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Response-state Learning for Transferable Vibrational Spectroscopic Characterization with Electron Prior

Zetong Li, Zhuosong Xie, Hengyu Fan, Jiaao Yu, Qiyao Hua, Zheng Lu, Liming Xu, Juanni Wu, Honglin Li

arXiv:2609.28935v1cs.LGphysics.chem-ph

TL;DR

Existing geometry-to-response models remain limited to represented chemical domains and lack an explicit mechanism for assessing response-state reliability or supporting corrections. SENK combines an equivariant response model with ENK refinement and electron-informed calibration, maintaining accurate response prediction across diverse molecular classes while selectively refining spectrally sensitive states.

  • Problem

    Existing geometry-to-response models remain limited to represented chemical domains and lack an explicit mechanism for assessing response-state reliability or supporting corrections.

    Source: Introduction · pp. 3–4

  • Method

    SENK pairs an equivariant transformer for response tensors with an ENK bridge for state-dependent refinement and reliability sensing.

    Source: Introduction · p. 4

  • Results

    SENK maintains accurate response prediction across diverse molecular classes while selectively refining spectrally sensitive features, with ENK adding consistent improvements across evaluated branches.

    Source: Discussion · pp. 19–20

  • Takeaways & Limitations

    SENK establishes a response-state framework for electron-informed calibration across diverse molecular classes.

    Source: Discussion · pp. 19–20

    Some materials evaluation involves system-specific retraining or fine-tuning.

    Source: materials · p. 16

Abstract

from arXiv · show

Vibrational spectral prediction can become inaccurate when localized stereoelectronic environments perturb intermediate response states and high-risk response units dominate characteristic spectral fingerprints, making prediction across external chemical space difficult. SO(3) Equivariant Neural Kalman Networks (SENK) form a response-state cascade that combines an equivariant transformer backbone for Hessian, dipole-derivative and polarizability-derivative learning, an Equivariant Neural Kalman bridge for state-dependent refinement and reliability sensing, and an NBO-informed electronic-prior pathway coupling consistency regularization with bounded, branch-specific guided spectral calibration. SENK outperforms DetaNet on QM9S and QMe14S while preserving full-spectrum IR and Raman fidelity from small molecules to drug-like systems. SENK remains stable and selectively improves spectrally sensitive features in biomolecular systems with complex stereoelectronic effects. It therefore integrates tensor prediction, reliability diagnosis and physics-informed calibration, supporting transferable vibrational spectroscopy from molecular systems to functional molecular materials.

Introduction

Vibrational response prediction is costly and can be disrupted by localized chemical environments, while existing models lack an explicit way to assess response reliability or support corrections. SENK combines tensor prediction, state-dependent reliability sensing and electron-informed calibration, and is evaluated across molecular and materials-relevant systems.

Introduction

  • Repeated electronic-structure and higher-order response calculations remain costly, while local response errors can produce conspicuous spectral deviations.These costs limit high-throughput applications to large molecules, biomolecules, condensed phases and heterogeneous systems.Source: Introduction · p. 3
  • Existing geometry-to-response advances lack an explicit mechanism for judging response-state reliability, interpreting local reliability loss or applying supported corrections.Prediction beyond represented chemical domains remains difficult where local groups, force constants, hydrogen bonding, substituents and polarization shape spectra.Source: Introduction · pp. 3–4
  • SENK treats response prediction as staged state estimation and calibration, using an equivariant transformer, an ENK refinement bridge and an NBO-informed electronic-prior pathway.The bridge supplies an environment-sensitive reliability signal; the prior pathway combines consistency regularization with bounded guided spectral calibration for eligible branches.Source: Introduction · p. 4
  • Evaluation spans QM9S and QMe14S, external drug-like molecules and bioactive peptides, plus zero-shot tests on molecular trimers and conjugated oligothiophenes.The materials-relevant tests examine full-spectrum IR/Raman prediction and conjugation-dependent Raman characterization against independent dilute-solution measurements.Source: Introduction · p. 4

Results

SENK establishes a stronger response-property baseline than DetaNet across QM9S and QMe14S, and its tensor predictions carry through to full-spectrum IR and Raman comparisons from small molecules to drug-like systems.

