Variational studies of ro-vibrational spectra of DT$^+$ and T$_2^+$ ions

A.K. Bekbaev; D.B. Zhomartova; D.T. Aznabayev; Vladimir I. Korobov

REVIEW 3 major objections 5 minor 22 references

Variational studies of ro-vibrational spectra of DT$^+$ and T$_2^+$ ions

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper reports the first high-precision theoretical ro-vibrational spectra for the last two hydrogen molecular ion isotopologues, DT+ and T2+, with nonrelativistic energies accurate to about ten to the minus twelve atomic units.

desk verdict Useful completion of the HMI isotopologue set, but the near-threshold T2+ states need convergence evidence before the digits enter metrology. read the letter →

arxiv 2507.12243 v1 pith:ILQ477LF submitted 2025-07-16 physics.atom-ph

D.T. Aznabayev , D.B. Zhomartova , A.K. Bekbaev , Vladimir I. Korobov This is my paper · ORCID

classification physics.atom-ph

keywords DT+T2+hydrogenmolecularionsrovibrationalspectravariationalcalculationnonrelativisticenergyhyperfinestructuretransitionamplitudes

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 the first high-precision theoretical spectra for the two remaining hydrogen molecular ion isotopologues, DT+ and T2+. By solving the three-body Schrödinger equation with Coulomb interactions in a variational exponential basis, the authors obtain nonrelativistic ro-vibrational energies through the highest bound 1sσg states, and through the 2pσu states of T2+, with twelve significant digits. They also compute electric dipole transition amplitudes and spontaneous emission rates, leading-order relativistic corrections including recoil terms, and coefficients of the effective hyperfine Hamiltonians. If the results hold up, they complete the set of hydrogen molecular ion isotopologues available for precision spectroscopy, with direct relevance to extracting the triton charge radius and testing quantum electrodynamics.

What carries the argument

The central object is the variational wave function expanded in exponential basis functions of the three interparticle distances, with complex exponents chosen in a pseudorandom way. For T2+, an explicitly symmetrized coordinate geometry enforces the gerade and ungerade permutation symmetry of the two identical tritons. This machinery supplies the bound-state energies, dipole matrix elements, and expectation values of the Breit-Pauli operators from which all tabulated quantities are derived.

What would settle it

Recompute the highest near-threshold levels, for example T2+ 1sσg v=34 or the 2pσu states, with an independent method or with substantially larger basis sets, and check whether the tabulated energies shift by more than the claimed $10^{-12}$ atomic units; alternatively, compare the predicted v(0→33) and v(0→34) transition frequencies against measured T2+ spectra once available.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is that the variational approach already used for other hydrogen molecular ions can be extended to the last two isotopologues, DT+ and T2+, producing converged nonrelativistic energies at the $10^{-12}$ level for the widest range of vibrational states, up to the highest bound 1sσg state and the 2pσu states of T2+. With these wave functions the authors obtain dipole transition amplitudes, spontaneous decay rates, spin-independent relativistic corrections at order $R_\infty\alpha^2$ including the transverse-photon recoil terms, and the coefficients of the effective hyperfine structure Hamiltonians for both ions. The assembled data give a solid theoretical basis for precision spectroscopy of tritium-bearing molecular ions.

Load-bearing premise

Every listed state, including the extremely weakly bound near-threshold states of T2+, is assumed to be fully converged in the variational basis, but the paper shows no convergence or basis-size dependence data.

Editorial extensions

If this is right

The 12-digit nonrelativistic energies provide the zero-order grid onto which QED, recoil, and hyperfine corrections can be attached for DT+ and T2+, making these ions usable for precision tests already performed on lighter hydrogen molecular ions.
Tabulated dipole moments and spontaneous emission rates let experimenters identify which ro-vibrational lines have usable intensity for laser spectroscopy and quantum logic readout.
With the hydrogen-atom Bethe logarithm approximation, transition energies can be predicted with relative theoretical error of about 10^-8, close enough to guide spectroscopy and to support triton charge radius extraction.
The effective hyperfine Hamiltonian coefficients allow hyperfine-resolved transition patterns to be computed, which is required to assign and drive individual hyperfine components.
Together with previous work on other isotopologues, this completes the high-precision theoretical spectroscopy needed for every hydrogen molecular ion isotopologue.

