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Spurious imaginary phonon modes silently underestimate MOF heat capacity.

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2026-08-03 03:37 UTC pith:SLTHL7MC

load-bearing objection Solid practical paper on how spurious imaginary phonon modes bias MOF heat capacities and MLIP benchmarks; the core MOF-74 result is convincing, but the five-MOF generalization needs an independent converged-DFT check. the 3 major comments →

arxiv 2602.07295 v3 pith:SLTHL7MC submitted 2026-02-07 cond-mat.mtrl-sci cond-mat.mes-hall

The impact of spurious imaginary phonon modes on thermal properties of Metal-organic Frameworks

Prathami Divakar Kamath , Kristin A. Persson This is my paper
classification cond-mat.mtrl-sci cond-mat.mes-hall PACS 65.40.Ba63.20.-e
keywords metal-organic frameworksimaginary phonon modesheat capacityphononsmachine learning interatomic potentialsthermal screeningtemperature swing adsorptiondensity functional theory
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper argues that imaginary phonon modes dismissed as artifacts still carry real heat-capacity weight. In MOFs, even 1% spurious imaginary modes causes Cv errors over 10% at low temperature and roughly double the mode fraction at 300 K, enough to reorder candidates in thermal screening. It shows that benchmarking machine-learned potentials against such DFT data can wrongly penalize models that predict clean phonon spectra. It then proposes a one-line post-processing correction that restores the missing contribution above twice the Debye temperature, reducing errors by roughly an order of magnitude.

Core claim

When imaginary modes are omitted from the harmonic Cv sum, each omitted mode introduces a finite negative error that depends on temperature and on which branches are imaginary—often the acoustic branches that dominate low-temperature heat capacity. For MOF-74, 1.03% imaginary modes underpredicts Cv by more than 10% below roughly the Debye temperature and by about 2% at 300 K. Across five MOFs with 0.71–5.75% imaginary modes, the 300 K Cv error ranges from −1.5% to −12.2%, consistently more than twice the imaginary-mode fraction. The paper's correction adds k_B T per omitted mode, i.e., Cv_corrected = Cv_imaginary + k_B T·(3N·%imaginary/100), for temperatures above twice the Debye temperature

What carries the argument

The harmonic phonon heat-capacity formula plus a correction term that restores the classical k_B contribution of each skipped imaginary mode. For temperatures above twice a proxy Debye temperature, the paper adds k_B T times the number of omitted modes (3N times the percentage of imaginary modes over 100) to the phonon-derived Cv. The proxy Debye temperature is estimated from the lowest optical mode at the zone center, and the correction is justified because MOF Debye temperatures are low, so ambient and TSA-relevant temperatures lie well above 2θ_D.

Load-bearing premise

The analysis assumes a fine-tuned machine-learned potential's zero-imaginary-mode spectra are a faithful stand-in for fully converged DFT, so the reported error multipliers and sub-0.8% correction residuals might not transfer to a true converged-DFT baseline.

What would settle it

Take one MOF with roughly 5% spurious imaginary modes, run a fully converged DFT phonon calculation with near-zero imaginary modes using tight relaxation and a large supercell, and compare its 300 K Cv with the corrected loose-convergence Cv; a residual above about 0.8% would falsify the claimed correction accuracy.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • Existing MOF thermal-screening rankings can shift by several percent once imaginary modes are corrected, changing which candidates look most attractive for temperature-swing adsorption.
  • MLIP benchmarks should use imaginary-mode-free references or apply the correction; otherwise models with clean spectra can appear 2–10 times worse than they actually are.
  • Keeping spurious imaginary modes below 0.5% keeps 300 K Cv errors under about 1%, giving high-throughput DFT a practical convergence target.
  • The correction is cheap enough to reprocess published phonon datasets in seconds without new DFT calculations.
  • Underpredicted heat capacity translates into underpredicted adsorbent-regeneration energy, so some MOFs may appear more energy-efficient than they truly are.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The error multiplier likely depends on which branches are imaginary, not just their count; a mode-fraction-only rule may underestimate errors when missing acoustic modes dominate the low-temperature response.
  • The same one-line correction could plausibly extend to phonon-derived entropy and free energy with analogous k_B-based terms, broadening the workflow beyond heat capacity.
  • Other soft framework materials with low Debye temperatures—covalent organic frameworks, flexible perovskites, or molecular crystals—may show the same amplification, and the correction is directly testable there.
  • Applying the correction to existing MOF heat-capacity datasets yields testable predictions for calorimetry, allowing experimental Cp measurements to discriminate between corrected and uncorrected rankings.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 8 minor