2. Full-spectrum prediction from small molecules to drug-like systems

  • For small molecules, both SENK configurations closely follow DFT references across fingerprint, carbonyl-associated and high-frequency stretching regions, while DetaNet shows local deviations.The near-coincidence of ENK-on and ENK-off spectra indicates that ENK preserves response states already represented reliably by the backbone.Source: §2 · p. 7
  • In drug-like systems, SENK retains principal IR and Raman peaks and overall spectral structure more faithfully than DetaNet in chemically crowded regions.DetaNet shows pronounced peak displacement and profile distortion in cefixime, and more localized shifts and intensity mismatches for atorvastatin and pomalidomide.Source: §2 · p. 7

1. Performance of SENK in property prediction

  • Across all three response branches, SENK’s ENK-off model outperformed DetaNet on QM9S, with mean absolute errors of 0.0091/0.0780/0.0267 versus 0.0182/0.2012/0.0822; ENK activation reduced them to 0.0081/0.0775/0.0262.Source: §1 · p. 6

3. Response-state sensing and electron-informed calibration across chemical space

ENK identifies localized response-state instability that is not fully captured by global molecular descriptors. SENK combines reliability with spectral sensitivity to prioritize consequential states, then applies bounded, branch-specific electronic-prior corrections where supported.

3.1 ENK identifies and refines localized response-state instability

  • ENK provides non-uniform local refinement across response branches and chemical environments, with response-state changes propagating differently across spectral windows.On external ChEMBL structures, uncertainty varies substantially even at comparable molecular sizes and heteroatom fractions, so those descriptors do not fully determine reliability.Source: §3.1 · p. 9

3.2 Response-state risk combines model reliability with spectral consequence

  • The response-risk score combines ENK-derived uncertainty with spectral sensitivity, prioritizing states where lower reliability and likely visible spectral consequences coincide.Across 4,989 structures, calibration magnitude varies with uncertainty and sensitivity, while subsequent error change depends on response branch and local environment.Source: §3.2 · p. 10

3.3 The electronic prior supplies selective, branch-specific calibration

  • SENK separates representation stabilization from targeted correction: EP/GSC applies bounded, branch-specific corrections where the electronic prior supports them.The response-specific updates improve selected dipole-derivative and off-diagonal Hessian measures, while correction direction and magnitude vary with branch and environment.Source: §3.3 · pp. 10–11
  • Pooled gains differ across chemical strata: C-N and aromatic-like C-C show +37.4% and +19.3%, whereas reported carbonyl-like C-O strata include -6.5%, -23.1%, and -27.9%.Source: §3.3 · p. 11

4. Response-state stability and electron-informed calibration in biomolecular systems

Biomolecular spectra are largely within the backbone’s learned response domain, but localized stereoelectronic or strongly polarized environments can create sensitive deviations. ENK and electronic-prior calibration refine those features while preserving unaffected spectral regions.

4. Response-state stability and electron-informed calibration in biomolecular systems

  • Oxytocin and Leu-enkephalin provide baseline-supported biomolecular cases, with predictions broadly consistent with experimental IR curves.Their local environments remain largely within the response domain learned by the backbone despite molecular size and flexibility.Source: §4 · p. 13
  • In Met-enkephalin, ENK shifts the affected O-H feature toward the reference and EP adds a -28 cm-1 calibration shift in both IR and Raman.NBO analysis supports an LP(S)→BD*(O-H) interaction associated with the affected Tyr-OH···Met-S contact; other spectral profiles are preserved.Source: §4 · p. 13
  • For 2onw:SSTSAA, SENK predicts the key spectral feature without transfer learning or system-specific fine-tuning; EP moves it from 2,526 to 2,493 cm-1.The total -33 cm-1 calibration shift occurs in both IR and Raman and remains mainly confined to salt-bridge-sensitive features.Source: §4 · pp. 13–14

5. Transferable vibrational characterization of molecular building units for organic

The section tests transfer in crystal-derived molecular trimers with intermolecular contacts and in a measured oligothiophene series. SENK retains strong full-spectrum agreement in the trimer test, while SENK-EP reproduces the oligothiophene Raman response ordering.

