Improving the predictive power of empirical shell-model Hamiltonians

B. A. Brown; B. C. He; J. A. Purcell; S. R. Stroberg; W. B. Walters

REVIEW 3 major objections 6 minor 1 cited by

Improving the predictive power of empirical shell-model Hamiltonians

T0 review · 3 major / 6 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read A better ab initio starting Hamiltonian and a train/test validation protocol yield the p35-i3 shell-model Hamiltonian, which reproduces $\pi j4$ spectra to 100–130 keV RMSD and extrapolates to neutron-rich nuclei with $Z \leq 36$ at…

desk verdict Useful demonstration that the IMSRG(3f2) starting point plus a train/test-guided SVD cutoff improves shell-model calibration in a sparse region, but the quoted 100–130 keV predictive error is optimistic because the same validation curve selects the cutoff. read the letter →

arxiv 2412.09917 v2 pith:LIJBQ2IZ submitted 2024-12-13 nucl-th

J. A. Purcell , B. A. Brown , B. C. He , S. R. Stroberg , W. B. Walters This is my paper · ORCID

classification nucl-th

keywords shellmodelempiricalHamiltoniansIMSRGsingularvaluedecompositionoverfittingnuclearspectraexoticisotopesπj4space

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 tries to show that empirical shell-model Hamiltonians can be made genuinely predictive even when calibration data are sparse. The authors combine a more accurate ab initio starting Hamiltonian, obtained with the IMSRG(3f2) approximation, with a training-and-testing protocol that chooses how many Hamiltonian parameter combinations to vary. Fitting to proton configurations in the $\pi j4$ model space between 78Ni and 100Sn, they obtain the p35-i3 Hamiltonian, which reproduces known spectra to a 100–130 keV root-mean-square deviation and extrapolates to neutron-rich nuclei with $Z \leq 36$ to about 300 keV. These tools matter because nuclei relevant to rare-isotope facilities and the r-process sit in model spaces where calibration data are far sparser than in the well-studied sd shell.

What carries the argument

The carrying mechanism is the singular-value decomposition (SVD) fit. The fit matrix $G$ — the symmetric product of wavefunction overlaps weighted by inverse data variances — is diagonalized to produce uncorrelated linear combinations of the Hamiltonian parameters, each with an error $d_i$; only combinations with error below a cutoff are varied, while the rest retain their ab initio values. The number of varied linear combinations (VLC) is chosen from the minimum of the validation RMSD curve obtained by randomly splitting known energies into 80% training and 20% testing sets, repeated 2000 times. The improved starting point is the IMSRG(3f2) factorization, which incorporates intermediate three-body operators in nested commutators at the same computational cost as the earlier IMSRG(2). The final product is the p35-i3 Hamiltonian with 35 varied linear combinations.

What would settle it

Measure the 100Sn binding energy and 99In excited states, whose values p35-i3 predicts; if the deviations exceed the roughly 300 keV found in the $Z \leq 36$ extrapolation test, the paper's claim that the validation protocol yields reliable extrapolation would be falsified.

Watch

Extended reading notes

Core claim

The central claim is that the p35-i3 Hamiltonian—built by averaging 78Ni- and 100Sn-referenced IMSRG(3f2) valence-space Hamiltonians and then adjusting 35 of the best-determined linear combinations of single-particle energies and two-body matrix elements by fits to energy data—predicts a subset of the experimental spectra for all nuclei in the $\pi j4$ space to within a 100–130 keV RMSD. A companion extrapolation test, training only on $Z > 36$ and evaluating $Z \leq 36$, reaches about 300 keV RMSD before overfitting sets in beyond 35 varied linear combinations. The paper argues that the IMSRG(3f2) starting point is closer to experiment than earlier ab initio Hamiltonians, needing far fewer adjusted linear combinations, and that the validation protocol protects against overfitting.

Load-bearing premise

The load-bearing premise is that the validation error measured on randomly held-out known levels, even after those same held-out levels helped choose the 35-combination cutoff, is an unbiased measure of how well the Hamiltonian will predict unmeasured exotic isotopes.

Editorial extensions

If this is right

The protocol should allow empirical Hamiltonians to be calibrated with considerably less data than the sd-shell cases needed, without losing predictive accuracy.
The IMSRG(3f2) starting point should reduce the number of parameters requiring adjustment, making the unadjusted remainder of the Hamiltonian more physically trustworthy.
For the $\pi j4$ region, the paper makes concrete predictions for unmeasured states, including the binding energy of 100Sn, excited states of 99In, and a 9/2+ state in 79Cu, which upcoming experiments can check.
The same train/test selection procedure could be applied to larger model spaces where full optimization is computationally heavy, improving extrapolation to exotic isotopes relevant to the r-process.

