Pith. sign in

REVIEW 3 major objections 4 minor 64 references

Precision measurement of the last bound states in H$_2$ and determination of the H + H scattering length

T0 review · 3 major / 4 minor · reviewed 2026-08-09 · deepseek-v4-flash

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

Pith's one-line read Precise Doppler-free spectroscopy of hydrogen's five highest bound levels converts one binding energy into the H+H s-wave scattering length, $a_s = 0.2724(5)\,a_0$, in agreement with theory.

desk verdict First complete set of v=14 H2 binding energies is a solid experimental result; the scattering-length uncertainty is understated because the model spread is 0.02 a0, not 0.0005. read the letter →

arxiv 2502.01877 v1 pith:WJGEX54M submitted 2025-02-03 physics.atom-ph physics.chem-ph

K.-F. Lai , W. Ubachs , M. Beyer This is my paper · ORCID
classification physics.atom-phphysics.chem-ph
keywords molecularhydrogenscatteringlengthbindingenergynonadiabaticperturbationtheoryDoppler-freespectroscopycoldatomcollisionshyperfinestructureFeshbachresonance
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 reports measurements of all five rotational levels $J=0$ through $J=4$ in the highest vibrational level $v=14$ of the X$^1\Sigma_g^+$ ground electronic state of H$_2$, reaching as close as $0.023(4)\,\mathrm{cm}^{-1}$ to the dissociation limit. From the binding energy of the last $J=0$ level, $144.807(5)\,\mathrm{cm}^{-1}$, it derives the s-wave scattering length of H+H collisions as $a_s = 0.2724(5)\,a_0$ using a direct semi-empirical scaling of a nonadiabatic perturbation theory potential. The $J=1$ binding energy gives the p-wave scattering volume $a_p = -134.0000(6)\,a_0^3$. These results matter because the H+H scattering length governs cold and ultracold hydrogen collisions, including recombination, hydrogen BEC studies, and precision hydrogen spectroscopy, and because the measurements provide a demanding test of relativistic and QED corrections in the molecular potential.

What carries the argument

The key machinery is the direct semi-empirical scaling method: the NAPT potential $V(R) = E_{\rm BO}(R) + E_{\rm AD}(R) + \delta E_{\rm NA}(R)$ is multiplied by a scaling factor $f$, the radial nuclear Schr\"odinger equation is solved for each $f$, and $f$ is adjusted until the computed binding energy of the ($v=14,J=0$) level matches the measured $144.807(5)\,\mathrm{cm}^{-1}$. The zero-energy scattering wavefunction is then propagated outward to $R=500\,a_0$ and matched to spherical Bessel functions to extract the phase shift and hence the scattering length $a_s$ at that $f$. The same connection between binding energy and scattering parameter for $J=1$ yields the p-wave volume. This converts one precisely measured bound-state energy into a scattering parameter while keeping the potential's shape fixed; the nonadiabatic mass functions $W_\parallel(R)$ and $W_\perp(R)$ are not scaled.

What would settle it

Recompute $a_s$ from the same measured binding energy using an independently constructed fully nonadiabatic potential that does not rely on the $f$-scaling ansatz; if the result differs from $0.2724$ by more than the quoted $0.0005\,a_0$ and instead approaches the roughly $0.02\,a_0$ spread between the Born-Oppenheimer and nonadiabatic values, the direct method's stated uncertainty budget would be refuted.

Watch

Extended reading notes

Core claim

The central claim is that the binding energies of all five $v=14$ bound levels of ground-state H$_2$ can be measured with uncertainties of $0.004$-$0.005\,\mathrm{cm}^{-1}$, and that the $J=0$ binding energy alone, fed through a uniformly scaled NAPT potential, fixes the zero-energy s-wave scattering length to $a_s = 0.2724(5)\,a_0$. The same procedure applied to $J=1$ yields the p-wave scattering volume $a_p = -134.0000(6)\,a_0^3$. The measured binding energy of the last bound level, X($v=14,J=4$) at $0.023(4)\,\mathrm{cm}^{-1}$, agrees with theoretical predictions and, once the hyperfine structure of the separated hydrogen atoms is included, places the level essentially at the intermediate $F_{1,2}=0,1$ dissociation limit, turning it into a very long-lived Feshbach-like quasi-bound state. The paper argues that these measurements verify the NAPT framework including $m\alpha^4$ relativistic and QED contributions, and that they establish the scattering parameters directly from experiment.

Load-bearing premise

The load-bearing premise is that a single uniform scaling of the NAPT potential's depth is enough to map the measured binding energy of the last $J=0$ level onto the correct zero-energy scattering length, so the potential's shape, including dispersion coefficients and nonadiabatic corrections, must already be accurate enough that no shape adjustment is needed.