R-3B69

  • R-3B69 comprises 69 crystal-derived trimers from 23 families; SENK was evaluated directly on supplied geometries, testing transfer to assemblies with intermolecular contacts.The benchmark provides double-harmonic IR and Raman references.Source: materials · pp. 16–17
  • For the acetamide trimer, predicted peak patterns persist across fingerprint and high-frequency regions, with local peak positions and intensities close to reference spectra.The comparison is shown for representative Raman and IR spectra against DFT.Source: materials · p. 17, Figure 6 · p. 18

Oligothiophene series

  • SENK-EP was applied without retraining to a homologous 4T–10T oligothiophene series measured experimentally in dilute solution, focusing on the in-phase C=C, R-like vibrational subspace.Source: materials · p. 17
  • For 4T/6T/8T/10T, experimental normalized Raman responses were 1.00/3.49/6.46/8.27 versus SENK-EP’s 1.00/2.84/4.13/8.33; predicted ordering matched (r=0.938, n=4).The predicted 10T/4T ratio of 8.33 closely matches the experimental value of 8.27.Source: materials · pp. 17–18

R-3B69

  • Across 69 trimers, SENK reached mean full-spectrum Raman rMSC 0.836 (median 0.895) and mean IR 0.705, while the MACE-OFF23/MACE-MDP pipeline reported mean R-3B69 Raman 0.835 and IR 0.764.Source: materials · p. 17, Figure 6 · p. 18

Discussion

SENK combines response-state learning, reliability sensing, and bounded electronic-prior calibration, with reported performance spanning drug-like and biomolecular systems, crystal-derived assemblies, and conjugated semiconductor building blocks. The discussion presents these tests as support for transfer across molecular and materials-relevant chemical space.

Discussion

  • SENK treats Hessian, dipole-derivative, and polarizability-derivative predictions as intermediate response states, combining equivariant learning with reliability sensing and branch-specific, bounded electronic-prior calibration.ENK identifies environments associated with reduced model confidence; EP/GSC corrects branches where a transferable calibration direction is supported.Source: Discussion · p. 19
  • Across QM9S and QMe14S, the equivariant backbone provides a strong response-state baseline, and ENK yields consistent additional improvements across evaluated response branches.The discussion also identifies broader chemical-space characterization as a way to guide targeted expansion of response-state coverage.Source: Discussion · pp. 19–20
  • In Met-enkephalin and 2onw:SSTSAA, sulfur-associated and salt-bridge environments are linked to consequential response deviations, and sequential ENK and EP corrections shift affected features toward reference while largely preserving broader profiles.The cases provide an electronic-structure context for localized stereoelectronic effects.Source: Discussion · pp. 19–20
  • Across 69 noncovalently interacting trimers, SENK preserves high full-spectrum Raman similarity, while ENK produces reproducible family-level fingerprint-region refinement.The discussion describes this as an external test on crystal-derived molecular assemblies.Source: Discussion · p. 20
  • The oligothiophene results show that learned polarizability response preserves an experimental conjugation-dependent Raman trend across semiconductor building blocks.The discussion places this result alongside transfer to crystal-derived assemblies.Source: Discussion · p. 20
  • The framework remains effective as molecular size and chemical complexity increase, and transfers without system-specific retraining to crystal-derived assemblies and π-conjugated semiconductor building blocks.The R-3B69 benchmark and oligothiophene Raman analysis provide the reported external tests.Source: Discussion · p. 20

Methods

The methods use QM9S and QMe14S for spectroscopic supervision and separate NBO-prior training data for electronic annotations. SENK predicts response tensors with an equivariant backbone, then adds environmental refinement and electronic-prior correction; the spectroscopic adaptation includes distinct derivative routes for Hessian blocks.

