arXiv subjects

O. Anatole von Lilienfeld

Publications and source records attributed to O. Anatole von Lilienfeld.

At least 19 recordsLinked to original sources

arXiv2026

A strategic roadmap for an atomistic machine-learning ecosystem

Data-driven machine learning (ML) techniques have become an essential tool in many domains of science. Their application to atomistic simulations of matter is particularly widespread and impactful. This success is due largely to the existence of a well-developed and established physics-based modeling framework, ranging from first-principles electronic-structure calculations to molecular dynamics and statistical sampling, into which ML was integrated naturally to reshape long-standing trade-offs between accuracy, efficiency, and scale. Nevertheless, this integration raises both conceptual and practical challenges, from choosing between data-centric and physics-based modeling approaches to adapting established software stacks to modern hardware accelerators and ML libraries. As the field evolves rapidly, fueled in part by widespread enthusiasm but also by tangible impact, it seems appropriate to take a moment to consider the current state of the art and open challenges, and reflect on what can be done to better coordinate efforts across the community. With this goal in mind, several members of this community met in Lausanne in January 2026 at CECAM to discuss algorithms, models, software and hardware infrastructure, and the most promising scientific applications that have become possible thanks to the use of artificial intelligence in atomic-scale simulations. This strategic roadmap paper summarizes the outcomes of these discussions, suggesting some long-term goals, and some concrete actions, to establish a healthy, sustainable and impactful atomistic ML ecosystem.

physics.chem-ph↗

arXiv2026

Alchemical harmonic approximation based potential for iso-electronic diatomics: Foundational baseline for $Δ$-machine learning

We introduce the alchemical harmonic approximation (AHA) of the absolute electronic energy for charge-neutral iso-electronic diatomics at fixed interatomic distance $d_0$. To account for variations in distance, we combine AHA with this Ansatz for the electronic binding potential, $E(d)=(E_{u}-E_s) \left(\frac{E_c-E_s}{E_u-E_s}\right)^{\sqrt{d/d_0}}+E_s$, where $E_u,E_c,E_s$ correspond to the energies of united atom, calibration at $d_0$, and sum of infinitely separated atoms, respectively. Our model covers the entire two-dimensional electronic potential energy surface spanned by distance and difference in nuclear charge from which only one single point (with elements of nuclear charge $Z_1,Z_2$ and distance $d_0$) is drawn to calibrate $E_c$. Using reference data from pbe0/cc-pVDZ, we present numerical evidence for the electronic ground-state of all neutral diatomics with 8, 10, 12, 14 electrons. We assess the validity of our model by comparison to legacy interatomic potentials (Harmonic oscillator, Lennard-Jones, and Morse) within the most relevant range of binding (0.7 - 2.5 A), and find comparable accuracy if restricted to single diatomics, and significantly better predictive power when extrapolating to the entire iso-electronic series. We also investigated $Δ$-learning of the electronic absolute energy using our model as baseline. This baseline model results in a systematic improvement, effectively reducing training data needs for reaching chemical accuracy by up to an order of magnitude from $\sim$1000 to $\sim$100. By contrast, using AHA+Morse as a baseline hardly leads to any improvement, and sometimes even deteriorates the predictive power. Inferring the energy of unseen CO converges to a prediction error of $\sim$0.1 Ha in direct learning, and $\sim$3 Ha with our baseline.

physics.chem-ph↗

arXiv2026

Approximate label symmetries improve data efficiency

Enforcing feature symmetries in machine learning (ML) models is a common strategy to mitigate data scarcity. Confirming expectations from statistical learning theory, we show that exact, as well as approximate, label symmetries can also improve data efficiency. We illustrate the idea for the s, p, d orbital densities of the electron in the hydrogen atom, for the three vibrational normal modes of the water molecule, and for its full 3D potential energy hypersurface. Resulting ML models of electron density and potential energies exhibit superior learning curves, demonstrating improved generalization efficiency. We observe that learning curves similarly improve even when label symmetries are not exact - up to the convergence floors set by the degree to which the symmetry is approximate. Further improvements are obtained for approximate label symmetries in the molecular potential energy surface, using a Hessian-based correction that suppresses the leading order term in the error.

