{"id":"a68b91b0-8013-44b7-8520-02cca6e1f95c","arxiv_id":"2411.14787","paper_version":4,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"A single parameter built from scattering lengths, hyperfine splitting, and van der Waals energy bins classifies the spin structure of weakly bound alkali dimers, and reduced potentials reproduce the spin properties.","lead":"This paper builds simplified versions of the interactions between pairs of ultracold alkali atoms and shows that they still describe how the molecules are bound and what their spin structure looks like. It also introduces one number per atomic species that tells whether the molecular spin is set by electron exchange or by hyperfine coupling.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Central classification rests on an ad hoc hyperfine cap and is validated only against the fitted reduced potentials, so the claimed universality of xi_ex is not yet quantitatively established.","rationale":"The paper is a solid, honest computational study. The reduced potentials convincingly reproduce a(B), r_e(B), bound-level energies, and spin-structure details such as the 85Rb2 avoided crossings, and the comparison in Fig. 6 against BO potentials for 7Li2 scattering spin components is a genuine independent check. However, the central classification claim goes beyond the fitted regime: xi_ex is proposed as a predictor of spin structure for each species, and its only quantitative validation is through gamma_ex computed from the same reduced potentials that were fitted to the low-energy parameters appearing in xi_ex. The reader's weakest_assumption identifies exactly this circularity and the ad hoc cap in Eq. (13); I agree. The reason the verdict stays CONDITIONAL rather than UNCHANGED is that the unverified universality is central to the paper's headline claim as stated ('this parameter can be used to classify the spin structure of vdW molecules for each atomic species'). The proposed concrete test - recomputing gamma_ex directly from BO wavefunctions and checking a same-parameters/different-short-range synthetic pair - would either confirm the universality or reveal that the classification is an artifact of the reduced-potential construction.","tokens_in":21266,"tokens_out":1706,"duration_ms":15433,"concrete_test":"Compute gamma_ex directly from the full Born-Oppenheimer coupled-channel wavefunctions (not from the reduced potentials) for all seven species, using the same N states and binding-energy window, and compare the resulting ordering with the xi_ex ordering from Table III and Fig. 10. In addition, test a synthetic species with the same (a_s, a_t, A_hf, r_vdW) but a different short-range singlet-triplet potential shape (e.g., modify the van der Waals C_8/C_10 coefficients or add a short-range barrier while keeping the scattering lengths fixed); if gamma_ex changes materially, the claimed universality of xi_ex fails.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The paper's central claim is that the dimensionless parameter xi_ex (Eq. 14) classifies the spin structure of vdW molecules across alkali species from a_s, a_t, A_hf, and binding energy. The load-bearing assumption is that weakly bound vdW spin structure is a universal function of these low-energy parameters, so that short-range details do not matter. This underlies the effective spin-exchange estimate tilde-E_ex = u_st E_bin (Eq. 12) and, especially, the effective hyperfine estimate tilde-E_hf = min(E_hf, E_bin/2) (Eq. 13). The cap at E_bin/2 is explicitly introduced as an upper bound ('we will utilize this upper bound for a rough characterization'), but there is no derivation showing that the hyperfine-induced level shift is actually bounded this way or that the resulting xi_ex tracks the true spin mixing. The validation (Table III, Fig. 10) is qualitative and circular in an important respect: the gamma_ex values used for comparison are computed from the reduced potentials, which were themselves fitted (via lambda_s, lambda_t, c_hf) to reproduce the same low-energy parameters that enter xi_ex. No independent test against the full BO potentials is shown for gamma_ex itself. Thus the ordering agreement in Fig. 10 could reflect the fitting procedure rather than a universal classification. The concern is not that the model is wrong, but that the central quantitative claim ('xi_ex can be used to classify the spin structure') is supported only by an internal consistency check.