1. Reference Data

  • QM9S supplies small-molecule spectroscopy references and QMe14S is the main target dataset, while SIMG and qcMol train the NBO-based electron prior.SIMG provides heterogeneous NBO graph supervision; qcMol supplies atom- and bond-resolved annotations, and the prior corpus includes PubChemQC/ZINC-derived molecules and PDBbind 2020 ligands.Source: §1 · p. 21

2. SENK architecture

  • SENK predicts four per-atom and per-edge spectroscopic quantities from geometry and passes them through harmonic normal-mode analysis to generate IR and Raman spectra.An SO(3)-equivariant EquiformerV2 backbone is paired with ENK confidence updates on selected channels and geometry-derived electronic-prior descriptors.Source: §2 · p. 21
  • The architecture couples geometric representation learning, environmental adaptation, and electronic-structure-guided tensor correction on a shared equivariant backbone.The electronic prior includes a trainable equivariant injection and bounded post-forward calibration for chemically localized residuals.Source: §2 · pp. 21–22

3. EquiformerV2 Adaptation for Spectroscopic Prediction

  • The Hessian diagonal blocks use a two-leaf mixed-derivative construction, whereas off-diagonal blocks use an edge-wise derivative route that preserves the interacting-pair curvature signal.Both routes share the equivariant representation while keeping the numerical structures of H_ii and H_ij separate.Source: §3 · p. 22
  • For spectroscopic adaptation, the polarizability branch uses EquiformerV2’s native rank-2 equivariant readout, while the complementary equivariant dipole-vector branch yields derivatives by differentiating molecular dipole with respect to Cartesian coordinates.Numerical-stability choices include non-finite-value filtering, gradient clipping, and position centering.Source: §3 · p. 22

4. Equivariant Neural Kalman (ENK) Filtering

ENK refines intermediate SO(3) representations through a learned confidence-weighted update before tensor readout. In the ENK-EP cascade, this refinement informs a subsequent selective, bounded correction using the NBO prior.

4. Equivariant Neural Kalman (ENK) Filtering

  • ENK is a learned confidence filter on the SO(3) latent representation, applied only to the implemented linear readout paths at l=0 and l=2.For each filtered degree, the state update, process uncertainty, and observation-noise logit are predicted from invariant features.Source: §4 · p. 23
  • When an atom-level electron prior is available, it modulates the observation-noise logit through a zero-initialized FiLM transformation.The stated design goal is to separate chemically meaningful feature signals from local geometric perturbations.Source: §4 · p. 23
  • An atom-resolved Kalman gain weights the observation against the learned state prior while preserving rotational equivariance.The gain is scalar for each atom and filtered degree and is broadcast over magnetic components; small gains favor the prior, while large gains preserve the observation.Source: §4 · p. 23
  • Bounded multiplicative updates and restriction to the linear l=0 and l=2 paths limit uncontrolled amplification through quadratic Clebsch-Gordan self-couplings.The paper characterizes ENK as a confidence-weighted representation update, not a generic low-pass denoiser.Source: §4 · p. 23
  • ENK judges local chemical environments in intermediate SO(3) features and partially corrects representations before the final tensor readout.The cascade then uses this judgment with the NBO prior for a bounded second-stage correction where the electronic proxy is informative.Source: §4 · p. 24

5. Multi-Stage Electron Prior (EP)

This section describes transferring a geometry-driven NBO predictor from SIMG to qcMol and integrating its electron prior into SENK with tensor-specific consistency constraints and bounded local spectral calibration. It also covers GSC, a selective second-stage correction that uses ENK to identify uncertain sites and the electron prior to guide site-specific adjustments.