Editorial extensions

If this is right

  • The H+H s-wave scattering length is determined experimentally as $a_s = 0.2724(5)\,a_0$, with an uncertainty small enough to distinguish between Born-Oppenheimer, adiabatic, nonadiabatic, and $m\alpha^4$-corrected levels of the potential.
  • The binding energy $0.023(4)\,\mathrm{cm}^{-1}$ for X($v=14,J=4$) establishes it as the last bound level of H$_2$; with hyperfine structure included it sits at the middle dissociation limit and acts as a Feshbach resonance with a lifetime too long to affect the observed spectra.
  • The p-wave scattering volume is determined experimentally as $a_p = -134.0000(6)\,a_0^3$.
  • The measured binding energies of all five levels confirm NAPT calculations with relativistic and QED contributions within about 1.5 combined uncertainties, validating the potential at large internuclear separation.
  • The revisited F1-X14 Q(1) frequency corrects an earlier $-0.01\,\mathrm{cm}^{-1}$ calibration offset, changing a previously reported value for the $J=1$ level.

Reading between the lines

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

  • If the f-scaling method is as robust as the paper claims, it could be applied to other hydrogen isotopologues such as D$_2$ and HD, and to other weakly bound diatomic molecules, yielding scattering lengths from a single high-precision binding-energy measurement without a full potential fit.
  • The quoted $0.0005\,a_0$ uncertainty excludes potential-curve uncertainties; the spread between the BO and full NAPT values is roughly $0.02\,a_0$, so a future calculation that quantifies potential-shape error could shift $a_s$ outside the quoted band.
  • The hyperfine-structure picture suggests that the near-threshold $J=4$ level may be magnetically tunable; a search for magnetic-field-dependent shifts of its energy or lifetime would test the Feshbach-resonance interpretation.
  • Because the $J=0$ and $J=1$ channels come from the same potential, the mutual consistency of the derived $a_s$ and $a_p$ with independent scattering calculations could serve as a cross-check of the direct method.
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 manuscript reports precision measurements of the binding energies of the five rotational levels J = 0–4 in the highest vibrational level v = 14 of the X^1Σ_g^+ ground state of H_2, using a three-step UV-laser scheme with H_2S photolysis, Doppler-free two-photon spectroscopy in the F–X system, and autoionization detection. The measured binding energies are compared with nonadiabatic perturbation theory (NAPT) calculations including relativistic and QED contributions, showing agreement within 1.5σ for most levels. From the J = 0 binding energy the authors extract an s-wave scattering length a_s = 0.2724(5) a_0 via a semi-empirical scaling method, and from the J = 1 binding energy a p-wave scattering volume a_p = -134.0000(6) a_0^3. The last bound level v = 14, J = 4 is measured at 0.023(4) cm^-1 below the hyperfineless dissociation limit, and its relation to the three hyperfine dissociation limits and possible Feshbach-resonance character is discussed.

Significance. If the experimental results stand, the paper provides the first complete spectroscopic characterization of all five bound states in the highest vibrational manifold of H_2 near the dissociation threshold, including a direct measurement of the very weakly bound J = 4 level. The combination-difference verification against NAPT is a valuable benchmark for molecular theory, and the corrected calibration of previously reported F1–X14 Q(1) frequencies is a useful community contribution. The extraction of the scattering length from near-threshold binding energies is an appealing application of the direct method, and the quoted precision (0.0005 a_0) would be remarkable. However, the central claim of a 0.0005 a_0 total uncertainty for a_s is not supported, because the extracted value varies by 0.022 a_0 across the BO, AD, NA, and mα^4 approximation levels, and the paper states explicitly that potential-curve uncertainties are not included. The experimental binding-energy measurements are carefully analyzed with ac-Stark extrapolation and error budgets, and those results are the strongest part of the paper.

