physics.chem-phOct 2, 2026

Electronic Density versus Geometry for Machine-Learned Molecular Absorption Spectra

Authors: Siddharth Dhanpal, Peter Elliott, Paolo Emilio Trevisanutto, Alin M. Elena, Gilberto Teobaldi

Organizations: Scientific Computing Department, Science and Technology Facilities Council, Rutherford Appleton Laboratory, Didcot OX11 0QX, United Kingdom. · Scientific Computing Department, Science and Technology Facilities Council, Daresbury Laboratory, Warrington, WA4 4AD, United Kingdom.

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Abstract

Molecular optical absorption spectroscopy provides a direct probe of electronic structure and is widely used for molecular identification, interpretation of photophysical behaviour, and planning of spectroscopy experiments. Calculating the absorption spectra using first-principle excited-state methods, however, is computationally demanding, at least compared to ground-state calculations, which limits their routine application across large molecular sets. Machine-learning (ML) surrogates can reduce this cost and allow rapid spectral prediction. However, their performance depends strongly on how molecular information is represented. Here, we compare using the ground-state electron density versus the molecular geometry as inputs to a ML model for predicting absorption spectra, for a training set of 6874 molecules selected from the QM7 dataset. For each of these molecules, the density was calculated using density functional theory (DFT) and the absorption spectrum was calculated using linear-response (LR) time-dependent DFT (TDDFT). Utilizing the ground-state density as the input to the ML model is motivated by the Hohenberg-Kohn and Runge-Gross theorems, and the fact that the ground-state density encodes information about bonding, charge localisation, and electronic delocalisation. Hence, it may be a more judicious starting point for the ML model compared to the geometry, as it effectively decouples the chemistry of the ground-state. The question we test is whether the benefits of using the density outweigh the (notprohibitive) penalty of requiring an additional single-point DFT calculation for the density. We find that the density-based convolutional neural network achieves a validation correlation of 0.9926, compared with 0.9795 for the best geometry-based graph model, reducing the residual decorrelation, by approximately 64%.