physics.chem-ph↗

arXiv2025

Calculated state-of-the art results for solvation and ionization energies of thousands of organic molecules relevant to battery design

We present high-quality reference data for two fundamentally important groups of molecular properties related to a compound's utility as a lithium battery electrolyte. The first property is energy changes associated with charge excitations of molecules, namely ionization potential and electron affinity. They were estimated for 7000 randomly chosen molecules with up to 9 non-hydrogen atoms C, N, O, and F (QM9 dataset) using the DH-HF, DF-HF-CABS, PNO-LMP2-F12, and PNO-LCCSD(T)-F12 methods as implemented in the Molpro software, and the aug-cc-pVTZ basis set. Additionally, we provide the corresponding atomization energies at these levels of theory, as well as the CPU time and disk space used during the calculations. The second property is solvation energies for 39 different solvents, which we estimate for 18361 molecules connected to battery design (Electrolyte Genome Project dataset), 309463 randomly chosen molecules with up to 17 non-hydrogen atoms C, N, O, S, and halogens (GDB17 dataset), as well as 88418 atoms-in-molecules of the ZINC database of commercially available compounds and 37772 atoms-in-molecules of GDB17. For these calculations we used the COnductor-like Screening MOdel for Real Solvents (COSMO-RS) method; we additionally provide estimates of gas-phase atomization energies, as well as information about conformers considered during the COSMO-RS calculations, namely coordinates, energies, and dipole moments.

physics.chem-ph↗

arXiv2025

AI4X Roadmap: Artificial Intelligence for the advancement of scientific pursuit and its future directions

Artificial intelligence and machine learning are reshaping how we approach scientific discovery, not by replacing established methods but by extending what researchers can probe, predict, and design. In this roadmap we provide a forward-looking view of AI-enabled science across biology, chemistry, climate science, mathematics, materials science, physics, self-driving laboratories and unconventional computing. Several shared themes emerge: the need for diverse and trustworthy data, transferable electronic-structure and interatomic models, AI systems integrated into end-to-end scientific workflows that connect simulations to experiments and generative systems grounded in synthesisability rather than purely idealised phases. Across domains, we highlight how large foundation models, active learning and self-driving laboratories can close loops between prediction and validation while maintaining reproducibility and physical interpretability. Taken together, these perspectives outline where AI-enabled science stands today, identify bottlenecks in data, methods and infrastructure, and chart concrete directions for building AI systems that are not only more powerful but also more transparent and capable of accelerating discovery in complex real-world environments.

physics.soc-ph↗

arXiv2025

Data-Error Scaling Laws in Machine Learning on Combinatorial Mutation-prone Sets: Proteins and Small Molecules

We investigate trends in the data-error scaling laws of machine learning (ML) models trained on discrete combinatorial spaces that are prone-to-mutation, such as proteins or organic small molecules. We trained and evaluated kernel ridge regression machines using variable amounts of computational and experimental training data. Our synthetic datasets comprised i) two naïve functions based on many-body theory; ii) binding energy estimates between a protein and a mutagenised peptide; and iii) solvation energies of two 6-heavy atom structural graphs, while the experimental dataset consisted of a full deep mutational scan of the binding protein GB1. In contrast to typical data-error scaling laws, our results showed discontinuous monotonic phase transitions during learning, observed as rapid drops in the test error at particular thresholds of training data. We observed two learning regimes, which we call saturated and asymptotic decay, and found that they are conditioned by the level of complexity (i.e. number of mutations) enclosed in the training set. We show that during training on this class of problems, the predictions were clustered by the ML models employed in the calibration plots. Furthermore, we present an alternative strategy to normalize learning curves (LCs) and introduce the concept of mutant-based shuffling. This work has implications for machine learning on mutagenisable discrete spaces such as chemical properties or protein phenotype prediction, and improves basic understanding of concepts in statistical learning theory.