major comments (3)
  1. [Section VI, Table IV and Fig. 7] The direct method yields a_s values of 0.2894, 0.2760, 0.2675, and 0.2724 a_0 at the BO, AD, NA, and mα^4 levels, respectively, a spread of 0.022 a_0 that is about 40 times the quoted uncertainty of 0.0005 a_0. Since the scaling factor f is adjusted to reproduce the experimental binding energy, the method absorbs a global depth error but cannot correct for shape errors in the long-range tail of the potential, which change a_s at fixed D. The statement that the semi-empirical method is "rather insensitive to small details of the potential energy curve" (Sec. VI, final paragraph) is contradicted by the variation in Table IV. The quoted uncertainty therefore represents only the experimental binding-energy contribution, not the total uncertainty. The authors should either report a model-dependent systematic uncertainty (e.g., a_s = 0.272(22) a_0 or a similar range) or provide a quantitative argument for why the mα^4 level is uniquely correct.
  2. [Section VII, Table V and Fig. 8] The same undercounting of model dependence applies to the p-wave scattering volume. The extracted a_p spans -133.4991 (BO) to -134.0000 (NA) a_0^3, a spread of 0.5 a_0^3, while the quoted uncertainty is 0.0006 a_0^3. This spread is not an experimental uncertainty but a model-dependent systematic effect. The abstract's headline value a_p = -134.0000(6) a_0^3 is therefore not supported as a total uncertainty. The authors should add a model-dependent error term, or at least discuss why the NA level is the appropriate final value and quantify the uncertainty from the choice of approximation level.
  3. [Section VI, Eqs. (2)–(6) and after Eq. (12)] The scaling ansatz assumes that multiplying only the potential V(R) by a factor f, while leaving the nonadiabatic W_∥(R) and W_⊥(R) functions unscaled, preserves the relationship between the binding energy of the last bound state and the zero-energy scattering length. This is the load-bearing model assumption. The authors assert that scaling the W functions has "negligible effect" but provide no numerical evidence. Given that the difference in a_s between the AD and NA levels (which differ by the W contributions) is about 0.0085 a_0, this assumption needs explicit testing. In addition, the relativistic correction E^(4,0)(R) is scaled by the same f, and the mα^5 and mα^6 terms are omitted because they violate Eq. (10). The omission of these terms could contribute to the model uncertainty, and the significance claimed for the mα^4 effect is not accompanied by an estimate of the truncation error. Please provide a quantitative sensitivity test of the scaling assumption and of the omitted higher-order corrections.
minor comments (4)
  1. [Section VI, paragraph after Eq. (10)] The phrase "they are of small a small amount" contains a typographical error; it should read "they are a small amount" or "they are small."
  2. [Abstract and Section VI] The notation "mα^4" is used in the abstract while the text uses both "mα^4" and "mα4"; please use a consistent notation (e.g., mα^4) throughout.
  3. [Section III, Table I footnote] The footnote for the revised F1-X14 Q(1) value states the previous value had an offset of -0.01 cm^-1 due to mis-assignment of an I_2 line. It would be helpful to give the magnitude of the correction and the reference for the I_2 line assignment.
  4. [Section VIII] The discussion of the hyperfine dissociation limits would benefit from a brief explanation of why the "hyperfineless" dissociation limit (36 118.069 605 (31) cm^-1) differs slightly from the D_0 value used in Sec. V (36 118.069 632 (26) cm^-1); the two values are quoted without explicit reconciliation.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the measured binding energy is an independent input, and the scaling factor f is fitted to that input to read off a_s from the same potential, which is a calibration rather than a reduction of the target to the input.

full rationale

The derivation chain is: measured transition frequencies -> independent F-state term values [37] and dissociation energy D0 [38] -> experimental binding energy D0_14(exp) -> scaling factor f -> computed a_s. The experimental binding energy is an independent external input, and a_s is obtained by solving the scaled Schrödinger equation and reading the zero-energy phase shift; the paper nowhere fits a_s to a_s data nor defines the binding energy in terms of a_s. The scaling factor f is the only adjustable parameter and is fixed by the measured D0_14, not by a_s, so the extracted scattering length is a genuine inference rather than a restatement of the input. The approximation-level spread in Table IV (BO/AD/NA/mα⁴) is explicitly acknowledged in Sec. VI as not included in the quoted uncertainty (“the uncertainties presented in Table IV only account for the uncertainty in the experimental value of the dissociation energy and do not take into account uncertainties from the potential energy curves and their computation”); this is a model-uncertainty caveat, not a circular step. The self-citations ([16,17,26]) supply experimental methods and earlier comparisons, while the NAPT potential and code [18,35,40] are external ab initio results, and D0 [38] is an independently measured constant with overlapping authors but not derived from the present target. No uniqueness theorem or ansatz is imported from the authors’ own prior work as the load-bearing justification. The central claim is therefore self-contained relative to its inputs.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The paper adds no new particles or forces. Its central derived quantity, a_s, is obtained by fitting a scaling factor f to the measured binding energy and reading the scattering length from the same scaled potential. The main free parameters are f and the choice of approximation level; the main domain assumptions are the accuracy of the NAPT potential and the reference term values.