Reading between the lines

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

Beyond the paper: because the same held-out data are used both to choose the 35-VLC cutoff and to report validation error, the 100–130 keV figure is likely optimistic for truly unmeasured nuclei; the $Z \leq 36$ test is a more honest estimate, though if that region contributed to cutoff selection it is not fully independent.
Beyond the paper: a stricter evaluation would reserve an entire region, such as all $Z \leq 36$ data, from both fitting and cutoff selection, then evaluate the chosen Hamiltonian on that region exactly once.
Beyond the paper: applying the same protocol in the sd shell, where abundant data exist, would let one simulate sparse calibration by withholding random subsets and check whether validation-selected cutoffs beat fixed cutoffs on truly withheld data.

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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

First, the paper’s real contribution is modest but concrete. It shows that the IMSRG(3f2) starting Hamiltonian is close enough to experiment that fitting fewer SVD linear combinations gets you to a good energy RMSD, and it packages that into a usable p35-i3 interaction for the πj4 space. Section III is a clean, standard exposition of the SVD fitting machinery. The idea of splitting data into training/testing to choose how many varied linear combinations to keep is borrowed from ML but new to this fitting context, and the 2000-split scatter plot in Fig. 3 is a useful visual.

The strongest part is the extrapolation test in Sec. V: train on Z>36, test on Z≤36, get about 300 keV at 35 VLC. That is a genuinely out-of-sample check and it supports the overfitting narrative.

Now the soft spots, in proportion. The headline '100–130 keV RMSD' is not an honest estimate of predictive error, because the 35-VLC cutoff was selected from the minimum of the green validation curve and then the same curve's minimum is quoted. That is selection on the validation set. The minimum over ~35 candidate cutoffs will sit below the true generalization error. The Z≤36 test doesn't repair this, since the random splits used to pick 35 already contain Z≤36 levels. So the central quantitative claim needs a nested holdout or a pre-registered frozen validation set.

Two secondary issues. Fig. 1 places the p35-i3 9/2+ energy in the 'experimental' panel, and some 99In levels are systematics-based extrapolations assigned 50 keV uncertainties rather than direct measurements. That is disclosed in the text, so it's not deceptive, but it means part of the agreement near 100Sn is not a fresh prediction. Also, the promised supplemental files don't exist yet, and there is no baseline comparison to JUN45 or other older Hamiltonians under the same train/test split, so we can't judge how much of the improvement comes from the protocol vs. the starting Hamiltonian.

That said, the paper is honest on its own terms and the math is right. I would send it out; the referee should push for an honest held-out evaluation and for the data release. The p35-i3 interaction will be useful to the community even if the methodology claim needs tightening.

Referee Report

3 major / 6 minor

Summary. The manuscript presents two developments intended to improve the predictive power of empirical shell-model Hamiltonians when calibration data are sparse. First, it uses an improved VS-IMSRG(3f2) starting Hamiltonian for the πj4 model space (protons in 0f5/2, 1p3/2, 1p1/2, 0g9/2), arguing that this starting point requires fewer phenomenological adjustments than the previous IMSRG(2) approximation. Second, it introduces a protocol based on singular-value decomposition (SVD) fitting combined with random 80/20 training/testing partitions, repeated 2000 times, to select the number of varied linear combinations (VLC) and to estimate predictive error. The authors select Nc = 35 varied linear combinations, producing the p35-i3 Hamiltonian, and report that it predicts a subset of experimental spectra for all nuclei in the πj4 space to within a 100-130 keV RMSD, with a neutron-rich extrapolation test (Z ≤ 36) reaching approximately 300 keV. The paper concludes that these developments enable more reliable extrapolation to exotic isotopes.

Significance. If the claims are correct, the paper would make a useful methodological contribution to the calibration of shell-model Hamiltonians in data-sparse regions, relevant for rare-isotope facilities and r-process studies. The use of a modern ab initio starting point, IMSRG(3f2), is timely, and the SVD formalism is standard and clearly presented. The 2000-split training/validation protocol is a sensible internal check and is a strength of the paper. The paper also provides the final Hamiltonian parameters and results in supplemental material, which aids reproducibility. However, the headline claim of predictive power is weakened by a selection-bias issue: the same validation data used to choose the 35-VLC cutoff are also used to quote the 100-130 keV RMSD. The extrapolation test in Sec. V is an important attempt, but it is contaminated because the hyperparameter choice was made using random splits that include Z ≤ 36 levels. These issues are fixable with a nested or frozen holdout procedure, but as presented the central quantitative claim is not yet fully supported.