physics.chem-ph↗

arXiv2025

Alchemical diastereomers from antisymmetric alchemical perturbations

The energy difference between two iso-electronic systems can be accurately approximated by the alchemical first order Hellmann-Feynmann derivative for the averaged Hamiltonian. This approximation is exact up to third order because even-order contributions cancel out. This finding holds for any iso-electronic compound pair (dubbed `alchemical diastereomers'), regardless of differences in configuration, composition, or energy, and consequently, relative energy estimates for all possible iso-electronic alchemical diastereomer pairs, require only O(1) self-consistent field cycles for any given averaging reference Hamiltonian. We discuss the relation to the Verlet algorithm, alchemical harmonic approximation (AHA) [J. Chem. Phys. 162, 044101 (2025)], relative properties such as forces, ionization potential or electron affinities, and Levy's formula for relative energies among iso-electronic systems that uses the averaged electron density of the two systems [J. Chem. Phys. 70, 1573 (1979)]. Numerical estimates accurately reflect trends in the charge-neutral iso-electronic diatomic molecule series with 14 protons (N$_2$, CO, BF, BeNe, LiNa, HeMg, HAl), with systematically increasing errors. Using alchemical Hellmann-Feynman derivatives for toluene, we demonstrate the concept's broader applicability by estimating relative energies for all 36 possible alchemical diastereomer pairs from vertical iso-electronic charge-neutral antisymmetric BN doping of toluene's aromatic ring, with mean absolute errors of a few milli-Hartrees.

physics.chem-ph↗

arXiv2025

Antisymmetry rules of response properties in certain chemical spaces

Understanding chemical compound space (CCS), a set of molecules and materials, is crucial for the rational discovery of molecules and materials. Concepts of symmetry have recently been introduced into CCS to account for near degeneracies and differences in electronic energies between iso-electronic materials. In this work, we present approximate relationships of response properties based on a first-principles view of CCS. They have been derived from perturbation theory and antisymmetry considerations involving nuclear charges. These rules allow approximate predictions of relative response properties of pairs of distinct compounds with opposite nuclear charge variations from a highly symmetric reference material, without the need for experiments or quantum chemical calculations of each compound. We numerically and statistically verified these rules for electric and magnetic response properties (electric dipole moment, polarizabilities, hyperpolarizabilities, and magnetizabilities) among charge-neutral and iso-electronic boron nitride-doped polycyclic aromatic hydrocarbon derivatives of naphthalene, anthracene, and pyrene. Our analysis indicates that, despite their simplicity, antisymmetry rule-based predictions are remarkably accurate, enabling dimensionality reduction of CCS. The rules predict the electric response properties more accurately than the magnetizabilities. The electric response properties in alchemical perturbation density functional theory were investigated to clarify the origin of this predictive power.

cond-mat.mtrl-sci↗

arXiv2025

Quantum mechanical dataset of 836k neutral closed shell molecules with upto 5 heavy atoms from CNOFSiPSClBr

