{"id":"9fcbd39f-00b4-42d0-819d-381c64715f53","arxiv_id":"2507.12243","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"The last two hydrogen molecular ion isotopologues, DT+ and T2+, receive high-precision variational ro-vibrational energies, dipole transition amplitudes, relativistic corrections, and hyperfine coefficients.","lead":"This paper computes highly precise energy levels, transition strengths, and spin interaction coefficients for the last two tritium-containing hydrogen molecular ions, DT+ and T2+. The numbers give experimenters the theoretical roadmap for laser spectroscopy that could determine the triton charge radius and test fundamental physics.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Near-threshold T2+ states lack convergence evidence, and the stated 1e-12 a.u. precision cannot support the kHz-level transition frequencies claimed for them.","rationale":"The reader's weakest assumption — that the variational basis is converged for every tabulated state, especially the weakly bound near-threshold levels — is exactly the right place to look. I agree with that concern. My read sharpens it with a quantitative, text-internal inconsistency: the paper's own stated numerical precision of 10^-12 a.u. converts to about 6.6 kHz in frequency, which is larger than the 0.001 MHz (1 kHz) digits printed in Tables IV–V and far larger than the 10^-8 relative error claimed for small near-threshold transition energies. This is not a disagreement with the field's consensus; it is a checkable mismatch between the uncertainty budget stated in Sec. VII and the significance of the published numbers. The absence of any convergence data or comparison with established H2+/HD+ calculations means the reader cannot tell whether the actual eigenvalues are good to 10^-15 a.u. (which would support the quoted digits) or only to the asserted 10^-12 a.u. (which would not). The proposed test — an enlarged-basis rerun with a second exponent set and full-precision output — would settle this directly. I do not see a reason to move the verdict away from CONDITIONAL: the paper is a competent application of a well-established variational method, and the requested convergence evidence could plausibly resolve the concern. But without that evidence, the near-threshold transition frequencies and the 1e-8 accuracy claim for them are not supported by the manuscript as written.","tokens_in":19749,"tokens_out":12434,"duration_ms":137067,"concrete_test":"Recompute the near-threshold T2+ states 1sσg v=34, L=0 and 2pσu v=1,2, L=0 with an enlarged basis (N=16000) and with a second, independently generated pseudorandom exponent set, reporting energies to at least 16 digits after the decimal. If any energy shifts by more than 5e-13 a.u. relative to Tables I–III, or if the implied v(2→34) transition frequency moves by more than 0.001 MHz from -176.232 MHz, the claimed 12-digit convergence is not established and the quoted near-threshold transition frequencies and 1e-8 relative error must be revised.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that the nonrelativistic energies in Tables I–III have about 10^-12 a.u. numerical precision (Sec. VII). This is ~6.6 kHz in frequency units, yet Tables IV–V quote transition energies to 0.001 MHz, including the near-threshold T2+ transition v(2→34) at -176.232 MHz, and the text claims a final relative theoretical error near 10^-8. For that transition, even the printed 12-decimal energies give rounding errors of about 3.3 kHz per level, i.e. ~6.6 kHz in the difference, a relative error of ~3.7e-5. The 1e-8 claim is therefore not supported by the stated numerical precision. The problem is compounded by the absence of any convergence study: the weakly bound 1sσg states with v≥29 and the 2pσu states with v=1,2 in T2+ lie within ~1e-4 a.u. of the T(n=1)+T+ threshold, exactly where the finite exponential basis of Eq. (2) is most likely to be incomplete. An outer-range truncation or an insufficient density of small exponents could shift these binding energies at the kHz level while leaving the total energy apparently stable to 12 decimals. No basis-size dependence, no second exponent-set comparison, and no validation against the known H2+/HD+ results is provided, so the asserted 12-significant-digit convergence for every tabulated state is unverified.