free parameters (2)
  • potential scaling factor f = Approximately 1.001 for BO, 1.0005 for AD, ~1 for NA; exact values not tabulated
    Uniformly scales V(R) in Eq. (3) to shift the computed binding energy D^J_14 to the measured value; the scattering length is then read at the same f. This is the adjustable parameter of the semi-empirical method.
  • approximation level choice (mα4) = Selected as final result
    The final a_s = 0.2724 uses BO+AD+NA+mα4. Higher QED terms (mα5, mα6) are omitted because they violate Eq. (10). The difference between NA and mα4 is 0.0049 a0, a systematic not included in the error.
assumptions (4)
  • domain assumption The NAPT potential curves V(R), W_parallel(R), W_perp(R) from Ref. [40] are accurate representations of the H2 interaction.
    Used as input to the radial Schrödinger equations (2)-(7); any error propagates into D and a.
  • ad hoc to paper A uniform scaling of V(R) by f is sufficient to reproduce the experimental binding energy; the W functions are kept unscaled.
    The paper states scaling W leads to nonphysical behavior and that its effect is negligible; this is a modeling assumption not independently justified.
  • domain assumption The long-range potential is dominated by the R^-6 dispersion term satisfying Eq. (10), so that aJ is well defined.
    Required to define the scattering length; higher QED terms decay as R^-3 and R^-2 and are excluded.
  • domain assumption Reference term values for F0 and F1 (Bailly et al.) and dissociation energy D0 (Beyer et al.) are accurate.
    Used to convert measured transition frequencies to binding energies; errors in these would shift all binding energies.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Precision measurement of the last bound states in H$_2$ and determination of the H + H scattering length." pith.science (2026). https://pith.science/paper/WJGEX54M

@misc{pith2026250201877,
  author       = {Pith},
  title        = {Pith review of: Precision measurement of the last bound states in H$_2$ and determination of the H + H scattering length},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WJGEX54M}},
  note         = {Machine review of arXiv:2502.01877}
}
abstract

The binding energies of the five bound rotational levels $J=0-4$ in the highest vibrational level $v=14$ in the X$^1\Sigma_g^+$ ground electronic state of H$_2$ were measured in a three-step ultraviolet-laser experiment. Two-photon UV-photolysis of H$_2$S produced population in these high-lying bound states, that were subsequently interrogated at high precision via Doppler-free spectroscopy of the F$^1\Sigma_g^+$ - X$^1\Sigma_g^+$ system. A third UV-laser was used for detection through auto-ionizing resonances. The experimentally determined binding energies were found to be in excellent agreement with calculations based on non-adiabatic perturbation theory, also including relativistic and quantum electrodynamical contributions. The $s$-wave scattering length of the H + H system is derived from the binding energy of the last bound $J=0$ level via a direct semi-empirical approach, yielding a value of $a_s$ = 0.2724(5) $a_0$, in good agreement with a result from a previously followed theoretical approach. The subtle effect of the $m\alpha^4$ relativity contribution to $a_s$ was found to be significant. In a similar manner a value for the $p$-wave scattering volume is determined via the $J=1$ binding energy yielding $a_p$ = -134.0000(6) $a_0^3$. The binding energy of the last bound state in H$_2$, the ($v=14$, $J=4$) level, is determined at 0.023(4) cm$^{-1}$, in good agreement with calculation. The effect of the hyperfine substructure caused by the two hydrogen atoms at large internuclear separation, giving rise to three distinct dissociation limits, is discussed.

Figures

Figures reproduced from arXiv: 2502.01877 by the authors.

Figure 1
Figure 1. FIG. 1. Effective potential energy curves of X [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Spectra of the F0-X14 Q(4) transition measured [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Spectra of the F1-X14 Q(0) transition measured [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figures from the paper (6 more)
Figure 5
Figure 5. Figure 5: FIG. 5. Calculated Franck-Condon factors of F [PITH_FULL_IMAGE:figures/full_fig_p004_5.png]
Figure 4
Figure 4. Figure 4: FIG. 4. Portions of autoionization spectra recorded from the [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]
Figure 6
Figure 6. Figure 6: FIG. 6. The experimentally determined binding energies of [PITH_FULL_IMAGE:figures/full_fig_p005_6.png]
Figure 7
Figure 7. Figure 7: shows the correlation between aJ and DJ 14 by varying f at different level of approximations for s￾wave scattering parameters. The slopes in [PITH_FULL_IMAGE:figures/full_fig_p007_7.png]
Figure 8
Figure 8. Figure 8: shows the relation of D1 14(f) and ap(f) at BO, adiabatic and nonadiabatic approximations. As listed in Table V, the relative variation of unscaled scattering volume among different approximations is much smaller than the s-wave scattering length. However, a similar ra…
Figure 9
Figure 9. Figure 9: FIG. 9. Potential energy curves for [PITH_FULL_IMAGE:figures/full_fig_p009_9.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