We introduce the Vector-QM24 (VQM24) dataset comprehensively covering all possible neutral closed-shell small organic and inorganic molecules with up to five heavy (\textit{p}-block) atoms: C, N, O, F, Si, P, S, Cl, Br. All valid stoichiometries, Lewis-rule-consistent graphs, and stable conformers (identified via GFN2-xTB) were enumerated combinatorially, yielding 577k conformational isomers spanning 258k constitutional isomers and 5,599 unique stoichiometries. DFT ($ω$B97X-D3/cc-pVDZ) optimizations were performed for all, and diffusion quantum Monte Carlo (DMC@PBE0(ccECP/cc-pVQZ)) energies are provided for 10,793 lowest-energy conformers with up to 4 heavy atoms. VQM24 includes structures, vibrational modes, rotational constants, thermodynamic properties (Gibbs free energies, enthalpies, ZPVEs, entropies, heat capacities), and electronic properties such as atomization, electron interaction, exchange-correlation, dispersion energies, multipole moments (dipole to hexadecapole), alchemical potentials, Mulliken charges, and wavefunctions. Machine learning models of atomization energies on this dataset reveal significantly higher complexity than QM9, with none achieving chemical accuracy. VQM24 offers a rigorous, high-fidelity benchmark for evaluating quantum machine learning models.

physics.chem-ph↗

arXiv2025

Hammett-Inspired Product Baseline for Data-efficient $Δ$-ML in Chemical Space

Data-hungry machine learning methods have become a new standard to efficiently navigate chemical compound space for molecular and materials design and discovery. Due to the severe scarcity and cost of high-quality experimental or synthetic simulated training data, however, data-acquisition costs can be considerable. Relying on reasonably accurate approximate legacy baseline labels with low computational complexity represents one of the most effective strategies to curb data-needs, e.g.~through $Δ$-, transfer-, or multi-fidelity learning. A surprisingly effective and data-efficient baseline model is presented in the form of a generic coarse-graining Hammett-Inspired Product (HIP) {\em Ansatz}, generalizing the empirical Hammett equation towards arbitrary systems and properties. Numerical evidence for the applicability of HIP includes solvation free energies of molecules, formation energies of quaternary elpasolite crystals, carbon adsorption energies on heterogeneous catalytic surfaces, HOMO-LUMO gaps of metallorganic complexes, activation energies for S$_\text{N}$2 reactions, and catalyst-substrate binding energies in cross-coupling reactions. After calibration on the same training sets, HIP yields an effective baseline for improved $Δ$-machine learning models with superior data-efficiency when compared to previously introduced specialised domain-specific models.

physics.chem-ph↗

arXiv2025

Combining Hammett $σ$ constants for $Δ$-machine learning and catalyst discovery

We study the applicability of the Hammett-inspired product (HIP) Ansatz to model relative substrate binding within homogenous organometallic catalysis, assigning $σ$ and $ρ$ to ligands and metals, respectively. Implementing an additive combination (c) rule for obtaining $σ$ constants for any ligand pair combination results in a cHIP model that enhances data efficiency in computational ligand tuning. We show its usefulness (i) as a baseline for $Δ$-machine learning (ML), and (ii) to identify novel catalyst candidates via volcano plots. After testing the combination rule on Hammett constants previously published in the literature, we have generated numerical evidence for the Suzuki-Miyaura (SM) C-C cross-coupling reaction using two synthetic datasets of metallic catalysts (including (10) and (11)-metals Ni, Pd, Pt, and Cu, Ag, Au as well as 96 ligands such as N-heterocyclic carbenes, phosphines, or pyridines). When used as a baseline, $Δ$-ML prediction errors of relative binding decrease systematically with training set size and reach chemical accuracy ($\sim$1 kcal/mol) for 20k training instances. Employing the individual ligand constants obtained from cHIP, we report relative substrate binding for a novel dataset consisting of 720 catalysts (not part of training data), of which 145 fall into the most promising range on the volcano plot accounting for oxidative addition, transmetalation, and reductive elimination steps. Multiple Ni-based catalysts, e.g. Aphos-Ni-P($t$-Bu)$_3$, are included among these promising candidates, potentially offering dramatic cost savings in experimental applications.

