory for reaction rates at any temperature: How a rigorous real-time derivation solves the crossover temperature problem , \ @noop journal journal J. Chem. Phys. \ volume 161 ( year 2024 ) NoStop

    2024

  37. [45]

    author author R. P. \ Bell ,\ @noop title The Tunnel Effect in Chemistry \ ( publisher Chapman and Hall ,\ address London ,\ year 1980 ) NoStop

    1980

  38. [46]

    author author M. H. \ Alexander ,\ title title Path-integral simulation of finite-temperature properties of systems involving multiple, coupled electronic states , \ 10.1016/S0009-2614(01)01012-0 journal journal Chem. Phys. Lett. \ volume 347 ,\ pages 436--442 ( year 2001 ) NoStop

    doi

  39. [47]

    Trenins \ and\ author J

    author author G. Trenins \ and\ author J. O. \ Richardson ,\ title title Nonadiabatic instanton rate theory beyond the golden-rule limit , \ 10.1063/5.0088518 journal journal J. Chem. Phys. \ volume 156 ,\ pages 174115 ( year 2022 a ) ,\ http://arxiv.org/abs/2202.08874 arXiv:2...

    arXiv

  40. [48]

    author author T. J. H. \ Hele ,\ title An Electronically Non-Adiabatic Generalization of Ring Polymer Molecular Dynamics ,\ @noop Master's thesis ,\ school Oxford University ( year 2011 ) NoStop

    2011

  41. [49]

    author author A. R. \ Menzeleev , author F. Bell , \ and\ author T. F. \ Miller III ,\ title title Kinetically constrained ring-polymer molecular dynamics for non-adiabatic chemical reactions , \ 10.1063/1.4863919 journal journal J. Chem. Phys. \ volume 140 ,\ pages 064103 ( y...

    doi

  42. [50]

    Holstein ,\ title title Studies of polaron motion: Part ii

    author author T. Holstein ,\ title title Studies of polaron motion: Part ii. the “small” polaron , \ https://doi.org/10.1016/0003-4916(59)90003-X journal journal Ann. Phys. - New York \ volume 8 ,\ pages 343--389 ( year 1959 ) NoStop

    1959 doi

  43. [51]

    author author A. Nitzan ,\ @noop title Chemical Dynamics in Condensed Phases: Relaxation, Transfer, and Reactions in Condensed Molecular Systems \ ( publisher Oxford University Press ,\ address Oxford ,\ year 2006 ) NoStop

    2006

  44. [52]

    author author J. E. \ Lawrence , author T. Fletcher , author L. P. \ Lindoy , \ and\ author D. E. \ Manolopoulos ,\ title title On the calculation of quantum mechanical electron transfer rates , \ 10.1063/1.5116800 journal journal J. Chem. Phys. \ volume 151 ,\ pages 114119 ( ...

    doi

  45. [53]

    author author I. R. \ Craig \ and\ author D. E. \ Manolopoulos ,\ title title Chemical reaction rates from ring polymer molecular dynamics. \ 10.1063/1.1850093 journal journal J. Chem. Phys. \ volume 122 ,\ eid 084106 ( year 2005 a ) NoStop

    doi

  46. [54]

    author author I. R. \ Craig \ and\ author D. E. \ Manolopoulos ,\ title title A refined ring polymer molecular dynamics theory of chemical reaction rates. \ 10.1063/1.1954769 journal journal J. Chem. Phys. \ volume 123 ,\ pages 034102 ( year 2005 b ) NoStop

    doi

  47. [55]

    Eyring ,\ title title The activated complex and the absolute rate of chemical reactions

    author author H. Eyring ,\ title title The activated complex and the absolute rate of chemical reactions. \ @noop journal journal Chem. Rev. \ volume 17 ,\ pages 65--77 ( year 1935 ) NoStop

    1935

  48. [56]

    Eyring ,\ title title The theory of absolute reaction rates , \ 10.1039/TF9383400041 journal journal Trans

    author author H. Eyring ,\ title title The theory of absolute reaction rates , \ 10.1039/TF9383400041 journal journal Trans. Faraday Soc. \ volume 34 ,\ pages 41--48 ( year 1938 ) NoStop

    1938 doi

  49. [57]

    author author P. G. \ Wolynes ,\ title title Imaginary time path integral Monte Carlo route to rate coefficients for nonadiabatic barrier crossing , \ 10.1063/1.453440 journal journal J. Chem. Phys. \ volume 87 ,\ pages 6559--6561 ( year 1987 ) NoStop

    doi

  50. [58]

    Trenins \ and\ author J

    author author G. Trenins \ and\ author J. O. \ Richardson ,\ title title Nonadiabatic instanton rate theory beyond the golden-rule limit , \ 10.1063/5.0088518 journal journal J. Chem. Phys. \ volume 156 ,\ pages 174115 ( year 2022 b ) NoStop

    doi

  51. [59]

    author author R. A. \ Marcus ,\ title title On the theory of oxidation-reduction reactions involving electron transfer. I , \ 10.1063/1.1742723 journal journal J. Chem. Phys. \ volume 24 ,\ pages 966--978 ( year 1956 ) NoStop