Figures & tables

Figure 1: Schematic of the 3D-CNN model mapping an electron-density cube to a normalized spectrum over LL frequency bins via a Softmax output.
Figure 2: Schematic of the geometry-based GNN model (SchNet / DimeNet++ / MACE) to predict a normalized spectrum.
Table 1: Validation performance for graph-based spectrum predictors (SchNet and DimeNet++). “Int.” denotes SchNet interaction blocks; DimeNet++ columns correspond to num_blocks (Blocks), num_radial (Radial), and maximum neighbors (MaxNN). Because loss functions have different numeric scales (e.g., Wasserstein vs KL/JS), correlation and MAE provide the most comparable cross-run summary
Exp.Hidden irrepsInt.ℓmax⁡\ell_{\max}LRDropoutVal lossVal corr
exp01128x0e + 128x1o238×10−58\times 10^{-5}0.100.01710860.979423
exp02128x0e + 128x1o338×10−58\times 10^{-5}0.100.01719740.979223
exp03128x0e + 128x1o438×10−58\times 10^{-5}0.100.01746570.978854
exp04192x0e + 192x1o238×10−58\times 10^{-5}0.100.01717630.979475
exp0596x0e + 96x1o + 96x2e238×10−58\times 10^{-5}0.100.01753850.979077
exp06128x0e (scalar control)238×10−58\times 10^{-5}0.100.01836230.977993
Table 2: MACE sweep on spectra prediction ( L=900L=900 bins). “Int.” denotes the number of interaction (message-passing) blocks
Training lossVal MAEVal MSEVal corrVal KLVal JSVal W1W_{1}
Wasserstein5.12×10−45.12\times 10^{-4}6.81×10−76.81\times 10^{-7}0.8264120.7238920.0690204.307083
Pearson corr.8.45×10−48.45\times 10^{-4}9.98×10−79.98\times 10^{-7}0.9773880.4352140.125681205.415649
JS divergence1.38×10−4\textbf{1.38}\times 10^{-4}6.17×10−8\textbf{6.17}\times 10^{-8}0.9793380.0173220.0041974.735388
KL divergence1.39×10−41.39\times 10^{-4}6.20×10−86.20\times 10^{-8}0.9793140.0172530.0042104.724957
MAE1.43×10−41.43\times 10^{-4}6.89×10−86.89\times 10^{-8}0.9767480.0204950.0047965.737404
MSE2.06×10−42.06\times 10^{-4}1.28×10−71.28\times 10^{-7}0.9596220.0451740.01039716.948169
Table 3: Loss-function comparison for the selected MACE spectrum model with hidden irreps 128×0e+128×1o128\times 0e+128\times 1o , two interaction blocks, ℓmax⁡=3\ell_{\max}=3 , rmax⁡=5.0r_{\max}=5.0 , eight radial Bessel functions, learning rate 8×10−58\times 10^{-5} , and dropout 0.10. For each row, the model is trained using the loss shown in the first column, while all other quantities are evaluated as validation metrics. JS and KL divergence give the best balanced performance across correlation, pointwise error, and distributional metrics
RankLayersKernelBaseFDropoutVal lossVal corrVal MAE
171580.206.66e-060.99267.14e-05
27760.107.18e-060.99227.54e-05
38540.101.84e-050.97941.44e-04
44760.302.59e-050.97571.57e-04
510980.302.98e-050.96821.79e-04
Table 4: Top-5 3D-CNN density-to-spectrum models ranked by validation correlation. The validation loss corresponds to the mean KL divergence reported on the validation set. Loss and MAE values are shown in scientific notation and rounded to three significant figures; correlations are rounded to four decimal places
Figure 3: Representative examples of predicted absorption spectra compared with the corresponding ground-truth spectra. Each panel shows one validation molecule using Simplified Molecular Input Line Entry System (SMILES) notation [ 52 ] : (a) C#CC#CCOC, (b) CNCC@@HCO, and (c) C=C(CC)NC=[N]. The horizontal axis is frequency in atomic units (a.u.) and the vertical axis is the spectral intensity. For each molecule, the model prediction (blue) is overlaid with the reference spectrum (orange), enabling a direct visual comparison of peak positions, relative amplitudes, and overall spectral shape across the full frequency range. The Pearson correlation coefficient between the predicted and true spectra is reported within each panel (Corr), computed over all frequency bins shown. These examples illustrate that the model captures the dominant spectral features and broad trends, while deviations remain in low amplitude peaks and the finer structure.
Figure 4: Comparison of two molecules with similar spectral structure and similar molecules, shown separately for the reference spectra and the model outputs. The top panel overlays the ground-truth absorption spectra for the two validation molecules C#CC#CCCO and C#CC#CCOC as a function of frequency (a.u.), highlighting their closely related peak positions and relative intensities. The bottom panel shows the corresponding model-predicted spectra for the same two molecules, plotted over the same frequency range. The predictions reproduce the overall bimodal profile present in both molecules and capture subtle inter-molecular differences. For instance, the peak and region near 0.3 ∼\sim a.u. has a larger amplitude for C#CC#CCOC in the ground truth, and the same trend is reflected in the model output.
ModelParametersTraining time (h)Hardware
SchNet228,5342.30NVIDIA A100 40 GB
DimeNet++6,545,8083.80NVIDIA A100 40 GB
MACE2,814,2113.22NVIDIA A100 40 GB
Density 3D-CNN401,295,28091.07NVIDIA A100 40 GB
Table 5: Computational characteristics of the selected best-performing spectrum-prediction models. Training times correspond to the estimated wall-clock duration of the complete training runs and exclude the wider hyperparameter searches. All models were trained using a single NVIDIA A100 GPU with 40 GB of memory
ModelLatencyThroughputGPU memory
 (ms molecule -1 )(molecules s -1 )(MB)
SchNet4.69213.15∼\sim 10
MACE12.4480.39106
Density 3D-CNN43.9522.752397
Table 6: Batch-size-one inference performance of the selected spectrum-prediction models on an NVIDIA A100 GPU. Latency denotes the average inference time per molecule, while throughput gives the corresponding number of molecular predictions per second. GPU memory denotes the peak CUDA memory allocated during the post-hoc inference benchmark
Figure 5: Validation performance for prediction of the spectral normalisation constant α\alpha , defined as the integrated discrete spectral weight within the selected frequency window. The left panel compares the predicted and reference values for molecules in the validation set. The solid line denotes ideal agreement, y=xy=x , and the coefficient of determination is R2=0.9628R^{2}=0.9628 . The right panel shows the distribution of residual errors, defined as predicted minus reference. Most predictions lie close to the reference line, although the residual distribution shows a small tendency towards negative errors, indicating slight underestimation for part of the validation set.
Figure 6: Effect of chemical categories on the median Pearson correlation between predicted and reference absorption spectra. Each point shows the difference in median Pearson correlation between molecules where the category is present and absent, computed as True group minus False group. Positive values indicate improved spectral-shape agreement for molecules containing the category, while negative values indicate reduced agreement. The annotations report the sample counts as ntrue/nfalsen_{\mathrm{true}}/n_{\mathrm{false}} .
Figure 7: Median Pearson correlation between predicted and reference absorption spectra grouped by chemical category. For each category, the False group contains molecules where the category is absent, while the True group contains molecules where the category is present. The annotations indicate the number of molecules in each group. Higher values correspond to better agreement between the predicted and reference spectra.
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