physics.chem-ph↗

arXiv2024

The Alchemical Integral Transform revisited

We recently introduced the Alchemical Integral Transform (AIT) enabling the prediction of energy differences, and guessed an Ansatz to parametrize space $\pmb{r}$ in some alchemical change $λ$. Here, we present a rigorous derivation of AIT's kernel $\mathcal{K}$ and discuss the parametrization $\pmb{r}(λ)$ in $n$ dimensions, i.e. necessary conditions, mathematical freedoms and additional constraints when obtaining it. Analytical expressions for changes in energy spectra and densities are given for a number of systems. Examples include homogeneous potentials like the quantum harmonic oscillator, Hydrogen-like atom, and Dirac well, both for one- and multiparticle cases, and a multiparticle system beyond coordinate scaling for harmonic potentials.

physics.chem-ph↗

arXiv2024

High-Tc superconductor candidates proposed by machine learning

We cast the relation between the chemical composition of a solid-state material and its superconducting critical temperature (Tc) as a statistical learning problem with reduced complexity. Training of query-aware similarity-based ridge regression models on experimental SuperCon data achieve average Tc prediction errors of ~5 K for unseen out-of-sample materials. Two models were trained with one excluding high pressure data in training ("ambient" model) and a second also including high pressure data ("implicit" model). Subsequent utilization of the approach to scan ~153k materials in the Materials Project enables the ranking of candidates by Tc while accounting for thermodynamic stability and small band gap. The ambient model is used to predict stable top three high-Tc candidate materials that include those with large band gaps of LiCuF4 (316 K), Ag2H12S(NO)4 (316 K), and Na2H6PtO6 (315 K). Filtering these candidates for those with small band gaps correspondingly yields LiCuF4 (316 K), Cu2P2O7 (311 K), and Cu3P2H2O9 (307 K).

cond-mat.supr-con↗

arXiv2024

Generalized convolutional many body distribution functional representations

Modern machine learning (ML) models of chemical and materials systems with billions of parameters require vast training datasets and considerable computational efforts. Lightweight kernel or decision tree based methods, however, can be rapidly trained, leading to a considerably lower carbon footprint. We introduce generalized convolutional many-body distribution functionals (cMBDF) as highly compute and data efficient atomic representations for accurate kernels that excel in low-data regimes. Generalizing the MBDF framework, cMBDF encodes local chemical environments in a compact fashion using translationally and rotationally invariant functionals of smooth atom centered Gaussian electron density proxy distributions weighted by interaction potentials. The functional values can be efficiently evaluated by expressing them in terms of convolutions which are calculated via fast Fourier transforms and stored on pre-defined grids. In the generalized form each atomic environment is described using a set of functionals uniformly defined by three integers; many-body, derivative, weighting orders. Irrespective of size/composition, cMBDF atomic vectors remain compact and constant in size for a fixed choice of these orders controlling the structural and compositional resolution. While being up to two orders of magnitude more compact than other popular representations, cMBDF is shown to be more accurate for the learning of various quantum properties such as energies, dipole moments, homo-lumo gaps, heat-capacity, polarizability, optimal exact-exchange admixtures and basis-set scaling factors. Applicability for organic and inorganic chemistry is tested as represented by the QM7b, QM9 and VQM24 data sets. Due to its compactness, model training and testing times are reduced from 23 hours to 8 minutes, implying a corresponding reduction in carbon footprint.

physics.chem-ph↗

arXiv2024

Alchemical insights into approximately quadratic energies of iso-electronic atoms

Accurate quantum mechanics based predictions of property trends are so important for materials design and discovery that even inexpensive approximate methods are valuable. We use the Alchemical Integral Transform (AIT) to study multi-electron atoms, and to gain a better understanding of the approximately quadratic behavior of energy differences between iso-electronic atoms in their nuclear charges. Based on this, we arrive at the following simple analytical estimate of energy differences between any two iso-electronic atoms, $ΔE \approx -(1 + 2γ\sqrt{N_e-1})\,ΔZ\, \bar{Z}$. Here, $γ\approx 0.3766 \pm 0.0020$ Ha corresponds to an empirical constant, and $N_e$, $ΔZ$, and $\bar{Z}$ respectively to electron number, and nuclear charge difference and average. We compare the formula's predictive accuracy using experimental numbers and non-relativistic, numerical results obtained via DFT (pbe0) for the entire periodic table up to Radon. A detailed discussion of the atomic Helium-series is included.