    1956 doi

  52. [60]

    author author M. E. \ Tuckerman ,\ @noop title Statistical Mechanics: Theory and Molecular Simulation \ ( publisher Oxford University Press ,\ address Oxford ,\ year 2010 ) NoStop

    2010

  53. [61]

    author author W. H. \ Miller , author Y. Zhao , author M. Ceotto , \ and\ author S. Yang ,\ title title Quantum instanton approximation for thermal rate constants of chemical reactions , \ 10.1063/1.1580110 journal journal J. Chem. Phys. \ volume 119 ,\ pages 1329--1342 ( year...

    doi

  54. [62]

    author author C. D. \ Schwieters \ and\ author G. A. \ Voth ,\ title title Extension of path integral quantum transition state theory to the case of nonadiabatic activated dynamics , \ 10.1063/1.479569 journal journal J. Chem. Phys. \ volume 111 ,\ pages 2869 ( year 1999 ) NoStop

    doi

  55. [63]

    Peters ,\ @noop title Reaction Rate Theory and Rare Events \ ( publisher Elsevier ,\ address Amsterdam ,\ year 2017 ) NoStop

    author author B. Peters ,\ @noop title Reaction Rate Theory and Rare Events \ ( publisher Elsevier ,\ address Amsterdam ,\ year 2017 ) NoStop

    2017

  56. [64]

    author author L. D. \ Landau ,\ title title Zur Theorie der Energie\"ubertragung. II , \ @noop journal journal Phys. Z. Sowjetunion \ volume 2 ,\ pages 46 ( year 1932 ) NoStop

    1932

  57. [65]

    Zener ,\ title title Non-adiabatic crossing of energy levels , \ 10.1098/rspa.1932.0165 journal journal Proc

    author author C. Zener ,\ title title Non-adiabatic crossing of energy levels , \ 10.1098/rspa.1932.0165 journal journal Proc. R. Soc. Lond. A \ volume 137 ,\ pages 696--702 ( year 1932 ) NoStop

    1932

  58. [66]

    author author L. D. \ Zusman ,\ title title Outer-sphere electron transfer in polar solvents , \ @noop journal journal Chem. Phys. \ volume 49 ,\ pages 295--304 ( year 1980 ) NoStop

    1980

  59. [67]

    Gladkikh , author A

    author author V. Gladkikh , author A. I. \ Burshtein , \ and\ author I. Rips ,\ title title Variation of the resonant transfer rate when passing from nonadiabatic to adiabatic electron transfer , \ 10.1021/jp044311y journal journal J. Phys. Chem. A \ volume 109 ,\ pages 4983--...

    doi

  60. [68]

    Garg , author J

    author author A. Garg , author J. N. \ Onuchic , \ and\ author V. Ambegaokar ,\ title title Effect of friction on electron transfer in biomolecules , \ 10.1063/1.449017 journal journal J. Chem. Phys. \ volume 83 ,\ pages 4491 ( year 1985 ) NoStop

    doi

  61. [69]

    author author J. T. \ Hynes ,\ title title Outer-sphere electron-transfer reactions and frequency-dependent friction , \ 10.1021/j100407a044 journal journal J. Phys. Chem. \ volume 90 ,\ pages 3701--3706 ( year 1986 ) NoStop

    1986 doi

  62. [70]

    Rips \ and\ author J

    author author I. Rips \ and\ author J. Jortner ,\ title title Dynamic solvent effects on outer‐sphere electron transfer , \ 10.1063/1.453184 journal journal J. Chem. Phys. \ volume 87 ,\ pages 2090--2104 ( year 1987 ) NoStop

    1987 doi

  63. [71]

    Sparpaglione \ and\ author S

    author author M. Sparpaglione \ and\ author S. Mukamel ,\ title title Dielectric friction and the transition from adiabatic to nonadiabatic electron transfer. I. Solvation dynamics in Liouville space , \ 10.1063/1.453922 journal journal J. Chem. Phys. \ volume 88 ,\ pages 3263...

    doi

  64. [72]

    Fang , author R

    author author W. Fang , author R. A. \ Zarotiadis , \ and\ author J. O. \ Richardson ,\ title title Revisiting nuclear tunnelling in the aqueous ferrous--ferric electron transfer , \ 10.1039/C9CP06841D journal journal Phys. Chem. Chem. Phys. \ volume 22 ,\ pages 10687--10698 (...

    arXiv

  65. [73]

    author author C. L. \ Vaillant , author M. J. \ Thapa , author J. Van\' c ek , \ and\ author J. O. \ Richardson ,\ title title Semiclassical analysis of the quantum instanton approximation , \ 10.1063/1.5123800 journal journal J. Chem. Phys. \ volume 151 ,\ pages 144111 ( year...

    doi

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.