QALPA: Property-guided diffusion modeling for efficient exploration of chemical spaces of flexible molecules

Michael Hanna; Julian Cremer; Zekiye Erarslan; Leonardo Medrano Sandonas

physics.chem-phSep 15, 2026

QALPA: Property-guided diffusion modeling for efficient exploration of chemical spaces of flexible molecules

Authors: Michael Hanna, Julian Cremer, Zekiye Erarslan, Leonardo Medrano Sandonas

Organizations: Faculty of Computer Science, TUD Dresden University of Technology, 01062 Dresden, Germany · Machine Learning & Computational Sciences, Pfizer Worldwide R&D, Berlin, Germany · Center for Advanced Systems Understanding (CASUS), Conrad-Schiedt-Straße 20, Görlitz 02826, Germany · Helmholtz Zentrum Dresden-Rossendorf, Bautzner Landstraße 400, Dresden 01328, Germany ·

Abstract

Exploring the chemical space of flexible molecules remains challenging because the vast number of possible compounds and conformations, together with the increasing cost and limited generalization of 3D generative models for larger and more complex molecules, restrict access to unexplored chemistry. Here, we introduce QALPA ("Quantum-Aware Learning for Property-space Augmentation"), a property-guided generative framework that combines an E(3)-equivariant diffusion model with active learning and efficient quantum-mechanical (QM) methods to iteratively explore targeted QM property manifolds. By coupling generation with physics-based evaluation, QALPA improves molecular sampling and model reliability in sparsely populated regions of chemical space. Our results show that training on complementary QM datasets spanning both small (QM7-X) and large (Aquamarine) drug-like compounds enables accurate molecular generation across a broad size range, improving transferability beyond the training distribution for complex property manifolds involving both extensive and intensive properties. As a proof of concept, QALPA coupled with the machine learning-augmented tight-binding method EquiDTB efficiently augments alloQM, a QM dataset introduced in this work, comprising 6,253 conformers of allosteric drug molecules, by populating sparse regions of the property landscape defined by the many-body dispersion energy and HOMO-LUMO energy gap. These results demonstrate that the integration of generative AI with efficient ML/QM methods offers a practical pathway toward augmenting sparse QM datasets and sustainably expanding the exploration of chemical space for molecular discovery.

Figures & tables

Figure 1: Scheme of the QALPA framework (“Quantum-Aware Learning for Property-space Augmentation”). The forward block trains the EQDIFF generative model on an initial dataset and generates molecules at each navigation coordinate associated with the target property values $P^{q}={P_{1}^{q},P_{2}^{q}}$ , where $q$ denotes the active learning (AL) iteration and $P_{1}$ and $P_{2}$ define the selected two-dimensional property manifold. The backward block optimizes the generated geometries using EquiDTB (or standard DFTB) and computes the target properties at either the DFT or DFTB level using FHI-aims or DFTB+ packages, respectively. After screening based on property accuracy and synthetic accessibility (SA) score, the selected molecules are added to the training dataset, and the AL loop proceeds until the predefined trajectory has been traversed, i.e. , $q>N_{\rm iter}$ , where $N_{\rm iter}$ is the total number of AL iterations.

Figure 2: Scalability analysis of EQDIFF models based on extensive properties. (a) Two-dimensional property space defined by the molecular polarizability $\alpha$ and many-body dispersion energy $E_{\mathrm{MBD}}$ for the QM7-X (blue circles) and Aquamarine (AQM, red triangles) datasets. Both properties increase in magnitude with molecular size, with QM7-X populating the small- $\alpha$ and near-zero- $E_{\mathrm{MBD}}$ region, while AQM extends toward larger $\alpha$ and more negative $E_{\mathrm{MBD}}$ . (b,c) Distributions of $\alpha$ and number of atoms, respectively, showing the smaller molecular size range covered by QM7-X and the broader range of larger molecules represented in AQM. (d) Relative error as a function of the number of atoms for molecules generated at a given target $\alpha$ using models trained on QM7-X, AQM, and the combined QM7X+AQM dataset (left to right).

Figure 3: Property-guided generation of flexible molecules with QALPA. (a) Density plot of the property space defined by the many-body dispersion energy $E_{\mathrm{MBD}}$ and HOMO–LUMO energy gap $E_{\mathrm{gap}}$ for the combined QM7X+AQM dataset. Screened molecules generated at each navigation coordinate (red circles) are shown, with the trajectory progressing from smaller molecules with larger $E_{\mathrm{gap}}$ toward larger molecules with smaller $E_{\mathrm{gap}}$ . Screened molecules at each AL iteration are represented by a distinct symbol and color. Three independent validation targets (T 1 –T 3 , black crosses) were defined to compare QALPA with the baseline model. (b) Number of molecules retained at each stage of the screening procedure across the AL iterations: initially generated (gray), passing the SA-score criterion (green), and passing the property-accuracy criterion (blue). (c) Average radius of gyration ( $R_{g}$ , circles) as a function of the Shannon entropy ( $H$ ) for the screened molecular set at each AL iteration. Circle color indicates the average number of non-H atoms. Relative errors for (d) $E\mathrm{MBD}$ and (e) $E\mathrm{gap}$ for molecules screened at validation targets T 1 –T 3 , shown as split violin plots comparing the QALPA (green) and baseline (purple) models.