physics.chem-ph↗

arXiv2024

Adaptive atomic basis sets

Atomic basis sets are widely employed within quantum mechanics based simulations of matter. We introduce a machine learning model that adapts the basis set to the local chemical environment of each atom, prior to the start of self consistent field (SCF) calculations. In particular, as a proof of principle and because of their historic popularity, we have studied the Gaussian type orbitals from the Pople basis set, i.e. the STO-3G, 3-21G, 6-31G and 6-31G*. We adapt the basis by scaling the variance of the radial Gaussian functions leading to contraction or expansion of the atomic orbitals.A data set of optimal scaling factors for C, H, O, N and F were obtained by variational minimization of the Hartree-Fock (HF) energy of the smallest 2500 organic molecules from the QM9 database. Kernel ridge regression based machine learning (ML) prediction errors of the change in scaling decay rapidly with training set size, typically reaching less than 1 % for training set size 2000. Overall, we find systematically lower variance, and consequently the larger training efficiencies, when going from hydrogen to carbon to nitrogen to oxygen. Using the scaled basis functions obtained from the ML model, we conducted HF calculations for the subsequent 30'000 molecules in QM9. In comparison to the corresponding default Pople basis set results we observed improved energetics in up to 99 % of all cases. With respect to the larger basis set 6-311G(2df,2pd), atomization energy errors are lowered on average by ~31, 107, 11, and 11 kcal/mol for STO-3G, 3-21G, 6-31G and 6-31G*, respectively -- with negligible computational overhead. We illustrate the high transferability of adaptive basis sets for larger out-of-domain molecules relevant to addiction, diabetes, pain, aging.

physics.chem-ph↗

arXiv2024

Boltzmann Generators and the New Frontier of Computational Sampling in Many-Body Systems

The paper by Noé et al. [F. Noé, S. Olsson, J. Köhler and H. Wu, Science, 365:6457 (2019)] introduced the concept of Boltzmann Generators (BGs), a deep generative model that can produce unbiased independent samples of many-body systems. They can generate equilibrium configurations from different metastable states, compute relative stabilities between different structures of proteins or other organic molecules, and discover new states. In this commentary, we motivate the necessity for a new generation of sampling methods beyond molecular dynamics, explain the methodology, and give our perspective on the future role of BGs.

physics.comp-ph↗

arXiv2024

Adaptive hybrid density functionals

Exact exchange contributions are known to crucially affect electronic states, which in turn govern covalent bond formation and breaking in chemical species. Empirically averaging the exact exchange admixture over compositional degrees of freedom, hybrid density functional approximations have been widely successful, yet have fallen short to reach high level quantum chemistry accuracy, primarily due to delocalization errors. We propose to `adaptify` hybrid functionals by generating optimal admixture ratios of exact exchange on the fly, i.e. specifically for any chemical compound, using extremely data efficient quantum machine learning models that carry negligible overhead. The adaptive Perdew-Burke-Ernzerhof based hybrid density functional (aPBE0) is shown to yield atomization energies with sufficient accuracy to effectively cure the infamous spin gap problem in open shell systems, such as carbenes. aPBE0 further improves energetics, electron densities, and HOMO-LUMO gaps in organic molecules drawn from the QM9 and QM7b data set. Obtained with aPBE0 in a large basis, we present a revision of the entire QM9 data set (revQM9) with an estimated quality vastly superior to the original containing on average, stronger covalent binding, larger band-gaps, more localized electron densities, and larger dipole-moments. While aPBE0 is applicable in the equilibrium regime, outstanding limitations include covalent bond dissociation when going beyond the Coulson-Fisher point.

physics.chem-ph↗