5.1 NBO Predictor and Multi-Stage Transfer from SIMG to qcMol

  • An EMPP-based NBO predictor separates geometric encoding from electronic supervision, using atom, bond, and lone-pair graph tokens with equivariant pair-aware message passing and decoupled output heads [57].The predictor learns electronic observables without forcing them into a single regression target.Source: §5.1 · p. 24
  • A three-stage curriculum pretrains on SIMG, jointly trains on SIMG and qcMol, then prioritizes qcMol with SIMG replay while broadening coverage to drug-like and protein-ligand chemistry.PubChemQC/ZINC contributes drug-like molecules, while PDBbind adds protein-ligand examples biased toward bioactive bound ligands.Source: §5.1 · pp. 24–25

5.2 Construction and Injection of the Electron Prior

  • From equilibrium coordinates and atomic numbers, the predictor produces NBO descriptors mapped to atom and edge priors, whose SO(3)-equivariant corrections enter deeper blocks and whose atom prior FiLM-modulates ENK noise.The atom prior summarizes population and valence-state cues; the edge prior encodes pairwise chemical information.Source: §5.2 · p. 25

5.3 NBO Consistency-Regularized EP Adaptation

  • EP-CR couples each spectroscopic tensor to its own physical consistency relation rather than a universal correction target, using detached NBO targets and consensus scores to downweight uncertain estimates.The adaptation keeps the inherited SENK backbone, ENK module, task readout, and NBO predictor fixed while training prior-injection and adaptation components.Source: §5.3 · pp. 26–27

5.4 NBO-Guided Spectral Calibration

  • GSC activates bounded local corrections when fused evidence indicates an electronic mismatch, NBO reliability, and a chemically unusual local environment, rather than relying on predefined motif lists.Its task-specific policy combines physical residuals, agreement among independent bond-order indicators, and bond-class support in the training set.Source: §5.4 · pp. 27–28
  • GSC combines ENK confidence and branch disagreement with novelty, hydrogen-bond, and donor-acceptor evidence, then uses a mode-aware gate to direct bounded Hessian updates to participating bonds.The principal formulation targets off-diagonal Hessian blocks, which control interatomic force coupling and normal-mode frequencies.Source: §5.4 · p. 28
  • The projector targets bond-stretching adjustments that control peak positions while leaving transverse bending and torsional couplings unchanged, with increments compensating toward acoustic-sum-rule consistency.Source: §5.4 · p. 28
  • GSC is a selective local second-stage correction: ENK identifies backbone uncertainty, and the electron prior guides how the uncertain site should be corrected.It is not a generic post-hoc rescaling layer.Source: §5.4 · p. 28
  • The calibration interface also applies to intensity-response tensors, but task-specific laws use bounded trace-free bond-directional dedipole refinement or delocalization-conditioned multiplicative depolar calibration.Dedipole refinement is paired with the charge-sum constraint.Source: §5.4 · p. 28
  • The reported configuration applies runtime GSC to H_ij, allows restricted dedipole refinement, and retains EP-CR for the polarizability-derivative branch without runtime Depolar-GSC.A matched runtime audit is reported in Supplementary Section 4.5.Source: §5.4 · p. 28

6. Evaluation and local-response analyses

The evaluation uses response, invariant, frequency, and peak error measures, alongside local-response analyses. Mode comparisons, motif strata, and molecule-level bootstrap resampling provide the stated analysis procedures.