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":20064,"tokens_out":6607,"duration_ms":74380,"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":[{"comment":"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.","section":"VII, Tables I–III"},{"comment":"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.","section":"VII, Tables IV–V"},{"comment":"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.","section":"VII, paragraph on uncertainties"}],"minor_comments":[{"comment":"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'.","section":"Table I and Sec. VII"},{"comment":"The headings '2πσu' should read '2pσu'.","section":"Tables IV and V"},{"comment":"There is a typo in the phrase 'transitions between the states of the same symmetry are fobbiden'; it should be 'forbidden'.","section":"Sec. VII"},{"comment":"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.","section":"Tables IV and V"},{"comment":"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).","section":"Sec. IV"}],"recommendation":"major_revision","confidential_remarks":"The paper presents potentially valuable data, but the lack of any convergence study for the near-threshold states is a serious gap for a precision-spectroscopy claim. The authors have the tools and expertise to provide such a study, so a major revision is appropriate rather than rejection. I would also encourage the editor to ask the authors to state explicitly which entries in Tables IV–V are affected by the numerical-precision inconsistency, and to revise the abstract's 'spectroscopic precision' claim to specify the range of transitions for which it holds."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThat 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.\n\nThe 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.\n\nThe 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.","headline":"Useful completion of the HMI isotopologue set, but the near-threshold T2+ states need convergence evidence before the digits enter metrology.","tokens_in":20561,"tokens_out":3900,"would_cite":true,"duration_ms":48868,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["DT+","T2+","hydrogen molecular ions","rovibrational spectra","variational calculation","nonrelativistic energy","hyperfine structure","transition amplitudes"],"falsifier":"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.","tokens_in":19566,"feed_emoji":"⚛️","tokens_out":5951,"duration_ms":63197,"temperature":0.7,"pith_summary":"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.","feed_headline":"DT+ and T2+ spectra now reach twelve digits","feed_subtitle":"All bound rovibrational states are covered, readying tritium ions for QED tests and charge-radius work.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the exponential variational expansion with randomly chosen exponents that generates all wave functions and energies in the paper.","marker":"[14]"},{"why":"Supplies the Breit-Pauli Hamiltonian used for the leading relativistic and spin-dependent corrections.","marker":"[16]"},{"why":"Supplies the CODATA recommended values used for conversion constants, nuclear magnetic moments, and the Rydberg frequency.","marker":"[17]"},{"why":"Supplies the deuteron quadrupole moment used in the DT+ hyperfine structure Hamiltonian.","marker":"[19]"},{"why":"Supplies the formulas and hydrogen ground-state Bethe logarithm used to estimate the leading radiative correction and the overall 10^-8 relative accuracy.","marker":"[21]"}],"fun_headline_variants":["DT+ and T2+ spectra: all bound states at 10^-12","Last hydrogen ions DT+ and T2+ fully solved","Complete ro-vibrational spectra for DT+ and T2+","Variational precision for tritium molecular ions","DT+ and T2+: high-precision ro-vibrational data"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["DT+ and T2+ spectra: all bound states at 10^-12","Last hydrogen ions DT+ and T2+ fully solved","Complete ro-vibrational spectra for DT+ and T2+","Variational precision for tritium molecular ions","DT+ and T2+: high-precision ro-vibrational data"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000661,"raw_usage":{"total_tokens":2947,"prompt_tokens":799,"completion_tokens":2148,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":415,"completion_tokens_details":{"reasoning_tokens":2059}},"tokens_in":415,"tokens_out":2148,"duration_ms":18956,"temperature":1.0,"reasoning_tokens":2059,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T16:51:21.220038+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Varshalovich, A.N","cited_arxiv_id":null,"evidence_quote":"Supplies the deuteron quadrupole moment used in the DT+ hyperfine structure Hamiltonian."},{"cited_title":"Delaunay, J.-P","cited_arxiv_id":null,"evidence_quote":"Supplies the exponential variational expansion with randomly chosen exponents that generates all wave functions and energies in the paper."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Breit-Pauli Hamiltonian used for the leading relativistic and spin-dependent corrections."},{"cited_title":"Bethe and E.E","cited_arxiv_id":null,"evidence_quote":"Supplies the CODATA recommended values used for conversion constants, nuclear magnetic moments, and the Rydberg frequency."},{"cited_title":"Bakalov, V.I","cited_arxiv_id":null,"evidence_quote":"Supplies the formulas and hydrogen ground-state Bethe logarithm used to estimate the leading radiative correction and the overall 10^-8 relative accuracy."}],"review_version":1}