64 extracted references · 63 canonical work pages

  1. [1]

    Precision measurement of the last bound states in H$_2$ and determination of the H + H scattering length

    The binding energy of the last bound state in H 2, the ( v = 14, J = 4) level, is determined at 0.023 (4) cm −1, in good agreement with calculation. The effect of the hyperfine substructure caused by the two hydrogen atoms at large internuclear separation, giving rise to three distinct dissociation limits, is discussed. I. INTRODUCTION The s-wave scatteri...

    work page Pith review arXiv 2025

  2. [2]

    R. E. Roberts, R. B. Bernstein, and C. F. Curtiss, Res- onance theory of thermolecular recombination kinetics: H+H+M → H2 + M, J. Chem. Phys. 50, 5163 (1969)

    work page 1969

  3. [3]

    The relativistic and QED effects to the scattering vol- ume cannot be evaluated using the perturbative approach as all terms decay slower than R−5 in Eq

    obtained by Jamieson and Dalgarno [46]. The relativistic and QED effects to the scattering vol- ume cannot be evaluated using the perturbative approach as all terms decay slower than R−5 in Eq. (10). Table V lists the relativistic and QED effects to p-wave scatter- ing volume using the perturbative approach. Since the asymptotic zero-energy wavefunction i...

    work page

  4. [4]

    M. J. Jamieson, A. Dalgarno, and J. N. Yukich, Elastic scattering of hydrogen atoms at low temperatures, Phys. Rev. A 46, 6956 (1992)

    work page 1992

  5. [5]

    J. P. Wittke and R. H. Dicke, Redetermination of the hy- perfine splitting in the ground state of atomic hydrogen, Phys. Rev. 103, 620 (1956)

    work page 1956

  6. [6]

    Palla, E

    F. Palla, E. E. Salpeter, and S. W. Stahler, Primordial star formation - The role of molecular hydrogen, Astroph. J. 271, 632 (1983)

    work page 1983

  7. [7]

    R. C. Forrey, Quadrupole association and dissociation of hydrogen in the early Universe, J. Phys. B: At. Mol. Opt. Phys. 49, 194002 (2016)

    work page 2016

  8. [8]

    D. G. Fried, T. C. Killian, L. Willmann, D. Landhuis, S. C. Moss, D. Kleppner, and T. J. Greytak, Bose- Einstein condensation of atomic hydrogen, Phys. Rev. Lett. 81, 3811 (1998)

    work page 1998

Show all 64 references
  1. [9]

    U. D. Jentschura, C. M. Adhikari, R. Dawes, A. Matveev, and N. Kolachevsky, Pressure shifts in high-precision hy- drogen spectroscopy. I. Long-range atom-atom and atom- molecule interactions, J. Phys. B 52, 075005 (2019)

    2019

  2. [10]

    Matveev, N

    A. Matveev, N. Kolachevsky, C. M. Adhikari, and U. D. 10 Jentschura, Pressure shifts in high-precision hydrogen spectroscopy: II. Impact approximation and Monte- Carlo simulations, J. Phys. B 52, 075006 (2019)

    2019

  3. [11]

    Szmytkowski, Analytical calculations of scattering lengths in atomic physics, J

    R. Szmytkowski, Analytical calculations of scattering lengths in atomic physics, J. Phys. A 28, 7333 (1995)

    1995

  4. [12]

    T. C. Killian, D. G. Fried, L. Willmann, D. Landhuis, S. C. Moss, T. J. Greytak, and D. Kleppner, Cold colli- sion frequency shift of the 1S-2S transition in hydrogen, Phys. Rev. Lett. 81, 3807 (1998)

    1998

  5. [13]

    G. F. Gribakin and V. V. Flambaum, Calculation of the scattering length in atomic collisions using the semiclas- sical approximation, Phys. Rev. A 48, 546 (1993)

    1993

  6. [14]

    TABLE IV

    The slope of the fitting is fairly identical for all approximations as the as and the D0 14 heavily depend on the BO potential energy. TABLE IV. Values for the s-wave scattering length, obtained for the present direct method as(exp) and from the previous theoretical approach [...