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies weakly bound van der Waals dimers of two identical bosonic alkali atoms (7Li through 133Cs). For each species it constructs reduced singlet and triplet potentials V*_S = V_S + C6 λ_S^6/r^12, with parameters tuned to reproduce the singlet and triplet scattering lengths and, after fine-tuning λ* and a hyperfine rescaling factor c_hf, the field-dependent scattering length a(B) and effective range r_e(B) over ±200 G (restricted to ±20 G for 133Cs). The reduced potentials are shown to reproduce near-threshold s- and d-wave bound levels, Feshbach-resonance features, and the spin structure of scattering and bound states, with the authors disclosing known failures (deep levels, high partial waves, and the 133Cs d-wave resonances at 31 and 33 G). At zero field the paper defines a spin-mixing parameter γ_ex from the average spin purity of all reduced-potential bound states, and introduces the dimensionless classification parameter ξ_ex = u_st E_bin / min(E_hf, E_bin/2), built from the singlet/triplet scattering lengths, the hyperfine constant, and the vdW energy-bin size. It shows that the ordering of ξ_ex across species and bins is consistent with γ_ex: exchange-dominated species (7Li, 23Na, 39K, 41K), hyperfine-dominated (85Rb, 87Rb), and intermediate (133Cs).","tokens_in":21666,"tokens_out":21525,"duration_ms":201797,"significance":"The reduced-potential construction is a genuinely useful, carefully benchmarked tool: the numerical comparisons are extensive, all tuning parameters are tabulated with sufficient precision for reproduction, and the authors are explicit about where the model fails (deep levels, large l, and the 133Cs d-wave regime). The paper also provides an independent validation of the effective exchange splitting: Fig. 9 compares Ẽ_ex with actual BO-potential singlet–triplet separations over six energy bins. If the ξ_ex classification holds, it is a novel and cheap predictor of molecular spin character from a_s, a_t, and A_hf alone, with direct relevance to the spin-propensity rules observed in ultracold three-body recombination, and it would identify the exchange-versus-hyperfine competition as the mechanism behind species-dependent spin structure. It is to the paper's credit that ξ_ex itself is not fitted to γ_ex, and that Eq. (13) is honestly described as a rough characterization; nonetheless, as detailed below, the classification claim currently rests on an internal consistency check rather than on an independent validation against the full Born-Oppenheimer physics.","major_comments":[{"comment":"The classification claim rests on the effective hyperfine energy Ẽ_hf = min(E_hf, E_bin/2) in Eq. (13), which the text introduces as a rough upper bound without derivation. This cap is load-bearing rather than cosmetic: for 133Cs, using E_hf instead of Ẽ_hf would change ξ_ex^(1) from 0.277 to about 0.0016, converting the paper's borderline 'mixed' example into a strongly hyperfine-dominated prediction, while for 23Na, 39K, 85Rb, and 87Rb the cap changes ξ by factors of roughly 1.05 to 57. Unlike Ẽ_ex, which is validated against actual BO singlet–triplet splittings in Fig. 9, no validation of Eq. (13) against actual hyperfine-induced level shifts is presented, even though the same machinery would produce it (for instance, by comparing BO molecular levels computed with A_hf = 0 and with the physical A_hf, bin by bin). I ask for this check to be added before the classification claim is accepted.","section":"Sec. IV.B–IV.C, Eqs. (13)–(14)"},{"comment":"The γ_ex values in Table III are computed from reduced-potential molecular states only (N ≈ 80–200 states, all partial waves up to l ≈ 20, binding energies up to several thousand E_vdW), and the authors note in a parenthetical that an analogous BO-based computation 'could be performed' but do not show it. The deep and high-l states that enter this average are precisely the regime where the reduced model's fidelity is weakest, as acknowledged in Secs. III.B–III.C (perceptible energy deviations for deeper levels; about 7 G shift of the d-wave avoided crossing in Fig. 7). At the same time, ξ_ex is evaluated from a_s, a_t, and A_hf — the same low-energy inputs used to fix λ_s, λ_t, and c_hf of the reduced potentials in Table II. The agreement in Fig. 10 is therefore an internal consistency check between two quantities that both derive from the same reduced model, rather than an independent test of the claim that weakly bound spin structure is a universal function of a_s, a_t, A_hf, and the bin size. I stress that the check is not strictly circular, since ξ_ex is not fitted to γ_ex and γ_ex is computed from multichannel wavefunctions; nevertheless, the missing external anchor is real. I request either (i) γ_ex computed from the BO potentials