PhysECD: A Physics-Constrained E(3)-Equivariant Framework for Electronic Circular Dichroism Spectrum Prediction

Yi Jiang; Letian Chen; Runhan Shi; Liangzhaoxuan Han; Tong Zhu; Yang Yang

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PhysECD: A Physics-Constrained E(3)-Equivariant Framework for Electronic Circular Dichroism Spectrum Prediction

Yi Jiang, Letian Chen, Runhan Shi, Liangzhaoxuan Han, Tong Zhu, Yang Yang

arXiv:2608.21892v1Aug 22, 2026physics.chem-phcs.LG

TL;DR

Assigning chiral molecules’ absolute configuration requires costly TDDFT calculations repeated across stereoisomers and conformations. PhysECD predicts the physical quantities underlying ECD spectra with parity-aware equivariant modeling, achieving a mean/median per-molecule spectral Pearson correlation of 0.642/0.822 on CMCDS while outperforming prior learned predictors.

Problem
Absolute-configuration assignment from ECD spectra requires costly TDDFT calculations repeated for each stereoisomer and low-energy conformation.
Method
PhysECD predicts excitation energies and electric and magnetic transition dipoles with parity-aware E(3)-equivariant features, then generates spectra through a physics-based Gaussian-broadening readout.
Results
0.642/0.822 mean/median per-molecule spectral Pearson correlation on CMCDS exceeds prior learned predictors, with gains consistent across three backbones.
Takeaways & Limitations
PhysECD provides an interpretable ECD predictor that respects chiroptical symmetry by construction and improves over prior learned predictors.
Takeaways & Limitations
CMCDS contains only single-stereocenter molecules, leaving multi-center chirality and conformational-ensemble evaluation for future work.

Abstract

from arXiv · show

The electronic circular dichroism (ECD) spectrum is a primary experimental probe for assigning the absolute configuration of chiral molecules, yet interpreting a measured spectrum requires time-dependent density functional theory (TDDFT) calculations that can cost hours per molecule and must be repeated for every candidate stereoisomer and conformation. We present PhysECD, a physics-constrained, parity-aware E(3)-equivariant framework that bypasses computationally expensive TDDFT and predicts ECD spectra directly from the 3D structure of an individual conformer. Instead of regressing the spectrum as an opaque sequence, PhysECD predicts the physical quantities that generate it: per-state excitation energies and electric and magnetic transition dipoles. These quantities determine the rotatory strength R -- the dot product of the two dipoles, a pseudoscalar that reverses sign under mirror reflection -- and yield the final spectrum through a differentiable Gaussian-broadening formula derived from the underlying physics. The parity structure of the equivariant features guarantees the correct chiroptical symmetry: reflecting a molecule exactly negates the predicted spectrum. On the CMCDS dataset, PhysECD attains a per-molecule spectral Pearson correlation of 0.642 (mean) / 0.822 (median), substantially exceeding prior learned predictors while remaining physically interpretable. Experiments across multiple backbones further show that the framework is backbone-agnostic, paving the way for real-time assignment of absolute configuration.

1 Introduction

ECD can identify molecular absolute configuration, but conventional TDDFT-based spectrum generation is costly and must be repeated across stereoisomers and conformations. PhysECD addresses this bottleneck by predicting physical transition quantities with parity-aware equivariance and constructing spectra through physics-based formulas, enforcing mirror-sign reversal by design.

Motivation: ECD links differential circularly polarized-light absorption to molecular absolute configuration, enabling stereoisomer assignment by comparing calculated and experimental spectra.The conventional workflow includes conformational searches, geometry optimization, TDDFT, Boltzmann weighting, and Gaussian broadening.
Motivation: Hours-long TDDFT calculations and repetition across stereoisomers and low-energy conformations create a computational bottleneck that worsens with molecular system size.This motivates machine-learning surrogates for rapid ECD prediction.
PhysECD framework: PhysECD predicts excitation energies and electric and magnetic transition dipoles, then derives molecular rotatory strengths and spectra through parameter-free physical aggregation and Gaussian broadening.Its E(3)-equivariant backbone produces parity-labeled features, while multitask heads predict per-atom, per-state quantities.
PhysECD framework: The rotatory strength R_n is a pseudoscalar because electric and magnetic transition dipoles have polar and axial parity, respectively, so reflection reverses its sign.This physical transformation provides the basis for parity-aware chiroptical prediction.
Results and properties: PhysECD exactly negates its predicted spectrum under mirror reflection, making correct chiroptical symmetry a construction-level property rather than a learned behavior.The framework also exposes physically meaningful intermediates and consistently outperforms baselines across three backbone architectures on CMCDS.