  7. [15]

    A. Sen, S. Chakraborty, and A. S. Ghosh, Scattering of hydrogen atoms at low temperature, Europhys. Lett. 76, 582 (2006)

    2006

  8. [16]

    Dalgarno, R

    A. Dalgarno, R. McCarroll, and D. R. Bates, Adia- batic coupling between electronic and nuclear motion in molecules, Proc. Roy. Soc. London. A 237, 383 (1956)

    1956

  9. [17]

    Wolniewicz, Nonadiabatic couplings in low-energy col- lisions of hydrogen ground-state atoms, Phys

    L. Wolniewicz, Nonadiabatic couplings in low-energy col- lisions of hydrogen ground-state atoms, Phys. Rev. A 68, 042717 (2003)

    2003

  10. [18]

    M. J. Jamieson, A. S. Cheung, and H. Ouerdane, De- pendence of the scattering length for hydrogen atoms on effective mass, Eur. Phys. J. D 56, 181 (2010)

    2010

  11. [19]

    Lai, , E

    K.-F. Lai, , E. J. Salumbides, W. Ubachs, and M. Beyer, Shape resonances in H 2 as photolysis reaction intermedi- ates, Phys. Rev. Lett. 127, 183001 (2021)

    2021

  12. [20]

    K.-F. Lai, E. J. Salumbides, M. Beyer, and W. Ubachs, Precision measurement of quasi-bound resonances in H2 and the H+H scattering length, Mol. Phys. 120, e2018063 (2022)

    2022

  13. [21]

    Czachorowski, M

    P. Czachorowski, M. Puchalski, J. Komasa, and K. Pachucki, Nonadiabatic relativistic correction in H 2, D2, and HD, Phys. Rev. A 98, 052506 (2018)

    2018

  14. [22]

    J. R. Gardner, R. A. Cline, J. D. Miller, D. J. Heinzen, H. M. J. M. Boesten, and B. J. Verhaar, Collisions of doubly spin-polarized, ultracold 85Rb atoms, Phys. Rev. Lett. 74, 3764 (1995)

    1995

  15. [23]

    C. C. Tsai, R. S. Freeland, J. M. Vogels, H. M. J. M. Boesten, B. J. Verhaar, and D. J. Heinzen, Two-color photoassociation spectroscopy of ground state Rb2, Phys. Rev. Lett. 79, 1245 (1997)

    1997

  16. [24]

    Crubellier, O

    A. Crubellier, O. Dulieu, F. Masnou-Seeuws, M. Elbs, H. Kn¨ ockel, and E. Tiemann, Simple determination of Na2 scattering lengths using observed bound levels at the ground state asymptote, Eur. Phys. J. D 6, 211 (1999)

    1999

  17. [25]

    A. D. Lange, K. Pilch, A. Prantner, F. Ferlaino, B. En- geser, H.-C. N¨ agerl, R. Grimm, and C. Chin, Determi- nation of atomic scattering lengths from measurements of molecular binding energies near Feshbach resonances, Phys. Rev. A 79, 013622 (2009)

    2009

  18. [26]

    L. D. Augustoviˇ cov´ a and V.ˇSpirko, Relating binding en- ergy and scattering length of weakly bound dimers of strontium, Ann. Phys. (Berlin) 533, 2000588 (2021)

    2021

  19. [27]

    T. M. Trivikram, M. L. Niu, P. Wcis lo, W. Ubachs, and E. J. Salumbides, Precision measurements and test of molecular theory in highly excited vibrational states of H2 (v = 11), Appl. Phys. B 122, 294 (2016)

    2016

  20. [28]

    T. M. Trivikram, E. J. Salumbides, C. Jungen, and W. Ubachs, Excitation of H 2 at large internuclear sepa- ration: outer well states and continuum resonances, Mol. Phys. 117, 2961 (2019)

    2019

  21. [29]

    K.-F. Lai, M. Beyer, E. J. Salumbides, and W. Ubachs, Photolysis production and spectroscopic investigation of the highest vibrational states in H 2 (X1Σ+ g v = 13 , 14), J. Phys. Chem. A 125, 1221 (2021)

    2021

  22. [30]

    K.-F. Lai, M. Beyer, and W. Ubachs, Spectroscopic study of the F 1Σ+ g outer well state in H 2, HD and D 2, J. Mol. Spectr. 393, 111778 (2023)

    2023

  23. [31]

    Steadman and T

    J. Steadman and T. Baer, The production and charac- terization by resonance enhanced multiphoton ionization of H 2 (v = 10 − 14) from photodissociation of H 2S, J. Chem. Phys. 91, 6113 (1989)

    1989

  24. [32]

    Hannemann, E

    S. Hannemann, E. J. Salumbides, and W. Ubachs, Re- ducing the first-order Doppler shift in a Sagnac interfer- ometer, Opt. Lett. 32, 1381 (2007)

    2007

  25. [33]