within the same binding-energy window, or (ii) a robustness test showing that a second reduced-potential realization with the same a_s, a_t, and A_hf but different n_s/n_t yields essentially the same γ_ex.","section":"Sec. IV, Eqs. (9)–(10), Table III"},{"comment":"For 85Rb, 87Rb, and 133Cs the reduced model rescales the hyperfine constant by c_hf = 0.9477, 0.9026, and 0.8710 to fit a(B) and r_e(B), yet ξ_ex in Table III is evaluated with the physical A_hf from Ref. [44]. The γ_ex values used to validate ξ_ex are therefore generated with a systematically different hyperfine interaction for the very species that define the hyperfine-dominated and intermediate regimes, and the 13% rescaling for 133Cs is not negligible compared with the margin by which Cs is classified as mixed (γ_ex = 0.641). The authors should either recompute γ_ex with the physical A_hf or demonstrate that the c_hf rescaling changes γ_ex by an amount too small to affect the qualitative ordering.","section":"Sec. III.A, Table II, and Sec. IV.C"}],"minor_comments":[{"comment":"The abstract's claim that ξ_ex 'can be used to classify the spin structure of vdW molecules for each atomic species' is stronger than what is shown: ξ_ex^(i) is bin-dependent, and for 23Na and 41K the first-bin values (0.619 and 0.394) lie in the intermediate range while their averaged γ_ex values (47.1 and 29.5) are firmly in the exchange-dominated class; I suggest stating explicitly that the classification is per energy bin (or per bin range), as the text itself explains.","section":"Abstract and Table III"},{"comment":"The statement that Ẽ_ex is a 'fairly good approximation' to ΔE_st is not quantified; reporting the mean and maximum relative deviation per bin (or a correlation measure) would make the validation robust, since the figure spans four orders of magnitude in both axes.","section":"Fig. 9"},{"comment":"The threshold σ_ex(r_ex) = 10 that defines the vdW-molecule regime is arbitrary; a brief statement of how r_ex and E_max^S shift for a different threshold (for example, 5 or 20) would indicate how robust the chosen binding-energy window is.","section":"Sec. II.A, Eq. (5)"},{"comment":"In the sentence describing 133Cs, 'ξex ∼ 1 and γex ∼ 1' is looser than the tabulated values (0.277 and 0.641); since the classification is intended to be qualitative, a phrase such as 'of order unity or below' would avoid inviting a quantitative reading.","section":"Sec. IV.C"},{"comment":"The brief remark that relating ξ_ex to Hund's cases 'merits further investigation' could be made more useful by adding one or two sentences on where the ξ_ex ≫ 1 and ξ_ex ≪ 1 limits plausibly sit in the standard Hund's-case classification of alkali dimers.","section":"Sec. IV.C, conclusion"}],"recommendation":"major_revision","confidential_remarks":"The two requested additions — an independent BO-based computation of γ_ex and a validation of the Eq. (13) hyperfine cap against actual hyperfine-induced level shifts — are natural extensions of computations the authors already perform (Figs. 5, 7, 9) and should be feasible within a revision. If the BO-based γ_ex were to disagree with Table III, the central classification claim would need to be correspondingly weakened, a possibility the paper's cautious conclusion partly anticipates. No concerns about novelty disclosure or citation patterns; the manuscript fits the journal's scope well."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper does two things. First, it constructs reduced singlet/triplet potentials for seven bosonic alkali dimers and shows, in detail, that they reproduce a(B), re(B), and the spin structure of near-threshold scattering and bound states without supporting the hundreds of bound states of the real potentials. That part is solid, carefully done, and practically valuable for three-body simulations. Second, it introduces xi_ex, a dimensionless parameter built from as, at, Ahf, and the vdW energy-bin size, and shows that it orders the species by molecular spin character (fafb vs [SI] purity) in agreement with the numerical gamma_ex ordering. The qualitative pattern is real: Li/Na/K sit on the strong-exchange side, Rb on the weak side, Cs in between, and xi_ex tracks that ordering across several orders of magnitude. That is a genuinely new contribution, not just an incremental extension.\n\nThe paper also earns credit for honesty. It openly reports where the reduced model fails: deep levels, d-wave states, and the Cs Feshbach resonances that require dipole-dipole coupling. The check of the effective exchange estimate against actual singlet-triplet splittings in Fig. 9 is a good, non-circular validation of that piece.