major comments (3)

[Sec. V, Fig. 3] The choice of Nc = 35 is made by taking the minimum of the green validation RMSD curve in Fig. 3, and the value of that curve (100-130 keV) is then quoted in Sec. VI as the predictive power of p35-i3. Because the validation data have already been used to select the model complexity, this error estimate is optimistically biased: the minimum over roughly 35 candidate cutoffs is expected to lie below the true generalization error even without overfitting. To support the claim, the authors should report a genuinely independent test error, for example by partitioning the data once into training, validation, and test sets, choosing Nc on the validation portion, and evaluating the final Hamiltonian on the untouched test portion (or by a nested cross-validation that accounts for the model-selection step).
[Sec. V, Fig. 5] The Z ≤ 36 extrapolation test does not fully remedy the selection-bias problem, because the same Nc = 35 was selected using random partitions that include Z ≤ 36 levels, so the target region has influenced hyperparameter selection. To claim extrapolative reliability, the cutoff should be fixed using only Z > 36 data — for example, by choosing Nc from a validation curve built exclusively from Z > 36 levels — and then applied to the Z ≤ 36 data as a true holdout. As written, the 300 keV figure is not an unbiased measure of extrapolation error.
[Sec. IV and Fig. 1] The manuscript blends measured and model-dependent input in the data used for fitting and validation. The caption of Fig. 1 states that the 9/2+ energy shown in the 'experimental' panel was obtained from the p35-i3 Hamiltonian itself, which is circular if that panel is presented as experimental. In addition, the 99In 3/2− and 5/2− excited states are not direct measurements but extrapolations based on 131In systematics, and Sec. IV states that they were assigned 50 keV uncertainties without the 150 keV theoretical error. These artificially small uncertainties make those states act as hard constraints, so agreement near 100Sn is not a fresh prediction. The authors should exclude such model-dependent points from the test and validation splits, or at minimum use realistic uncertainties and clearly flag them as extrapolated input.

minor comments (6)

[Sec. III heading] The heading 'SVD Proceedure' contains a typo; it should read 'SVD Procedure'.
[Fig. 1 caption] The sentence 'The 9/2+ energy shown in the experimental panel was obtained from our p35–i3 Hamiltonian' is confusing; the panel should be relabeled so that theory-generated points are not presented as experimental data.
[Sec. IV] The data selection is described narratively but the actual list is only in the supplemental material; a table in the main text listing the included and excluded levels with their adopted Jπ values and uncertainties would improve transparency.
[Sec. V, Fig. 5] The text says 'The calculations presented in FIG. 5 sampled from the full range of data 28 ≤ Z ≤ 50', but Fig. 5 actually shows the partition with training restricted to A > 86 (Z > 36); the wording should be clarified to distinguish the interpolation experiment (Fig. 3) from the extrapolation experiment (Fig. 5).
[Sec. II, Eq. (1)] In the text after Eq. (2), the notation is dense: the matrix G is used for the fit matrix and also referred to as the error matrix without a derivation; a short paragraph linking G^{-1} to parameter covariances would help readers not familiar with the SVD fitting literature.
[Sec. II] There is a minor capitalization inconsistency: 'states for 99in' should be '99In'.

Circularity Check

3 steps flagged · score 4.0 of 10

The 35-VLC cutoff is selected from the same validation curve whose minimum is then reported as the 100-130 keV predictive RMSD, and model-generated or extrapolated inputs partly enter the 'predicted' set.

fitted input called prediction [Sec. V (Fig. 3) and Sec. VI]
"The green curve highlights the predictive capacity of each Hamiltonian as the number of VLC's is increased. It can be seen that each Hamiltonian approaches an energy-RMSD minimum (a maximum for the predictive power) at about 35 VLC. ... The resulting Hamiltonian predicts a subset of the experimental spectra for all nuclei in the πj4 space to within a 100-130 keV RMSD."

The number 35 is selected as the minimum of the validation (green) RMSD curve obtained by randomly holding out 20% of the data. Reporting the value of that same minimized curve as the Hamiltonian's predictive RMSD is a fitted-hyperparameter-renamed-as-prediction step: the minimum over roughly 35 candidate cutoffs is a lower envelope, not an unbiased estimate for data that never influenced model selection. The Z<=36 extrapolation (about 300 keV) is a more honest external check, but the 35-VLC choice was still informed by random splits containing Z<=36 levels, so the headline 100-130 keV number is optimistically biased.
self definitional [Fig. 1 caption]
"The 9/2+ energy shown in the experimental panel was obtained from our p35–i3 Hamiltonian."