    H.Kn¨ ockel (Leibniz University, Hannover)

    We used the ”IodineSpec” program, kindly provided to us by Dr. H.Kn¨ ockel (Leibniz University, Hannover). See also B. Bodermann, H. Kn¨ ockel and E. Tiemann, Eur. Phys. J. D 19, 31, (2002)

    2002

  26. [34]

    Komasa, In search for the negative polarizability states - the EF 1Σ+ g state of hydrogen molecule, Adv

    J. Komasa, In search for the negative polarizability states - the EF 1Σ+ g state of hydrogen molecule, Adv. Quant. Chem. 48, 151 (2005)

    2005

  27. [35]

    Ko los and L

    W. Ko los and L. Wolniewicz, Polarizability of the hydro- gen molecule, J. Chem. Phys. 46, 1426 (1967)

    1967

  28. [36]

    Y. Zhao, Z. Luo, Y. Chang, Y. Wu, S. Zhang, Z. Li, H. Ding, G. Wu, J. S. Campbell, C. S. Hansen, S. W. Crane, C. M. Western, M. N. R. Ashfold, K. Yuan, and X. Yang, Rotational and nuclear-spin level depen- dent photodissociation dynamics of H2S, Nat. Comm. 12, 4459 (2021)

    2021

  29. [37]

    Y. Zhao, J. Chen, Z. Luo, Z. Li, S. Yang, Y. Chang, F. An, Z. Chen, J. Yang, G. Wu, W. Zhang, X. Hu, D. Xie, H. Ding, K. Yuan, and X. Yang, Photodissoci- ation of H 2S: A new pathway for the production of vi- brationally excited molecular hydrogen in the interstellar medium, J...

    2022

  30. [38]

    H2SPECTRE ver 7.4 Fortran source code, University of Warsaw, Poland (2022)

    2022

  31. [39]

    E. J. Salumbides, D. Bailly, A. Khramov, A. L. Wolf, K. S. E. Eikema, M. Vervloet, and W. Ubachs, Improved laboratory values of the H 2 Lyman and Werner lines for constraining time variation of the proton-to-electron mass ratio, Phys. Rev. Lett. 101, 223001 (2008)

    2008

  32. [40]

    Bailly, E

    D. Bailly, E. Salumbides, M. Vervloet, and W. Ubachs, Accurate level energies in the EF 1Σ+ g , GK 1Σ+ g , H 1Σ+ g , B1Σ+ u , C 1Πu, B’ 1Σ+ u , D 1Πu, I 1Πg, J 1∆g states of H 2, Mol. Phys. 108, 827 (2010)

    2010

  33. [41]

    Beyer, N

    M. Beyer, N. H¨ olsch, J. Hussels, C.-F. Cheng, E. J. Salumbides, K. S. E. Eikema, W. Ubachs, C. Jungen, and F. Merkt, Determination of the interval between the ground states of para- and ortho-H 2, Phys. Rev. Lett. 123, 163002 (2019)

    2019

  34. [42]

    Landau, Quantum Mechanics II: A Second Course in Quantum Theory (Wiley, 2008)

    R. Landau, Quantum Mechanics II: A Second Course in Quantum Theory (Wiley, 2008)

    2008

  35. [43]

    Komasa, M

    J. Komasa, M. Puchalski, P. Czachorowski, G. Lach, and K. Pachucki, Rovibrational energy levels of the hydro- gen molecule through nonadiabatic perturbation theory, Phys. Rev. A 100, 032519 (2019)

    2019

  36. [44]

    B. R. Johnson, New numerical methods applied to solving the one-dimensional eigenvalue problem, J. Chem. Phys. 67, 4086 (1977)

    1977

  37. [45]

    M. J. Jamieson and A. Dalgarno, How a change in the interaction potential affects the s-wave scattering length, 11 J. Phys. B 31, L219 (1998)

    1998

  38. [46]

    Puchalski, J

    M. Puchalski, J. Komasa, P. Czachorowski, and K. Pachucki, Complete α6m corrections to the ground state of H 2, Phys. Rev. Lett. 117, 263002 (2016)

    2016

  39. [47]

    Puchalski, J

    M. Puchalski, J. Komasa, and K. Pachucki, Relativistic corrections for the ground electronic state of molecular hydrogen, Phys. Rev. A 95, 052506 (2017)

    2017

  40. [48]

    Si lkowski, K

    M. Si lkowski, K. Pachucki, J. Komasa, and M. Puchalski, Leading-order QED effects in the ground electronic state of molecular hydrogen, Phys. Rev. A 107, 032807 (2023)

    2023

  41. [49]