\n\nThe soft spots are exactly where the stress-test puts them. The cap at E_bin/2 in Eq. (13) is an ad hoc upper bound, and the validation of xi_ex against gamma_ex is partly circular: gamma_ex is computed from the reduced potentials, which were fitted to the same low-energy parameters that feed xi_ex. There is no independent BO-level computation of gamma_ex for the full spectrum. So the claim that xi_ex can classify spin structure is supported as a qualitative ordering, but not as a quantitative predictor. The authors do call it a rough characterization, which softens the problem, but the abstract says \"can be used to classify\" without that caveat. The absence of deposited code or data is also a real but minor issue; the method is described clearly enough to reproduce with effort.\n\nNone of this is fatal. The central physical picture, that spin structure is governed by the competition between exchange and hyperfine energies, holds up. The paper deserves a serious referee. My recommendation: send it to peer review, and ask the authors for two things: a direct comparison of gamma_ex from reduced versus full BO potentials, and either a derivation or a systematic numerical test of the E_bin/2 hyperfine cap. If those come out clean, the paper becomes a standard reference for few-body modelers.","headline":"A practically useful reduced-potential toolkit with a genuinely new but only qualitatively validated classification parameter; the quantitative universality claim needs a direct independent check.","tokens_in":682,"tokens_out":767,"would_cite":true,"duration_ms":42200,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"The spin structure of weakly bound alkali dimers is set by a single dimensionless exchange-to-hyperfine ratio, and this paper shows that the ratio classifies all seven bosonic alkali species.","keywords":["van der Waals molecules","alkali dimers","spin structure","reduced interaction potentials","hyperfine interactions","spin-exchange interaction","Feshbach resonances","energy bins"],"falsifier":"Calculate the zero-field spin-mixing parameter $\\gamma_{\\rm ex}$ for $^{133}$Cs$_2$ using the full Born-Oppenheimer potentials including the magnetic dipole-dipole interaction, and compare it with the reduced-potential value in Table III; a difference substantially larger than the few-percent agreement found for the other species would show that the $\\xi_{\\rm ex}$ classification fails in the intermediate regime it is meant to capture.","tokens_in":21064,"feed_emoji":"⚛️","tokens_out":13225,"duration_ms":114404,"temperature":0.7,"pith_summary":"This paper aims to establish that, at zero magnetic field, the spin structure of weakly bound diatomic van der Waals molecules formed by two identical bosonic alkali atoms is governed by the competition between electronic spin exchange and hyperfine interactions, and that a single dimensionless parameter $\\xi_{\\rm ex}$---built from the singlet and triplet scattering lengths, the atomic hyperfine constant, and the universal size of the van der Waals energy bin---orders all seven species. It also constructs reduced interaction potentials that keep only 5--7 $s$-wave bound states yet reproduce the scattering length, effective range, binding energies, Feshbach resonances, and spin structure computed with the full Born-Oppenheimer potentials. If these claims hold, $\\xi_{\\rm ex}$ becomes a cheap predictor of molecular spin purity, and the reduced potentials become a practical stand-in for full coupled-channel calculations in three-body reaction studies.","feed_headline":"One ratio predicts spin structure of ultracold alkali molecules","feed_subtitle":"The exchange-versus-hyperfine ratio orders seven alkali species from atomic-spin Rb to molecular-spin Li.","key_machinery":"The load-bearing machinery is the reduced potential $V_S^*(r)=V_S(r)+C_6\\lambda_S^6/r^{12}$, which adds a tunable $1/r^{12}$ repulsion to the real Born-Oppenheimer singlet and triplet potentials so that only 5--7 $s$-wave bound states survive, with $\\lambda_S$ fixed by matching $a_s$ and $a_t$ and then fine-tuned (at most about 2%) against $a(B)$ and $r_e(B)$. The classification uses the universal vdW energy bins---fixed intervals of the near-threshold spectrum labeled by the vdW length and energy---together with the quantum-defect function $u(a)=\\tan^{-1}[\\bar a/(a-\\bar a)]/\\pi$, which gives the position of an $s$-wave level inside its bin. From these, $u_{st}=\\min(|u(a_s)-u(a_t)|,1-|u(a_s)-u(a_t)|)$ measures the effective exchange splitting, $\\tilde E_{\\rm hf}=\\min(E_{\\rm hf},E_{\\rm bin}/2)$ caps the effective hyperfine energy, and their ratio $\\xi_{\\rm ex}=\\tilde E_{\\rm ex}/\\tilde E_{\\rm hf}$ is the classification parameter. The numerical benchmark is $\\gamma_{\\rm ex}$, computed from Eq.