Reading between the lines

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

A comparison the paper does not show: low-lying DT+ and T2+ energies should reduce smoothly toward the known H2+ and HD+ limits as the nuclear masses change, so a table of such differences would give an inexpensive convergence check.
The authors note that spin-spin interaction breaks g/u symmetry near the T(n=1) threshold; a natural next step, which they defer, is to quantify how this shifts the near-threshold binding energies and enables nominally forbidden E1 transitions between gerade states.
The stated 10^-8 theoretical accuracy depends on replacing the state-specific Bethe logarithm by the hydrogen ground-state value; the wave functions and operator expectation values reported here are exactly what a future state-specific Bethe-logarithm calculation would need.
If the triton charge radius is to be extracted from DT+ or T2+ spectra, the hyperfine coefficients and dipole amplitudes given here can be used to design transitions with minimal sensitivity to the deuteron quadrupole moment and nuclear magnetic moments.

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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

The paper does what it says: it gives the first high-precision variational nonrelativistic energies for DT+ and T2+, along with transition amplitudes, leading relativistic corrections, and hyperfine coefficients. The Hamiltonian, Breit-Pauli operators, and hyperfine effective Hamiltonians are standard and correctly written. The Korobov exponential expansion is established, and the basis sizes (N=12000–14000, octuple precision) are in line with what is needed for 10^-12 a.u. energies. This fills the last gap in the HMI isotopologue set, so the dataset will be a useful reference for planning spectroscopy and for triton-radius extraction work.

That said, the precision claims are not fully backed up. The main missing piece is convergence evidence. There is no basis-size dependence study, no comparison with a second exponent set, and no validation against known H2+/HD+ results. For the near-threshold T2+ states—especially the weakly bound 1sσg states with v≥29 and the 2pσu states with v=1,2—the finite exponential basis is exactly where incompleteness would show silently in the total energy while shifting the binding energy at the kHz level. The tables in the main text only include T2+ ungerade states up to v=2; the high gerade states appear only in the transition tables, which makes independent checking hard.

The stress-test note about the transition-frequency digits is partly fair and partly not. The paper does not claim kHz-level absolute accuracy; it claims a relative theoretical error near 10^-8, and the radiative correction is explicitly approximated using the hydrogen ground-state Bethe logarithm. But Tables IV and V quote transition energies to 0.001 MHz, and with 10^-12 a.u. rounding (≈6.6 kHz) in the printed energies, those digits cannot be justified unless the internal values are kept to more digits than shown. That is probably cosmetic if they computed with higher precision, but the discrepancy should be stated or the tables rounded appropriately.

The paper is honest about its limitations, which I appreciate. The hyperfine tables are a useful resource, and the transition amplitudes/rates are a nice addition. For the subfield of precision molecular spectroscopy, this is a worthwhile contribution. I would send it to peer review, but the referee should ask for convergence evidence for the near-threshold states and a clarification of the precision of the derived transition frequencies.

Referee Report

3 major / 5 minor

Summary. The manuscript reports variational calculations of nonrelativistic ro-vibrational energies, dipole transition amplitudes, leading-order relativistic corrections, and effective hyperfine-structure Hamiltonian coefficients for the two tritium-bearing hydrogen molecular ions DT+ and T2+. The method is the authors' established exponential variational expansion with pseudorandom exponents, run with basis sizes N=12000–14000 in octuple precision. The paper claims twelve-significant-digit nonrelativistic energies, leading-order relativistic and radiative corrections, and a final relative theoretical error near 10^-8 for transition energies, and it tabulates transition frequencies for T2+ g/u transitions, including near-threshold weakly bound states.

Significance. If the numerical results are correct, this is the first high-precision theoretical dataset for DT+ and T2+, the last hydrogen molecular ion isotopologues lacking such data, and it would be useful for planned experiments on triton charge radius extraction and QED tests. The underlying formalism is standard, the Hamiltonian and Breit-Pauli operators are written correctly, and the variational method is well established from prior work on H2+ and HD+. The paper also provides useful operator expectation values and hyperfine coefficients in the Supplemental Material. However, the central precision claims rest on convergence behavior that is asserted but not demonstrated, and the quoted transition frequencies for near-threshold states are not compatible with the stated numerical uncertainty.

major comments (3)