6. Evaluation and local-response analyses

  • MAE and RMSE were calculated for response quantities, invariants, frequencies, and peaks.Percentage gain relative to a stated baseline is defined so that positive gain denotes lower error.Source: §6 · p. 29
  • Local motifs are overlapping geometry-defined descriptive strata rather than mutually exclusive functional-group assignments.The supplied proxy definitions include carbonyl-like C-O, amide-like C-N, aromatic-like C-C, O/N-H donor, halogen, and high-Z categories.Source: §6 · p. 29
  • Mode analyses use target or DFT modes as anchors and retain independently matched ENK-on and ENK-off modes only when both normalized vector overlaps are at least 0.5.Stretching, bending, and torsional labels use displacement-derived internal-coordinate scores; peak records are selected from reference modes.Source: §6 · p. 29
  • For the Main Fig. 3c comparison, the trained ENK mapping is tested against an identity mapping with model state and remaining calculations fixed.Molecules are the resampling clusters; 2,000 bootstrap replicates yield pooled local MAEs and percentage gains, with 95% intervals from the 2.5th and 97.5th percentiles.Source: §6 · pp. 29–30

7. Materials-relevant molecular building-unit evaluation

The materials-relevant evaluation covers noncovalent molecular assemblies and oligothiophene Raman analysis. It applies SENK to released benchmark geometries and compares selected predicted Raman responses with experimental values.

7.1 R-3B69 noncovalent molecular-assembly benchmark

  • R-3B69 evaluates transfer of SENK to noncovalently interacting molecular assemblies on 69 crystal-derived trimers from 23 molecular families, using released geometries without benchmark-specific retraining or fine-tuning.Reference spectra are double-harmonic IR and Raman spectra calculated at ωB97M-D3(BJ)/def2-TZVPPD; paired ENK effects were averaged over the three configurations within each molecular family.Source: §7.1 · p. 30
  • The R-3B69 analysis compares full-range and separately windowed spectra, using independently normalized stick spectra broadened with a Lorentzian of 30 cm^-1 full width at half maximum.The windows are 0-400, 400-1800, and 2800-4000 cm^-1.Source: §7.1 · p. 30

7.2 Oligothiophene Raman analysis

  • The oligothiophene analysis compares SENK Raman responses for β-n-hexyl 4T, 6T, 8T, and 10T with experimental differential Raman cross-sections measured at 1064 nm.Predicted responses are normalized to 4T; experimental spectra are digitized from published spectra, and Pearson r is descriptive for four normalized values.Source: §7.2 · pp. 30–31
  • R-like and Z-like oligothiophene patterns are selected by projection scores within specified Raman windows, retaining up to five modes accounting for 80% of the squared score.The analysis uses a 30° inter-ring twist and reports no DFT calculation, basis-set treatment, multiplicity specification, or solvent model.Source: §7.2 · p. 31

8. Spectral Conversion Methods

Predicted response tensors are converted into IR and Raman spectra using DetaNet’s harmonic-analysis and line-broadening procedure, with minimal numerical safeguards. The study uses publicly available datasets and generates no new training dataset.

8. Spectral Conversion Methods

  • The conversion follows DetaNet’s harmonic-analysis and line-broadening procedure to derive normal modes, IR intensities, and Raman activities from predicted response tensors.The Hessian is assembled, symmetrized, mass weighted, and diagonalized; the same chain-rule contractions, Raman invariants, Lorentzian broadening kernel, and unit-conversion coefficients are used.Source: §8 · p. 31
  • Minimal safeguards address ill-conditioned predicted Hessians through non-finite-value cleanup, tiny diagonal regularization, and sign-safe handling of negative eigenvalues.These steps precede eigendecomposition and filtering of non-physical modes; no new spectral post-processing design was introduced.Source: §8 · pp. 31–32

Data Availability

  • QM9S, QMe14S, SIMG, and qcMol adaptation data are available through Figshare, Hugging Face, and the qcMol Download Center.No new training dataset was generated in the study.Source: Data Availability · p. 32
  • External chemical-space structures were obtained from the Raman-ChEMBL repository.Source: Data Availability · p. 32
  • Reference spectra are publicly available through the MACE-MDP dataset release on Zenodo.Source: Data Availability · p. 32
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