~(9) as a ratio of averaged spin purities in the two bases.","core_discovery":"At zero magnetic field, the spin composition of a weakly bound alkali vdW molecule is set by which interaction wins: electronic spin exchange favors molecular states of definite total electronic and nuclear spin ($[SI]$ basis), while hyperfine coupling favors states of definite atomic hyperfine spins ($(f_a f_b)$ basis). The paper quantifies this with a spin-mixing parameter $\\gamma_{\\rm ex}$ obtained from the average spin purity of 80--200 molecular levels in the two bases, finding $\\gamma_{\\rm ex}\\gg 1$ for $^7$Li$_2$, $^{23}$Na$_2$, $^{39}$K$_2$, and $^{41}$K$_2$; $\\gamma_{\\rm ex}\\ll 1$ for $^{85}$Rb$_2$ and $^{87}$Rb$_2$; and $\\gamma_{\\rm ex}\\sim 1$ for $^{133}$Cs$_2$. Its central claim is that this ordering is qualitatively captured by the exchange parameter $\\xi_{\\rm ex}=u_{st}E_{\\rm bin}/\\min(E_{\\rm hf},E_{\\rm bin}/2)$, where $u_{st}$ measures the separation of singlet and triplet levels within the universal vdW energy bins and $E_{\\rm hf}$ is the atomic hyperfine splitting; no short-range potential details beyond the scattering lengths and hyperfine constant are needed. The same reduced potentials that underlie this analysis reproduce, for most species over $\\pm 200$ G, the scattering length, effective range, molecular binding energies, and the spin structure of scattering and bound states, making them a practical surrogate for full Born-Oppenheimer calculations.","pith_inferences":["Extending the same ratio construction to heteronuclear alkali dimers, replacing the two scattering lengths and the two hyperfine constants with the combined species values, is a natural testable step that the authors do not take; a generalized $\\xi_{\\rm ex}$ would predict where heteronuclear molecules fall between the homonuclear classes.","At finite magnetic field the Zeeman energy enters as a third competing scale, so one may expect a field-dependent generalization $\\xi_{\\rm ex}(B)$ to predict where spin purity changes and where Feshbach resonances alter the molecular character; the paper only notes this extension as future work.","The empirical success of matching only $a(B)$ and $r_e(B)$ to build potentials suggests a constructive recipe for other long-range-interacting systems, such as polar or Rydberg dimers, where the short-range potential is hard to compute but low-energy scattering parameters are known.","Because $\\xi_{\\rm ex}$ uses only experimentally measurable quantities, the classification could be checked without molecular spectroscopy: measure the scattering lengths by Feshbach spectroscopy, use the known hyperfine constants, and compare the predicted ordering with state-resolved product distributions from three-body recombination."],"forward_implications":["For any alkali species, the spin purity of weakly bound vdW molecules can be predicted from just the singlet and triplet scattering lengths and the hyperfine constant, without solving the full coupled-channel problem.","Three-body recombination simulations can use the shallow reduced potentials instead of deep Born-Oppenheimer potentials, cutting the numerical burden while preserving Feshbach resonances, binding energies, and spin propensity rules.","The $\\xi_{\\rm ex}$ ordering gives a rule of thumb for which spin basis to use when interpreting spectra or reactions: the atomic $(f_a f_b)$ basis for rubidium, the molecular $[SI]$ basis for lithium, sodium, and potassium, and a mixed description for cesium.","Because $\\xi_{\\rm ex}$ grows with energy-bin index for the lighter species, deeper vdW levels of lithium, sodium, and potassium are predicted to become increasingly $[SI]$-dominated, whereas rubidium and cesium levels stay in the same spin regime across bins.","The reduced potentials support bound states up to $l=20$ and reproduce $s$- and $d$-wave avoided crossings, so near-threshold multichannel physics in these dimers is largely insensitive to short-range potential details."],"supporting_citations":[{"why":"Supplies the singlet and triplet Born-Oppenheimer potentials whose scattering lengths define the target values for the reduced-model construction.","marker":"[33]"},{"why":"Provides Feshbach-spectroscopy interaction potentials for sodium used to fix its scattering lengths.","marker":"[34]"},{"why":"Supplies the beyond-Born-Oppenheimer interaction potentials for potassium that set the