[VII, Tables I–III] The central claim of 'twelve significant digits' for the nonrelativistic energies is not supported by any convergence evidence in the manuscript. There is no basis-size dependence, no comparison with the known H2+ or HD+ results, and no error estimate. This is especially critical for the near-threshold T2+ states (1sσg v=29–34 and 2pσu v=0–2), whose binding energies are below 1e-4 a.u. and for which the finite exponential basis of Eqs. (2)–(4) is most likely to be incomplete. The authors should provide a convergence study (energies versus N, or a second independent basis set) and a benchmark against previously published H2+/HD+ values; without this, the asserted precision is unverified.
[VII, Tables IV–V] The transition frequencies quoted to 0.001 MHz are inconsistent with the stated 1e-12 a.u. numerical precision of the energy levels. Since 1e-12 a.u. is approximately 6.6 kHz, a difference of two such energies carries an uncertainty of order 10 kHz, so the last printed digits in entries such as v(2→34) = -176.232 MHz (Table IV) are not meaningful. Moreover, for this near-threshold transition the relative error implied by the 1e-12 a.u. precision is about 3.7e-5, far above the claimed 'relative theoretical error of about 10^-8' in Sec. VII. The 1e-8 figure can at best apply to transitions of order 10^6 MHz; the error budget for each quoted transition frequency and the range of validity of the 1e-8 statement should be clarified.
[VII, paragraph on uncertainties] The radiative correction is evaluated by replacing the state-specific Bethe logarithm β(L,v) with the hydrogen ground-state value β(1S)=2.9841, following Ref. [21]. This is an uncontrolled approximation whose per-state error is not quantified, yet the manuscript uses it to claim a final relative theoretical error of 10^-8. Since the α^3 lnα contribution is of order 1e-6 a.u., the resulting shifts can be comparable to the claimed precision for some transitions. The authors should either justify the 'at least two digits accuracy' assertion for the specific states considered or provide an explicit uncertainty estimate for the radiative correction in each tabulated transition.

minor comments (5)

[Table I and Sec. VII] The table heading 'gerade 1sσu states' is incorrect; the ground state of the hydrogen molecular ion is the 1sσg (gerade) state, not 1sσu. The same mislabeling appears in Sec. VII and in the discussion of Eq. (4), where the ungerade state is also called '2sσu' instead of '2pσu'.
[Tables IV and V] The headings '2πσu' should read '2pσu'.
[Sec. VII] There is a typo in the phrase 'transitions between the states of the same symmetry are fobbiden'; it should be 'forbidden'.
[Tables IV and V] The sign convention for the tabulated 'transition energy' is not defined; for example, v(2→34) is listed as a negative frequency, but it is not clear from the level energies whether the initial or final state is higher. A sentence defining ΔE = E_upper − E_lower, or the equivalent, would remove ambiguity.
[Sec. IV] The transition amplitudes in the Supplemental Material are computed with basis sets of N=3000–4000, but no convergence information is given for these matrix elements. A brief statement of the expected accuracy of the dipole moments would be helpful, especially for the unusually large values (e.g., d=33.89 a.u. in Table IV).

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the reported energies and transition data are obtained by solving the Coulomb three-body Schrödinger equation with a variational basis, and no target observable is fitted as an input.

full rationale

The paper's central derivation is a direct numerical solution of the nonrelativistic three-body Coulomb Hamiltonian, Eq. (1), using the variational exponential expansion of Eqs. (2)-(4). The variational parameters are internal computational quantities (pseudorandom exponents and optimization intervals), not physical constants adjusted to the tabulated energies or transition frequencies. The transition amplitudes, relativistic corrections, and hyperfine coefficients are all computed as expectation values over the resulting wavefunctions, using standard operators and CODATA22 fundamental constants. No energy level, transition frequency, or hyperfine splitting from experiment is used as input, and no fitted parameter is renamed as a prediction. The self-citations, notably [14] for the variational method and [21] for leading-order QED correction formulas, are methodological or formulaic references; they are not invoked as uniqueness theorems, nor do they define the target quantities in terms of themselves. The absence of an explicit convergence study for the highest near-threshold states is a numerical-completeness and uncertainty concern, but it is not a circularity. Accordingly, the derivation is self-contained with respect to the circularity criteria: the outputs are not equivalent to the inputs by construction.

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

No physical constants are fitted to measured spectra. The numerical basis parameters are computational tuning; if the variational calculation is converged, the final observables do not depend on them. The main assumptions are the three-body Coulomb Hamiltonian, the standard Breit-Pauli expansion, and the approximate treatment of radiative corrections with the hydrogen-atom Bethe logarithm.

free parameters (1)

Variational exponent intervals and pseudorandom seeds
Eq. (3) generates the basis exponents from optimized intervals [A1,A2], [A1',A2'] and prime seeds; these are numerical tuning parameters whose values are not reported. In a converged variational calculation the final energies do not depend on them, so they are not physical free parameters of the result.