A level generated by the p35-i3 Hamiltonian is placed in the panel labeled 'Experimental' and compared with IMSRG and shell-model results. Agreement between p35-i3 and this point is therefore tautological: the 'experimental' point is not independent data. The text later calls it an extrapolation to be confirmed, but its placement as an experimental anchor makes the visual comparison partly self-confirming.

1 more flagged steps

fitted input called prediction [Sec. IV (data selection) and Sec. VI (conclusions)]
"In order to constrain the SPE at the beginning and the end of the πj4 model space, smaller uncertainties of 50 keV (without the theoretical error of 150 keV) were used for the 3/2− and 1/2− excited states observed for 79Cu and for the 3/2− and 5/2− excited states extrapolated for 99In."

The 3/2− and 5/2− states of 99In are not measured 99In levels but systematics-based extrapolations (from 131In), yet they are entered into the SVD fit with 50 keV uncertainties. The conclusion that the resulting Hamiltonian 'predicts' spectra for all nuclei in the πj4 space, including 99In, therefore includes agreement with data that are themselves inputs to the fit, weighted more strongly than their actual model-dependence. This is a partial fitted-input-called-prediction entanglement rather than a fully independent validation.

full rationale

The paper's SVD train/test protocol is a genuine external check: within each of 2000 random 80/20 partitions the Hamiltonian parameters are fitted without the held-out levels, and the Z<=36 split is a separate out-of-sample extrapolation test. Self-citation of Refs. [3] and [6] is not load-bearing here, because the IMSRG(3f2) starting point is implemented and assessed through the paper's own TBME and RMSD comparisons rather than by invoking an unverified uniqueness or correctness claim from the cited authors. The partial circularity is in the reporting and input selection. The 35-VLC complexity is chosen as the minimum of the validation RMSD curve, and the value of that minimized curve is then quoted as the Hamiltonian's predictive RMSD; this is a fitted-hyperparameter-renamed-as-prediction step. In addition, a 9/2+ level generated by p35-i3 is displayed in the 'experimental' panel of Fig. 1, and systematics-based extrapolated 99In states with 50 keV uncertainties are used as SVD fit inputs while the conclusions count agreement with 99In among the 'predicted' spectra. These steps do not make the derivation equivalent to its inputs, but they do make the headline 100-130 keV number optimistic and partly self-referential. Score 4 reflects partial circularity with a substantial independent extrapolation benchmark remaining.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

No new particles, forces, or conserved quantities are introduced. The p35-i3 Hamiltonian is a fitted empirical object, not a postulated entity. Most free content is the shell-model parameter set, the validation-tuned cutoff, and the ab initio starting-point choices that import assumptions from IMSRG and the underlying NN+3N interaction.

free parameters (5)

sigma_th theoretical uncertainty = 150 keV
Chosen by hand to make final chi-square about unity; authors state 100-200 keV has minimal effect, but no quantitative robustness is shown.
Number of varied linear combinations (SVD cutoff) = 35
Selected from the minimum of the validation-set RMSD curve; this is a model-complexity parameter tuned to the same data used to evaluate prediction.
Artificial uncertainties for boundary states = 50 keV for 79Cu 3/2-, 1/2- and 99In 3/2-, 5/2-
Weights reduced from roughly 150 keV to 50 keV specifically to constrain SPE at both ends of the model space, including extrapolated (not measured) 99In levels.
SPE and TBME of p35-i3 = 4 SPE + 65 TBME, listed in supplemental [28]
The empirical Hamiltonian parameters are the fitted output; they are adjusted to 189 energy data and are the object of the paper.
IMSRG starting choices
Harmonic oscillator frequency 12 MeV, emax 12, NO2B approximation, averaged 78Ni and 100Sn references. These are chosen by hand and not varied.

assumptions (6)