Summary. The paper argues that spurious imaginary phonon modes—routinely discarded in harmonic heat-capacity calculations—cause a systematic, nontrivial underestimation of C_v in metal-organic frameworks. Using MOF-74 as a case study, it compares a DFT phonon spectrum with 1.03% imaginary modes (Moosavi et al.) against a spectrum with ~0% imaginary modes (Wieser et al.) and finds C_v errors that exceed 10% at low temperatures and saturate at roughly 2× the percentage of imaginary modes near 300 K. The paper then proposes a simple post-processing correction (Eq. 2) that adds k_B per imaginary mode for T > 2θ_D, reporting a reduction in the MOF-74 error from 1.7% to 0.23% at 300 K. The generalization to five QMOF MOFs uses MACE-MP-MOF0 as a surrogate 0%-imaginary-mode baseline for DFT data with 0.71–5.75% imaginary modes, obtaining C_v underestimations of 1.5–12.2% and post-correction residuals below 0.8%. The paper also discusses implications for MLIP benchmarking, showing that MACE-MP-MOF0's apparent 2.53% overestimation against unconverged DFT drops to 0.77% against corrected or 0%-imaginary-mode data. The code and data are made publicly available.

Significance. If the claims hold, this is a practically useful contribution. It quantifies a common but rarely assessed approximation in phonon-based MOF thermodynamics, provides a cheap correction that can be applied to standard Phonopy outputs, and alerts the community to a benchmarking pitfall when MLIPs are compared against DFT data containing spurious imaginary modes. The MOF-74 case study is clear and internally consistent: the comparison between a 1.03%-imaginary-mode DFT spectrum and a converged 0%-imaginary-mode DFT spectrum from an independent group directly demonstrates the phenomenon. The paper also ships reproducible code and data, which strengthens its value. However, the broader quantitative claims—the 2× amplification factor, the 0.5% screening threshold, and the <0.8% post-correction residuals—are currently validated against a machine-learned surrogate rather than against fully converged DFT for the five QMOF MOFs. Because low-frequency phonon behavior is exactly the quantity being corrected, this missing anchor is a central gap that must be addressed before the general claims can be accepted.

major comments (3)
  1. [Systematic dependence of C_v errors on % Imaginary Modes / Fig. 4] The generalization rests on using MACE-MP-MOF0 as the 0%-imaginary-mode baseline: the text states this explicitly ('Using MACE-MP-MOF0 as a substitute for DFT for the 0% imaginary mode baseline'). MACE-MP-MOF0 is an MLIP developed co-located with this work, and although the five QMOF MOFs are outside its training set, no independent converged-DFT phonon calculation is provided for these structures. A surrogate can reproduce room-temperature C_v well (as in Table 1) while still misplacing low-frequency phonon modes, which is precisely the regime where the imaginary-mode correction acts. Consequently, the reported 1.5–12.2% deviations, the 2× amplification factor, and the <0.8% corrected residuals are not yet established as statements about true DFT behavior; they are statements about MACE-MP-MOF0-relative behavior. Please provide at least one converged-DFT 0%-imaginary-mode baseline for o
  2. [Eq. (2) and Discussion of thresholds] The correction term adds exactly k_B per imaginary mode for T > 2θ_D. This is an ansatz: for a genuinely imaginary harmonic mode the usual oscillator partition function is not defined, and the physically appropriate contribution depends on the character of the mode (e.g., a near-zero-frequency spurious mode vs. a true dynamical instability). The paper does not justify the k_B-per-mode assumption from the actual eigenvectors or frequencies of the spurious modes, nor does it quantify the uncertainty in the empirical 'error > 2× %imaginary' rule, the 0.5% threshold, or the claimed order-of-magnitude residual reduction. Because the threshold is meant to guide practical screening, please report the individual ratios for the five MOFs, provide error bars or sensitivity analysis, and state clearly which regimes of imaginary-mode magnitude the correction is designed for.
  3. [Abstract / Introduction / Results] The paper claims that spurious imaginary modes 'lead to incorrect ranking of MOFs in thermal-property-based screening,' but no actual ranking or rank-reversal analysis is presented. Figure 4 shows percentage errors in C_v, and the paper notes that MOFs can differ by 1–2% in C_v, but it does not demonstrate a concrete screening scenario where the order of candidates changes after correction. Since the ranking claim is part of the central motivation, please include a concrete rank-order comparison (e.g., a small set of MOFs ranked by corrected vs. uncorrected C_v) or temper the claim to a qualitative risk statement.
minor comments (8)
  1. [Abstract] Typo: 'green house gases' should be 'greenhouse gases'.
  2. [Figure 1 caption] 'prox Debye temperature' should be 'proxy Debye temperature'.
  3. [Table 2] The column header 'θD (K) % imaginary modes from MACE-MP-MOF0 from DFT31' is confusing: it is unclear which numeric column corresponds to which quantity. The row for qmof-08b5558 also appears to contain an extra '121' that is not explained.
  4. [Methods] Typo: 'Helmoltz' should be 'Helmholtz'.
  5. [Eq. (2)] The notation 'C_v(T) imaginary' is awkward; use a proper subscript, e.g., C_v^imag(T). Also clarify in the text that Eq. (2) is applied only for T > 2θ_D and how θ_D is defined in practical cases with imaginary modes present.
  6. [Figure 4 captions] The captions say '0% imaginary-modes data from MACE-MP-MOF0'; it would be clearer to state explicitly that this is a surrogate baseline, not DFT, to avoid overstating the validation.
  7. [Results, 'Comparisons of DFT data with MLIP...'] The sentence 'the remaining errors after applying corrections to the DFT data may partially reflect the intrinsic accuracy of the model' is an important caveat and should appear earlier, in the main results section, not only as an aside.
  8. [Introduction] The discussion of ad hoc imaginary-mode cutoffs would benefit from citing the specific thresholds used in Refs. 32–36, rather than only listing them, so readers can see the range of practices.