    M. J. Jamieson and A. Dalgarno, How a change in the in- teraction potential affects the p-wave scattering volume, J. Phys. B 45, 061001 (2012)

    2012

  42. [50]

    Komasa, K

    J. Komasa, K. Piszczatowski, G. Lach, M. Przybytek, B. Jeziorski, and K. Pachucki, Quantum electrodynamics effects in rovibrational spectra of molecular hydrogen, J. Chem. Theory Comput. 7, 3105 (2011)

    2011

  43. [51]

    Selg, A quasi-bound rovibrational state of hydrogen molecule resulting from hyperfine proton-electron spin- spin interaction, Eur

    M. Selg, A quasi-bound rovibrational state of hydrogen molecule resulting from hyperfine proton-electron spin- spin interaction, Eur. Phys. Lett. 96, 10009 (2011)

    2011

  44. [52]

    Selg, Observable quasi-bound states of the H 2 molecule, J

    M. Selg, Observable quasi-bound states of the H 2 molecule, J. Chem. Phys, 136, 114113 (2012)

    2012

  45. [53]

    E. M. Roueff and H. Abgrall, The quasi bound spectrum of H2, Mol. Phys. 121, e2122887 (2023)

    2023

  46. [54]

    Dabrowski, The Lyman and Werner bands of H 2, Can

    I. Dabrowski, The Lyman and Werner bands of H 2, Can. J. Phys. 62, 1639 (1984)

    1984

  47. [55]

    Roncin and F

    J.-Y. Roncin and F. Launay, Atlas of the vacuum ultra- violet emission spectrum of molecular hydrogen, J. Phys. Chem. Ref. Data, Monograph 4 (1994)

    1994

  48. [56]

    J. E. Harriman, M. Twerdoch, M. B. Milleur, and J. O. Hirschfelder, Hyperfine splitting of interaction energy of 2 hydrogen atoms, Proc. Nat. Ac. Sci. 57, 1558 (1967)

    1967

  49. [57]

    Ko los and L

    W. Ko los and L. Wolniewicz, Potential-energy curves for the X 1Σ+ g , b 3Σ+ u , and C 1Πu states of the hydrogen molecule, J. Chem. Phys. 43, 2429 (1965)

    1965

  50. [58]

    S. G. Karshenboim, Precision physics of simple atoms: QED tests, nuclear structure and fundamental constants, Physics Reports 422, 1 (2005)

    2005

  51. [59]

    Cheng, J

    C.-F. Cheng, J. Hussels, M. Niu, H. L. Bethlem, K. S. E. Eikema, E. J. Salumbides, W. Ubachs, M. Beyer, N. H¨ olsch, J. A. Agner, F. Merkt, L.-G. Tao, S.-M. Hu, and C. Jungen, Dissociation energy of the hydrogen molecule at 10−9 accuracy, Phys. Rev. Lett. 121, 013001 (2018)

    2018

  52. [60]

    H¨ olsch, M

    N. H¨ olsch, M. Beyer, E. J. Salumbides, K. S. E. Eikema, W. Ubachs, C. Jungen, and F. Merkt, Benchmarking the- ory with an improved measurement of the ionization and dissociation energies of H2, Phys. Rev. Lett. 122, 103002 (2019)

    2019

  53. [61]

    Hussels, N

    J. Hussels, N. H¨ olsch, C.-F. Cheng, E. J. Salumbides, H. L. Bethlem, K. S. E. Eikema, C. Jungen, M. Beyer, F. Merkt, and W. Ubachs, Improved ionization and dis- sociation energies of the deuterium molecule, Phys. Rev. A 105, 022820 (2022)

    2022

  54. [62]

    H¨ olsch, I

    N. H¨ olsch, I. Doran, F. Merkt, J. Hussels, C.-F. Cheng, E. J. Salumbides, H. L. Bethlem, K. S. E. Eikema, M. Beyer, W. Ubachs, and C. Jungen, Ionization and dissociation energies of HD and dipole-induced g/u- symmetry breaking, Phys. Rev. A 108, 022811 (2023)

    2023

  55. [63]

    Puchalski, A

    M. Puchalski, A. Spyszkiewicz, J. Komasa, and K. Pachucki, Nonadiabatic relativistic correction to the dissociation energy of H 2, D2, and HD, Phys. Rev. Lett. 121, 073001 (2018)

    2018

  56. [64]

    Puchalski, J

    M. Puchalski, J. Komasa, A. Spyszkiewicz, and K. Pachucki, Dissociation energy of molecular hydrogen isotopologues, Phys. Rev. A 100, 020503 (2019)

    2019

Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.