domain assumption VS-IMSRG with NO2B approximation and 3f2 factorized correction provides an accurate ab initio effective Hamiltonian for the πj4 space
Main input starting point; depends on truncation of many-body operators, the EM 1.8/2.0 interaction, HF reference, and refs. [3,6].
domain assumption The πj4 model space (proton 0f5/2, 1p3/2, 1p1/2, 0g9/2 with N=50 closed) contains the low-lying structure; intruder and cross-shell states are excluded
Data selection excludes intruder states and uncertain J-pi values throughout Sec. IV.
domain assumption Nucleus-independent SPE and TBME are sufficient for the fit
Used in Eq. (1) and the fitting procedure; standard in empirical fits but an approximation.
standard math Wavefunctions used to compute operator overlaps beta are adequately updated by iteration to convergence
SVD algorithm assumes linear response around current wavefunctions and converges after iteration.
ad hoc to paper The extrapolated 99In 3/2- and 5/2- levels, derived from 131In systematics, can stand in for measured data
Used as constraints with 50 keV uncertainty to pin SPE at the 100Sn end; not experimental data.
domain assumption Chi-square weighting with sigma_th=150 keV and sigma_k = sqrt(sigma_exp^2 + sigma_th^2) is a valid statistical model of theory error
Theory uncertainty is assumed uncorrelated and state-independent; this determines the SVD weights and the well-determined cutoff.

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Cite this review

Pith. "Pith review of Improving the predictive power of empirical shell-model Hamiltonians." pith.science (2026). https://pith.science/paper/LIJBQ2IZ

@misc{pith2026241209917,
  author       = {Pith},
  title        = {Pith review of: Improving the predictive power of empirical shell-model Hamiltonians},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LIJBQ2IZ}},
  note         = {Machine review of arXiv:2412.09917}
}

read the original abstract

We present two developments which enhance the predictive power of empirical shell-model Hamiltonians for cases in which calibration data are sparse. A recent improvement in the ab initio derivation of effective Hamiltonians leads to a much better starting point for the optimization procedure. In addition, we introduce a protocol to avoid overfitting, enabling a more reliable extrapolation beyond available data. These developments will enable more robust predictions for exotic isotopes produced at rare isotope beam facilities and in astrophysical environments.

Figures

Figures reproduced from arXiv: 2412.09917 by the authors.

**Figure 2.** FIG. 2. Comparison of the TBME obtained with the two ver [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** FIG. 3. Results of the SVD fits as a function of the number of varied linear combinations (VLC) of parameters. The left-hand [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 6.** Figure 6: FIG. 6. Comparison of the IMSRG(3f2) and p35-i3 TBME. [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]

**Figure 5.** Figure 5: FIG. 5. Results obtained when the fitted data set is restricted [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

Starting from the best available Hamiltonian pa- rameters ⃗ xs we construct and diagonalize the G- matrix to obtain Dii eigenvalues and the orthonor- mal basis for our parameter space

work page

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Explicitly: yi = dici = di pX l=1 el AT il Simultaneously, linear combinations of abinitio Hamiltonian parameters are determined from equation (4)

Mutually independent SVD parameters yi are de- termined in the fit from equation (4). Explicitly: yi = dici = di pX l=1 el AT il Simultaneously, linear combinations of abinitio Hamiltonian parameters are determined from equation (4). Explicitly: ⃗ y∗ = AT ⃗ x∗ − → y∗ i = pX l=1 x∗ l AT il

work page

[3] [3]

One defines a cutoff criterion δ, and updated linear combinations ⃗ ya are defined by only adopting well- determined values yi with respect to this cutoff, and leaving abinitio values y∗ i for the rest: ya i = ( yi (di ≤ δ) y∗ i (di > δ) The number of well-determined linear combinations is Nd

work page

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With the updated set of model parameters ⃗ ya one recovers the Hamiltonian parameters by inverting the rotation: ⃗ xa = (AT )−1⃗ ya

work page

[5] [5]

This process is repeated until convergence

This set of Hamiltonian parameters takes the place of ⃗ xs as input to the first step of this algorithm, and is used to obtain the next set of parameters ⃗ xb. This process is repeated until convergence. 5 IV. EXPERIMENT AL DA T A FOR N = 50 ISOTONES For the data used as input for the SVD fits we chose levels which have reliable excitation energies and J ...

work page

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S. R. Stroberg, H. Hergert, S. K. Bogner, and J. D. Holt, Annu. Rev. Nucl. Part. Sci. 2019. 69, 307 (2019)

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For all of these Hamiltonians the root-mean-square deviation (RMSD) between the calculated and experimental en- ergy data is 150-200 keV

for the {0f7/2, 0f5/2, 1p3/2, 1p1/2} (f p) model space, and JUN45 [8] and jj44b (Appendix A of [9]) for the {0f5/2, 1p3/2, 1p1/2, 0g9/2} (jj 44) model space. For all of these Hamiltonians the root-mean-square deviation (RMSD) between the calculated and experimental en- ergy data is 150-200 keV. This is to be compared with the results of abinitio type calc...

work page

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Honma, T

M. Honma, T. Otsuka, T. Mizusaki, and M. Hjorth- Jensen, Phys. Rev. C 80, 064323 (2009)

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