Circularity Check

1 steps flagged

Five-MOF generalization and correction validation are anchored to the same-group MACE-MP-MOF0 model as the 0% imaginary-mode baseline; the MOF-74 central claim remains independently grounded.

specific steps
  1. self citation load bearing [Results, 'Systematic dependence of C_v errors on % Imaginary Modes']
    "Using MACE-MP-MOF0 as a substitute for DFT for the 0% imaginary mode baseline, Fig 4 shows that DFT data with spurious imaginary modes varying from 0.71% to 5.75% leads to underestimations in C_v at 300K from -1.5% to -12.2% respectively. ... None of the selected MOFs were included in the training of the MACE-MP-MOF0 model. Therefore, the remaining errors after applying corrections to the DFT data may partially reflect the intrinsic accuracy of the model."

    MACE-MP-MOF0 is the authors' own MLIP (ref. [27], co-authored by Kamath and Persson). It provides the '0% imaginary mode' baseline for all five QMOF MOFs used to generalize the correction and to quantify error amplification. Since these five MOFs are outside the training set, the comparison is a genuine extrapolation test, but it is still measured against a same-group surrogate, not an independent converged-DFT spectrum. The paper itself concedes that the correction residuals 'may partially reflect the intrinsic accuracy of the model,' so the <0.8% residual and the 1.5-12.2% error range are not fully independently established. This makes the five-MOF generalization and the subsequent MACE-vs-Yue benchmarking partially self-referential, although the MOF-74 case is separately anchored to Wie

full rationale

The paper's central claim that spurious imaginary modes cause systematic C_v underestimation is independently grounded for MOF-74: the 0% imaginary-mode reference there comes from Wieser et al.'s converged DFT, not from the authors' model, and Table 1 further shows reasonable agreement with experimental C_p. Equation (2) is a physically motivated classical correction (k_B T per omitted imaginary mode) and is not fitted to the validation data, so it does not reduce to a fit by construction. The main circularity concern is limited to the generalization across five QMOF MOFs and the MLIP-benchmark demonstration: both rely on MACE-MP-MOF0, a same-group MLIP, as the 0% imaginary-mode ground truth. The paper acknowledges this limitation explicitly, noting that residuals may reflect model accuracy rather than true DFT error. Because the central effect and the MOF-74 correction are independently supported, the circularity is partial and does not invalidate the core message, but the five-MOF generalization and the claimed correction accuracy are not fully independent of the authors' own model.

Axiom & Free-Parameter Ledger

2 free parameters · 4 axioms · 0 invented entities

No new physical entities are introduced. The central claim rests on a small set of domain assumptions: the spurious nature of the imaginary modes, the k_B-per-mode correction model, and the MACE-MP-MOF0 surrogate baseline. The latter is the most fragile because the validation of the correction across five MOFs depends on it.