Figure 4: Influence of geometry optimization on QALPA-generated molecules. Distributions of (a) C–C bond lengths and (b) C–C–C bond angles for screened molecules generated by QALPA models trained without geometry optimization (wo-opt, blue) and with geometry optimization included (w-opt, red). Geometry optimization calculations were carried out using the DFTB3+MBD method. The distributions include all screened molecules across the six navigation coordinates. Variation of the relative error in (c) $E_{\mathrm{MBD}}$ and (d) $E_{\mathrm{gap}}$ as a function of the radius of gyration ( $R_{g}$ ) for screened molecules generated at each AL iteration across the three validation targets (T 1 –T 3 ). The lateral panels show the corresponding distributions of relative errors for the two QALPA models.

Figure 5: Benchmarking the EquiDTB26 model. (a) Boxplots of the mean absolute errors (MAEs) for atomic forces (

F_{\rm at}

) predicted by the best-performing EquiDTB26 model across molecules of increasing chemical complexity, ranging from molecules containing a single unique element to those containing seven unique elements. The analysis uses a subset of the QCML dataset comprising

\approx

5.7 million equilibrium and non-equilibrium conformations of small molecules containing up to eight non-hydrogen atoms. (b) Comparison of EquiDTB26, the previously developed EquiDTB25 model, and standard DFTB3 for force predictions across three validation benchmarks involving molecular dimers (S66x8 and QUID) and flexible molecules (paracetamol, zaprinast, and ligand 2Q5k). Boxplots show the force MAEs on a logarithmic scale, with the corresponding mean values indicated above each box.

Figure 6: Application of the QALPA framework to augment a quantum-mechanical (QM) dataset of allosteric drug molecules. (a) Workflow: conformers of allosteric drugs are generated and processed using EquiDTB for geometry optimization and property evaluation to construct the alloQM dataset ( $6{,}253$ conformers), which is combined with QM7-X to form the alloQMX training set ( $47{,}790$ conformers). (b) Explored property space defined by the many-body dispersion energy $E_{\mathrm{MBD}}$ and HOMO–LUMO energy gap $E_{\mathrm{gap}}$ , with points colored according to the number of atoms. Red circles indicate the navigation coordinates along the planned trajectory (AL iterations 1–6). The chemical and structural diversity of the screened molecules generated across the six iterations is compared with that of alloQM in terms of (c) number-of-atom distributions, (d) elemental composition, and (e) all angle distributions. The dashed line at 120.7 ∘ indicates the most frequently occurring angle in the alloQM dataset.

Appendix figures & tables13 assets

Supplementary material from the paper’s appendix.

Appendix

Dataset	Property	1	2	3	4	5	6	7
QM7-X	$N$	15	20	25	26	–	–	–
	$\alpha$	83.5278	97.0245	110.5213	113.2206	–	–	–
	$E_{\rm MBD}$	-0.2332	-0.3485	-0.4639	-0.4869	–	–	–
AQM	$N$	15	20	25	50	90	100	102
	$\alpha$	102.2349	136.4424	170.6499	341.6874	615.3474	683.7624	697.4454
	$E_{\rm MBD}$	-0.1761	-0.3596	-0.5431	-1.4605	-2.9283	-3.2952	-3.3686

Appendix

Table S 1: List of validation targets in the property space defined by the molecular polarizability

\alpha

and many-body dispersion energy

E_{\rm MBD}

for EQDIFF models trained on QM7-X, AQM, and the combined QM7-X+AQM datasets.

Property	1	2	3	4	5	6
$N$	23	41	44	57	69	78
$E_{\rm MBD}$	-0.5	-1.0	-1.4	-1.9	-2.2	-2.5
$E_{\rm gap}$	4.0	3.1	2.8	2.7	3.2	3.5

Appendix

Table S 2: List of navigation coordinates in the property space defined by the many-body dispersion energy

E_{\rm MBD}

and HOMO–LUMO energy gap

E_{\rm gap}

for QALPA models trained on the combined QM7-X+AQM dataset, considering either with or without geometry optimization at each active learning iteration.

Property	T 1	T 2	T 3
$N$	50	60	75
$E_{\rm MBD}$	-1.4	-1.9	-2.4
$E_{\rm gap}$	4.6	3.7	4.7

Appendix

Table S 3: List of additional validation targets (T) in the property space defined by the many-body dispersion energy

E_{\rm MBD}

and HOMO–LUMO energy gap

E_{\rm gap}

for QALPA models trained on the combined QM7-X+AQM dataset, considering either with or without geometry optimization at each active learning iteration.