assumptions (4)

domain assumption Nonrelativistic three-body Coulomb Hamiltonian (Eq. 1) governs the bound states; nuclear size and finite nuclear mass polarization are not included in H0 at this order.
Standard starting point for hydrogen molecular ion calculations; the paper solves H0 to obtain the tabulated nonrelativistic energies.
domain assumption Breit-Pauli Hamiltonian, Eqs. (7)-(9), gives the complete leading relativistic and hyperfine corrections.
Standard QED expansion for molecular hydrogen ions; cited to Ref [16] and used without modification.
ad hoc to paper The Bethe logarithm for a specific ro-vibrational state can be replaced by the hydrogen ground-state value β(1S)=2.9841 when estimating radiative corrections.
Section VII explicitly uses this substitution to claim about 1e-8 relative accuracy for total transition energies; it is accurate to about two digits but is an approximation.
domain assumption Deuteron and triton masses used in Eq. (1) are known input constants from CODATA or similar sources.
Energies depend on M1 and M2, but the paper does not list the exact mass values used, which limits exact reproduction.

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Cite this review

Pith. "Pith review of Variational studies of ro-vibrational spectra of DT$^+$ and T$_2^+$ ions." pith.science (2026). https://pith.science/paper/ILQ477LF

@misc{pith2026250712243,
  author       = {Pith},
  title        = {Pith review of: Variational studies of ro-vibrational spectra of DT$^+$ and T$_2^+$ ions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ILQ477LF}},
  note         = {Machine review of arXiv:2507.12243}
}

abstract

In this work we study the last two hydrogen molecular ion isotopologues: DT$^+$ and T$_2^+$, for which high-precision calculations have not yet been made. We obtain the nonrelativistic solutions of the Schr\"odinger equation for the wide range of vibrational states up to the highest possible vibrational $1s\sigma_g$ state which are of spectroscopic precision. Transition amplitudes for electric dipole transitions, leading-order relativistic corrections and coefficients of the effective hyperfine structure Hamiltonians for both DT$^+$ and T$_2^+$ are also calculated.

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Reference graph

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Pith tools

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

[1] [21]

Bakalov, V.I

D. Bakalov, V.I. Korobov and S. Schiller, Magnetic field effects in the transitions of the HD + molecular ion and precision spectroscopy. J. Phys. B: At. Mol. Opt. Phys.44, 025003 (2011). 6

work page 2011

[2] [1]

Parametersβ i andγ i are obtained in a similar way

+A ′ 1],(3) ⌊x⌋designates the fractional part ofx,p α andq α are some prime numbers, [A 1, A2] and [A ′ 1, A′ 2] are real variational intervals which need to be optimized. Parametersβ i andγ i are obtained in a similar way. When ”gerade” 1sσ u weakly bound and ”ungerade” 2pσ u vibrational states in T + 2 molecular ion are consid- ered, we use other coordi...

work page

[3] [2]

J.-Ph. Karr, S. Schiller, V.I. Korobov, S. Alighanbari, Determination of a set of fundamental constants from molec- ular hydrogen ion spectroscopy: a modeling study. arXiv:2505.05615, submitted to PRL

arXiv

[4] [3]

Alighanbari, G.S

S. Alighanbari, G.S. Giri, F.L. Constantin, V.I. Korobov, and S. Schiller, Precise test of quantum electrodynamics and determination of fundamental constants with HD + ions, Nature581, 152 (2020)

work page 2020

[5] [4]

Patra, M

S. Patra, M. Germann, J.-Ph. Karr, M. Haidar, L. Hilico, V.I. Korobov, F.M.J. Cozijn, K.S.E. Eikema, W. Ubachs, and J.C.J. Koelemeij, Proton-electron mass ratio from laser spectroscopy of HD + at the part-per-trillion level, Science369, 1238 (2020)

work page 2020

[6] [5]

Kortunov, S

I. Kortunov, S. Alighanbari, M.G. Hansen, G.S. Giri, S. Schiller, and V.I. Korobov, Proton-electron mass ratio by high-resolution optical spectroscopy of ion ensemble in the resolved-carrier regime. Nature Phys.17, 569 (2021)

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[7] [6]

Alighanbari, M

S. Alighanbari, M. R. Schenkel, V. I. Korobov, S. Schiller, High-accuracy laser spectroscopy of H + 2 and the proton- electron mass ratio. to appear in Nature

work page

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Epelbaum, H.-W

E. Epelbaum, H.-W. Hammer, U.-G. Meißner, Modern theory of nuclear forces, Rev. Mod. Phys.81, 1773 (2009)

work page 2009