potassium scattering lengths.","marker":"[35]"},{"why":"Establishes the hyperfine, rotational, and vibrational structure of the $a\\,^3\\Sigma_u^+$ state of $^{87}$Rb$_2$, supporting the hyperfine treatment used in the model.","marker":"[36]"},{"why":"Gives coupled-channel potentials and Feshbach resonances for cesium, including the dipole-dipole d-wave resonances that the reduced model must bypass.","marker":"[37]"},{"why":"Provides the experimental atomic hyperfine constants used as input to the effective hyperfine energy.","marker":"[44]"},{"why":"Defines the vdW length and energy scales and the energy-bin structure that underlie $E_{\\rm bin}$ and the reduced-potential design.","marker":"[45]"},{"why":"Derives the universal energy-bin quantization for van der Waals interactions that justifies placing levels via the function $u(a)$.","marker":"[49]"},{"why":"Supplies the WKB formula for $u(a)$ used to locate an s-wave level within its van der Waals energy bin.","marker":"[57]"},{"why":"Gives the mean scattering length $\\bar a \\approx 0.9560\\,r_{\\rm vdW}$ that enters the definition of $u(a)$.","marker":"[58]"}],"fun_headline_variants":["One exchange parameter ranks spin structure of alkali vdW molecules","Spin of alkali vdW molecules set by exchange vs hyperfine ratio","Single ratio predicts molecular spin character across alkali species","Alkali molecule spin from exchange parameter across seven species"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the zero-field spin structure of these weakly bound long-range molecules depends only on the singlet and triplet scattering lengths, the atomic hyperfine energy, and the universal vdW energy-bin size, so that species-specific short-range effects such as the dipole-dipole interactions in cesium do not change the spin purity; if such details matter, the classification fails in the intermediate regime it is meant to describe.","fun_headline_variants_meta":{"raw":{"variants":["One exchange parameter ranks spin structure of alkali vdW molecules","Spin of alkali vdW molecules set by exchange vs hyperfine ratio","Single ratio predicts molecular spin character across alkali species","Alkali molecule spin from exchange parameter across seven species"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000532,"raw_usage":{"total_tokens":2622,"prompt_tokens":1065,"completion_tokens":1557,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":681,"completion_tokens_details":{"reasoning_tokens":1488}},"tokens_in":681,"tokens_out":1557,"duration_ms":11361,"temperature":1.0,"reasoning_tokens":1488,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T14:53:40.850298+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Calculate the zero-field spin-mixing parameter $\\gamma_{\\rm ex}$ for $^{133}$Cs$_2$ using the full Born-Oppenheimer potentials including the magnetic dipole-dipole interaction, and compare it with the reduced-potential value in Table III; a difference substantially larger than the few-percent agreement found for the other species would show that the $\\xi_{\\rm ex}$ classification fails in the intermediate regime it is meant to capture.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the singlet and triplet Born-Oppenheimer potentials whose scattering lengths define the target values for the reduced-model construction."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides Feshbach-spectroscopy interaction potentials for sodium used to fix its scattering lengths."},{"cited_title":"Tiemann, P","cited_arxiv_id":null,"evidence_quote":"Supplies the beyond-Born-Oppenheimer interaction potentials for potassium that set the potassium scattering lengths."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the hyperfine, rotational, and vibrational structure of the $a\\,^3\\Sigma_u^+$ state of $^{87}$Rb$_2$, supporting the hyperfine treatment used in the model."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives coupled-channel potentials and Feshbach resonances for cesium, including the dipole-dipole d-wave resonances that the reduced model must bypass."},{"cited_title":"Tempest and S","cited_arxiv_id":null,"evidence_quote":"Defines the vdW length and energy scales and the energy-bin structure that underlie $E_{\\rm bin}$ and the reduced-potential design."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the WKB formula for $u(a)$ used to locate an s-wave level within its van der Waals energy bin."},{"cited_title":"Haze, J.-L","cited_arxiv_id":null,"evidence_quote":"Gives the mean scattering length $\\bar a \\approx 0.9560\\,r_{\\rm vdW}$ that enters the definition of $u(a)$."}],"review_version":1}