2 Related work

Prior work spans direct learned prediction of ECD and other chiroptical spectra, chirality-aware kernels, and invariant or equivariant molecular-property models. TDDFT remains the standard theoretical route, while CMCDS provides a large-scale benchmark for data-driven ECD prediction.

Learned ECD and chiroptical prediction: ECDFormer decomposes ECD spectra into peak number, positions, and signs before rendering the spectrum from these extracted properties.It combines a geometry-enhanced GNN with a transformer encoder.
Learned ECD and chiroptical prediction: ChiDeK represents central and axial chirality using chiral-determinant kernels for chirality-aware property prediction.
Equivariant networks for molecular properties: Invariant models such as SchNet use reflection-even geometric features, whereas equivariant networks propagate directional tensorial features based on irreducible representations.
TDDFT and ECD datasets: TDDFT is the standard method for calculating theoretical ECD spectra, and CMCDS supplies CAM-B3LYP/6-31G(d) reference spectra for more than ten thousand monochiral organic molecules.

3 Method

PhysECD predicts ECD spectra from a molecule’s 3D conformation by learning excitation energies and transition dipoles, then deriving rotatory strengths and spectra through physics-based, differentiable readouts. Its parity-aware E(3)-equivariant design enforces exact enantiomer sign reversal, while multi-term training combines spectral shape, magnitude, energy, and dipole-norm objectives.

Problem definition: PhysECD targets the ECD spectrum ∆ε(λ) over 180–400 nm by predicting the lowest 20 excitation energies and rotatory strengths underlying discrete transitions.The continuous spectrum is obtained by broadening these transitions.
Data representation and enantiomer symmetry: Reflecting a molecule changes the axial magnetic dipole by det(P) = −1 while leaving the polar electric dipole without an extra sign, so the predicted enantiomer spectrum is exactly negated.This implements the expected chiroptical symmetry through the dipole parity structure.
Parity-aware equivariant backbone: Its parity-aware E(3)-equivariant backbone separates invariant scalar features from tensor channels containing polar 1o and axial 1e representations.Equiformer, DetaNet, and MACE can be used interchangeably behind the same downstream head.
Multi-task atomic heads: Multi-task atomic heads predict excitation-energy contributions, velocity electric dipoles, and magnetic dipoles, which physics-governed pooling aggregates into molecular quantities.The readout enforces En ≥ 3.0 eV and uses the velocity-form convention without learnable readout parameters.
Training objective: The loss combines Pearson-correlation shape, scaled-ellipticity magnitude, excitation-energy MSE, and a minimum-norm regularizer with fixed weights 1.0, 1.0, 0.1, and 0.2, respectively.The regularizer resolves unobservable dipole gauge freedom, while direct dipole and rotatory-strength supervision proved practically unlearnable.

4 Experiments

Experiments on CMCDS evaluate PhysECD under controlled data splits, shared training settings, ablations, and symmetry tests. PhysECD achieves strong spectral correlation and exactly reproduces the enantiomer sign flip.

Dataset and metric: 10,004 unique molecules are split into 9,003/500/501 train/validation/test sets, keeping each molecule and mirror image within the same split.The split differs from ECDFormer’s released split, and parity makes enantiomer augmentation provably redundant.
Main results: 0.642/0.822 mean/median per-molecule spectral PCC is achieved by PhysECD on the CMCDS test set.ECDFormer and ChiDeK are re-evaluated using released recipes, the same split, and identical ground-truth broadening.
Loss ablation: Removing the energy loss term collapses performance, while shape and magnitude losses are complementary and min-norm regularization further lowers shape loss.The ablation removes each of the four loss terms in turn.
Backbone ablation: Identical readouts, heads, losses, data, and training protocols isolate equivariant backbone inductive bias while matching each backbone to approximately 1.7M parameters.This controls capacity and training confounds in the backbone ablation.
Chiroptical symmetry: < 10−6 relative error is observed when reflecting every test molecule, confirming numerical-precision spectrum negation by construction.This verifies the predicted enantiomer sign flip without requiring the model to learn it from data.