free parameters (2)
  • Acceptable imaginary-mode threshold (0.5%) = 0.5% imaginary modes
    Proposed in Results: 'maintaining spurious imaginary modes below a threshold of 0.5% limits errors in Cv to under 1%'. Chosen by hand from a five-MOF empirical trend; no derivation.
  • Error amplification factor (~2×) = ~2
    Empirical ratio reported at 300K: 'errors almost 2× the percentage of imaginary modes at 300 K and higher'. Observed across studied MOFs; not predicted from a first-principles model.
axioms (4)
  • domain assumption Imaginary modes in the analyzed DFT phonon datasets are spurious artifacts, not genuine dynamical instabilities.
    The correction treats them as missing low-frequency modes. If some were real instabilities, the system should relax and the correction would not apply. The paper argues this for its selected MOFs but does not prove it for all.
  • ad hoc to paper At T > 2θ_D each missing imaginary mode contributes exactly k_B to C_v.
    Used to derive Eq. 2: Cv_corrected = Cv_imaginary + k_B T (3N %imaginary/100). Justified only by a classical-limit argument ('By assuming a full k_b contribution at T∼2θ_D'); no rigorous derivation for finite T.
  • domain assumption MACE-MP-MOF0 phonon spectra with ~0% imaginary modes are accurate surrogates for fully converged DFT for the five QMOF MOFs.
    The generalization of the correction across five MOFs (Fig. 4) is validated against this MLIP, not against converged DFT. The paper acknowledges this in the text: 'Using MACE-MP-MOF0 as a substitute for DFT for the 0% imaginary mode baseline'.
  • standard math The harmonic approximation and Phonopy's C_v formula adequately describe MOF heat capacities.
    Used throughout (Eq. 1); standard in the field. The paper compares to quasi-harmonic C_p only via MACE-MP-MOF0.

pith-pipeline@v1.3.0-alltime-deepseek · 11163 in / 12860 out tokens · 122098 ms · 2026-08-03T03:37:31.992699+00:00 · methodology

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read the original abstract

Metal-organic Frameworks (MOFs) have emerged as potential candidates for direct air capture (DAC) of green house gases and water. Thermal properties of MOFs, such as their heat capacity, are used to determine the energy penalty associated with the adsorbent retrieval during the Temperature Swing Adsorption process. To aid exploration of the vast experimental design space of MOFs for such applications, computational methods like Density Functional Theory (DFT) or surrogate machine learning models trained on DFT data have been developed for obtaining phonon-derived heat capacities of MOFs. However, the high cost of explicit phonon computation in large and flexible nanoporous MOFs often necessitates the use of small supercells or lower convergence criteria which decrease predictive accuracy. These approximations often result in spurious imaginary phonon modes which are commonly ignored in practice. At present, there is no clear consensus in the literature on what magnitude of negative frequency or what fraction of imaginary modes can be considered acceptable. Here, we systematically demonstrate that spurious imaginary phonon modes can introduce substantial errors in heat capacity estimates, leading to incorrect ranking of MOFs in thermal-property-based screening. We further show that benchmarking machine learning interatomic potentials (MLIPs) against DFT datasets containing spurious imaginary modes can misrepresent models that predict physically meaningful phonon spectra for dynamically stable MOFs. Finally, we introduce a simple, rapid post-processing workflow that can be applied to standard phonon calculations to effectively correct heat capacity estimates and account for spurious imaginary modes in MOFs.

Figures

Figures reproduced from arXiv: 2602.07295 by Kristin A. Persson, Prathami Divakar Kamath.

Figure 1
Figure 1. Figure 1: (a)The density of states (DOS) of the two phonon spectra obtained with and without spurious [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The phonon band diagram for MOF-74 obtained from the force constants reported in Moosavi [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: (a)The % deviations in Cv obtained from DFT data with 1.03% spurious imaginary modes relative to 0% become significantly lower at T ≫ θD after adding corrections (b) A constant underestimation ∼ 2% can be seen in Cv at temperatures around 300K due to the imaginary modes in the red curve. Accounting for the imaginary modes in the corrected contribution curve leads to a nearly perfect overlap with the blue 0… view at source ↗
Figure 4
Figure 4. Figure 4: (a) The deviations (%) in Cv obtained from DFT data with increasing spurious imaginary modes in different MOFs relative to the 0% imaginary-modes data from MACE-MP-MOF0. (b) The deviations (%) in the corrected Cv obtained from post-processing the DFT data are 10× smaller for all MOFs relative to the 0% imaginary-modes data from MACE-MP-MOF0. as one of the top-performing models in MOFSim￾Bench,43 with a rep… view at source ↗
Figure 5
Figure 5. Figure 5: The Cv curve from DFT data containing 1.03% spurious imaginary modes shows a constant underestimation relative to other DFT30 and MACE-MP-MOF0 data containing ∼ 0% imaginary modes. The corrected contribution to the DFT data bridges this gap (a) (b) [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The Cv curves obtained from DFT with and without corrections for imaginary modes compared against MACE-MP-MOF0 and the MLP from Yue et al.31 for (a) ”qmof-08b5558” and (b)”qmof-574737f” mon computational approximations, such as insuf￾ficient structural relaxation, inadequate supercell sizes in finite-difference calculations, and broken translational symmetry — issues that are particu￾larly prevalent in MOF… view at source ↗

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