Fig. S 1: Property-guided generation of flexible molecules with QALPA, considering geometry optimization at every AL loop. (a) Density plot of the property space defined by the many-body dispersion energy $E_{\mathrm{MBD}}$ and HOMO–LUMO energy gap $E_{\mathrm{gap}}$ for the combined QM7X+AQM dataset. Screened molecules generated at each navigation coordinate (red circles) are shown, with the trajectory progressing from smaller molecules with larger $E_{\mathrm{gap}}$ toward larger molecules with smaller $E_{\mathrm{gap}}$ . Screened molecules at each AL iteration are represented by a distinct symbol and color. Three independent validation targets (T 1 –T 3 , black crosses) were defined to compare QALPA model with the baseline model. (b) Number of molecules retained at each stage of the screening procedure across the AL iterations: initially generated (gray), passing the SA-score criterion (green), and passing the property-accuracy criterion (blue).

Fig. S 2: Comparison of the atomic-force ( $F_{\rm atom}$ ) distributions at each iteration of the QALPA active learning workflow, with and without geometry optimization. Mean values are reported above each boxplot.

Fig. S 3: Property-guided generation of flexible molecules with QALPA, considering geometry optimization at every AL loop. Relative errors for (left panel) $E\mathrm{MBD}$ and (right panel) $E\mathrm{gap}$ for molecules screened at validation targets T 1 –T 3 , shown as split violin plots comparing the QALPA (green) and baseline (purple) models.

Fig. S 4: Elemental composition and molecular-size distribution of the QCML dataset. (a) Number of molecular conformations containing each chemical element. (b) Distribution of the number of atoms per molecular conformation.

Model	Energy MAE	Force MAE
	(kcal/mol/atom)	(kcal/mol/Å)
500k	0.132	1.716
1M	0.087	1.228
2M	0.080	1.150
3M	0.065	0.987
4M (EquiDTB26)	0.059	0.905

Appendix

Table S 4: Mean absolute errors (MAEs) in the test set for energy and force predictions of models trained on different subsets of the QCML dataset. Energy errors are reported per atom.

Fig. S 5: Variation of the mean absolute error (MAE) of atomic forces as a function of the intermolecular distance scaling factor $q$ for molecular dimers from the (a) S66 $\times$ 8 and (b) QUID benchmarks. Results are shown for EquiDTB25 and EquiDTB26 using PBE0+MBD reference atomic forces. Error values represent averages over all dimers at each distance scale.

Fig. S 6: Correlation plots between PBE0+MBD reference atomic forces and those predicted by EquiDTB25 and EquiDTB26 for 200 molecular conformations extracted from molecular dynamics trajectories at 300 K of (a) paracetamol, (b) ligand 2Q5k, and (c) zaprinast.

Model	Flexible Molecules			S66 $\times$ 8	QUID
	Zaprinast	Ligand 2Q5K	Paracetamol
EquiDTB25	1.059	0.433	0.366	0.522	–
500k	0.885	0.519	0.567	0.630	2.228
1M	0.711	0.399	0.453	0.432	1.588
2M	0.894	0.379	0.363	0.390	1.556
3M	0.700	0.349	0.393	0.436	1.484

Appendix

Table S 5: Mean absolute error (MAE) of the atomic forces (kcal/mol/Å) for the validation benchmarks obtained with EquiDTB25 and QCML-trained correction models of increasing training-set size. The best-performing model for each benchmark is highlighted in bold.

Label	Property name	Dim
$\vec{R}$	XYZ atomic coordinates	N,3
Z	Atomic numbers	N
$\text{E}_{\text{band}}$	Band energy	1
$\text{E}_{\text{H0}}$	Reference density energy	1
$\text{E}_{\text{scc}}$	Self-consistent charge energy	1
$\text{E}_{\text{3rd}}$	Third-order correction energy	1

Appendix

Table S 6: List of quantum-mechanical properties included in the alloQM dataset, which contains 6,253 conformations of 241 unique allosteric drug molecules. These features were calculated using the semi-empirical third-order DFTB method (DFTB3) supplemented with a treatment of many-body dispersion (MBD) for van der Waals interactions. Structures were optimized using the EquiDTB26 model.

Property	1	2	3	4	5	6
$N$	23	41	44	57	69	78
$E_{\rm MBD}$	-0.5	-1.0	-1.4	-1.9	-2.2	-2.5
$E_{\rm gap}$	4.0	3.1	2.8	2.7	3.2	3.5

Appendix

Table S 7: List of navigation coordinates in the property space defined by the many-body dispersion energy

E_{\rm MBD}

and HOMO–LUMO energy gap

E_{\rm gap}

for QALPA model trained on the alloQM dataset.

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MolPaQ: Modular Quantum-Classical Patch Learning for Interpretable Molecular Generation

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Department of Computer Science, Tulane University, New Orleans, LA, USA.

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