5 Conclusion and limitations

PhysECD learns the physical quantities underlying ECD spectra and combines them with a parameter-free, parity-aware readout, producing an interpretable model that respects chiroptical symmetry and is far faster than TDDFT. Its current evaluation is limited to single-stereocenter molecules, leaving multi-center chirality and conformational ensembles for future benchmarks.

Conclusion: PhysECD predicts ECD spectra from learned generating quantities combined through a parameter-free, parity-aware readout.This design yields an interpretable model that respects chiroptical symmetry.
Conclusion: PhysECD improves over prior learned predictors while remaining orders of magnitude faster than TDDFT.
Limitations: CMCDS contains only single-stereocenter molecules, limiting current evaluation of multi-center chirality and conformational ensembles.Extending the framework to these settings is identified as future work.
Limitations: A larger TDDFT benchmark with multistereocenter molecules and explicit conformational search is under construction.It will test diastereomer discrimination and ensemble averaging at scale.

A Implementation and training details

PhysECD uses a shared AdamW training protocol across all backbones, with square-root-scaled learning rates, warmup and cyclic cosine annealing, gradient clipping, and validation shape-loss model selection.

Optimization protocol: All backbones use identical training with AdamW, weight decay 10−5, and a square-root-scaled learning rate.The learning rate follows lr = base_lr√(B/B0).
Learning-rate schedule: Training uses 5-epoch linear warmup followed by decoupled cyclic cosine annealing with period 398 epochs (Tmax = 199).Decoupling the cosine period from run length keeps the waveform fixed when the epoch count changes.
Training and selection: 1000 epochs of training use gradient-norm clipping at 1.0 and seed 42, with model selection based on validation shape loss (1 −PCC).The validation shape loss is defined as one minus the Pearson correlation coefficient.

B Backbone architectures and the controlled ablation

The controlled comparison uses three native equivariant backbones matched at approximately 1.7M parameters and constrained to the same physical output contract. Each model processes whole-molecule structures and predicts electric and magnetic dipole features through parity-specific channels.

Controlled ablation: Three backbones are matched to approximately 1.7M parameters by adjusting width only.The configurations are listed in Table 4.
Shared physical contract: All backbones implement the same (S, T) contract, combining invariant scalar features with equivariant tensor features.S contains 0e scalars, while T provides the equivariant channels used for dipole prediction.
Shared physical contract: A 1o equivariant channel feeds the electric dipole µ, while a 1e pseudovector channel feeds the magnetic dipole m.These parity-specific channels preserve the physical distinction between the two transition dipoles.
Receptive field: Each backbone uses its native equivariant form with a whole-molecule receptive field spanning structures up to 25.9 Å in diameter.The passage specifies a receptive field covering the entire molecule.

C Dataset deduplication and split protocol

CMCDS supplies TDDFT-derived excited-state and transition-dipole data, while PhysECD uses a fixed, deduplicated parent-molecule split and generates enantiomers within each split by exact reflection. This protocol prevents exact duplicates and mirror partners from crossing splits, unlike the ECDFormer split procedure.

PhysECD split protocol: 90%/5%/5%: PhysECD partitions 10,004 unique parent molecules into 9,003/500/501 train, validation, and test sets with a fixed seed.Each parent’s enantiomer is generated within the same split by exact reflection, with induced parity transformation applied to the labels.
PhysECD split protocol: The protocol keeps each molecule and its mirror image in the same split and prevents exact duplicates from being shared across splits.This follows from deduplicating parent molecules before splitting and generating enantiomers within their assigned split.
Comparison with the ECDFormer split: ECDFormer instead randomly shuffles a flat list of spectra and slices it by ratio without grouping by SMILES, molecule id, or enantiomer-group id.Its records are not deduplicated, so identical molecules or enantiomer partners can appear across different splits.

D Proof of the exact redundancy of enantiomer augmentation for PhysECD

For a parity-aware E(3)-equivariant PhysECD model, enantiomer augmentation is exactly redundant: reflected outputs and labels transform consistently, every loss term is invariant, and mirror copies add no gradient signal. Consequently, full-batch training from the same initialization follows the same trajectory and reaches the same optimum with or without augmentation.

Equivariant transformations: Under reflection, excitation energies remain invariant, electric dipoles transform as vectors, and magnetic dipoles transform as axial vectors.The transformations are En(g·x) = En(x), µn(g·x) = Pµn(x), and mn(g·x) = det(P)Pmn(x).
Chiroptical symmetry: The rotatory strength and broadened ECD spectrum exactly reverse sign under reflection, matching the reflected ground-truth labels.The model therefore satisfies Rn(g·x) = −Rn(x) and ∆ε(g·x) = −∆ε(x).
Loss invariance: Every training loss term is invariant under reflection, including energy, dipole-norm, spectral-shape, and magnetic-transition terms.This follows from orthogonality preserving norms, Pearson correlation being invariant to a common sign flip, and SmoothL1 depending on prediction-target differences.
Optimization consequence: Mirror samples have identical loss functions and gradients to their originals, so full-batch gradient descent follows an identical trajectory to an identical optimum.The mirror copies therefore contribute no information beyond the original samples.

E Physics readout and broadening: constants and derivations · F Additional ECD prediction examples

PhysECD derives three convention-fixed numerical factors for rotatory strength, Gaussian band normalization, and molar ellipticity, integrating them into a differentiable, unfit spectrum construction. Additional examples illustrate ECD predictions.

E Physics readout and broadening: constants and derivations: PhysECD uses three numerical factors fixed by operator and unit conventions.These factors govern the physics readout and spectrum construction.
E Physics readout and broadening: constants and derivations: Rotatory strength follows R_n = C_R(µ_n·m_n)/E_n after combining Gaussian’s velocity convention with unit conversions.The velocity-form expression contains 1/(2ω_n), with ω_n = E_n/E_h.
E Physics readout and broadening: constants and derivations: 2.296 × 10^-39 is the cgs prefactor used for standard Gaussian band normalization.Because R_cgs = 10^-40R, this determines the normalization for E and σ in eV.
E Physics readout and broadening: constants and derivations: Molar ellipticity is derived by combining the Beer–Lambert relation with radian-to-degree and mol-to-dmol conversions.This supplies the third convention-dependent factor in the readout.
E Physics readout and broadening: constants and derivations: All three factors enter the differentiable spectrum in closed form, allowing spectrum-loss gradients to reach E_n, µ_n, m_n, and the network without fitted normalization.The construction is therefore directly differentiable with respect to the predicted physical quantities.
F Additional ECD prediction examples: Figure 3 presents additional ECD prediction examples.The figure is identified as covering additional ECD prediction examples.

G Prediction bottleneck of the transition dipoles and rotatory strength

Predicting magnetic transition dipoles and rotatory strengths is a central bottleneck because magnetic dipoles are physically difficult to represent and direct supervision failed to generalize. PhysECD therefore learns the final smoothed spectral shape rather than microscopic transition moments for each excited state.

Transition-dipole bottleneck: The sign of R_n is dictated by the magnetic transition dipole m_n, which is harder to predict than the electric dipole μ_n.Magnetic dipoles face gauge-origin dependence in approximate quantum methods and require accurate modeling of the wavefunction’s complex phase.
Transition-dipole bottleneck: Direct supervision on magnetic dipoles or rotatory strengths completely failed to generalize in preliminary experiments.This result motivated avoiding direct prediction of these microscopic quantities.
Framework design: The framework instead learns the final observable: the smoothed spectral shape, rather than precise transition moments for individual excited states.This design directly addresses the optimization bottleneck associated with microscopic transition-moment prediction.