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Fourier series of atomic radial distribution functions: A molecular fingerprint for machine learning models of quantum chemical properties
O. Anatole von Lilienfeld, Raghunathan Ramakrishnan, Matthias Rupp, Aaron Knoll
TL;DR
The combinatorial size of chemical compound space makes compact molecular representations desirable for machine learning across molecules. The paper introduces a Fourier-based atomic radial distribution fingerprint and evaluates it for predicting molecular enthalpies of atomization, finding performance comparable to the Coulomb matrix at smaller training sizes but slower improvement with larger sets.
Problem
The combinatorial explosion of chemical compound space makes screening prohibitive and motivates compact representations for detecting patterns in increasingly available simulation data.
Method
The study introduces FR, a Fourier series of atomic radial distribution functions using atomic coordinates and nuclear identities while remaining invariant to translation, rotation, and atom indexing.
Results
FR model errors decrease with increasing training-set size, but for larger sets its MAE and RMSE improve more slowly than those of the sorted Coulomb matrix.
Takeaways & Limitations
FR provides a differentiable, invariant molecular representation that accounts for composition and geometry without requiring predefined bonding or orbital information.
Takeaways & Limitations
The one-dimensional Fourier fingerprint is not unique because modifying the cosine argument loses information, including distinctions among non-enantiomeric homometric compounds.
Abstract
from arXiv · showhide
We introduce a fingerprint representation of molecules based on a Fourier series of atomic radial distribution functions. This fingerprint is unique (except for chirality), continuous, and differentiable with respect to atomic coordinates and nuclear charges. It is invariant with respect to translation, rotation, and nuclear permutation, and requires no pre-conceived knowledge about chemical bonding, topology, or electronic orbitals. As such it meets many important criteria for a good molecular representation, suggesting its usefulness for machine learning models of molecular properties trained across chemical compound space. To assess the performance of this new descriptor we have trained machine learning models of molecular enthalpies of atomization for training sets with up to 10k organic molecules, drawn at random from a published set of 134k organic molecules. We validate the descriptor on all remaining molecules of the 134k set. For a training set of 5k molecules the fingerprint descriptor achieves a mean absolute error of 8.0 kcal/mol, respectively. This is slightly worse than the performance attained using the Coulomb matrix, another popular alternative, reaching 6.2 kcal/mol for the same training and test sets.
I. INTRODUCTION
Chemical compound space is too combinatorial for broad first-principles screening, motivating compact molecular descriptors for machine learning. The paper introduces a fingerprint designed to represent molecular composition and geometry without assuming bonding patterns.
- Motivation: More than 10^60 small stable organic molecules illustrate why exhaustive screening across chemical compound space becomes prohibitive.The underlying search space has 4N degrees of freedom for N atoms.
- Motivation: Machine learning offers a route to infer electronic-structure solutions for new materials as additional training data become available.These methods are presented as promising tools for exploring chemical compound space from first principles.
- Descriptors: Descriptors are crucial because they encode chemical identity through composition and atomic configuration for predictive models across chemical compound space.The success of learning potential energy surfaces depends on how well these input variables are represented.
- Descriptors: Bond-based descriptors are limited when bonding is unknown or changes, including bond breaking, ligand exchange, proton hopping, tautomerism, and noncovalent interactions.The cited limitation motivates descriptors that do not require pre-specified chemical topology.
- Contribution: The proposed fingerprint uniquely represents molecular composition and geometry except for chirality while remaining differentiable and invariant to rotation, translation, and atom indexing.It uses only atomic coordinates and nuclear identities and is intended for machine learning across chemical compound space.
II. METHOD
The method frames molecular descriptors as representations of quantum-mechanical information that should preserve relevant distinctions while removing only property-invariant redundancies. It uses uniqueness and invariance as central design criteria for first-principles machine-learning descriptors.
- Descriptor framework: A descriptor should represent the variables entering the modeled quantum-mechanical relation in a form suitable for correlating with the target property.The paper distinguishes first-principles, coarsened, and integrated descriptor classes.
- Descriptor framework: First-principles descriptors preserve Hamiltonian information and are intended to support learning quantum observables such as energies, forces, and electronic properties.Coarsened descriptors capture structural features, whereas integrated descriptors encode properties correlated with a target.
- Uniqueness: Uniqueness is required because distinct systems mapped to one descriptor can have different observables but receive identical model predictions.The reductio argument concludes that artificial descriptor degeneracies create errors that additional training data cannot remedy.
- Uniqueness: Non-unique descriptors can therefore produce prediction errors that cannot be eliminated by adding more training data.This conclusion follows from identical descriptor inputs forcing identical predictions for systems with different observables.
- Invariance: Good descriptors should remove rotational, translational, and nuclear-permutation redundancies without sacrificing uniqueness.The paper presents this balance as necessary for compact representations, while noting that parity sensitivity depends on the modeled property.
Desirable properties
The paper identifies descriptor properties needed for reliable quantum-chemical machine learning, emphasizing global, differentiable, symmetry-aware, compact, and smooth representations. These criteria guide the design and assessment of the Fourier radial descriptor against alternatives.
- Core properties: Descriptors should scale consistently with system size, treat symmetric atoms equivalently, and encode the whole compound rather than only local structure.These properties are described as size-extensivity, symmetry behavior, and completeness or globality.
- Core properties: Useful descriptors also require analytic evaluation, differentiability with respect to charges and coordinates, uniform length, and coverage of relevant physical-chemistry ranges.These features support analysis, response properties, advanced learning methods, and comparisons across differently sized molecules.
- Descriptor comparison: Table I compares Signature, coordinate, Coulomb-matrix, diagonalized Coulomb-matrix, and Fourier radial-distribution descriptors against the stated requirements.The table uses ✓ and ¬ to indicate whether each requirement is fulfilled.
- Descriptor comparison: The Coulomb matrix satisfies many requirements but lacks direct nuclear-permutation invariance and has dimensionality that scales quadratically with atom count.The formal dimensionality comparison does not imply that all listed grid or matrix entries are independent variables.
- Smoothness: Descriptor smoothness affects both model selection and data efficiency, because excessive smoothness can reduce accuracy where properties change rapidly.The paper notes that local smoothness may address differing smoothness across descriptor-space regions.
- Design criteria: The FR design is guided by first-principles and nuclear-permutation invariance, translational and rotational invariance, uniqueness, and differentiability.These are presented as the required and desired features developed in the subsequent construction.
B. First principles Ansatz: The external potential
The descriptor begins from the external potential and replaces it with a Gaussian model of nuclear charge densities that remains in one-to-one correspondence with the potential while being atom-index invariant.
- The electron density uniquely determines the electronic wavefunction and, through the energy functional, the total potential energy of a molecule.
- The external potential is uniquely related to atomic Cartesian coordinates and nuclear charges, but its translational and rotational dependence makes it unsuitable as a promising descriptor.
- The construction replaces the external potential with a sum of Gaussian nuclear charge densities centered at atomic coordinates and weighted by atom-specific nuclear charges.
- The Gaussian width is global across atoms and molecules, although it could instead be defined in an atom-type-specific way.
- The resulting nuclear charge density remains one-to-one with the external potential and invariant to atom indexing, despite abandoning physical non-Gaussian features for computational convenience.
C. 3-D Fourier transform
The three-dimensional Fourier representation removes translational dependence but retains rotational dependence, and preliminary aligned-pair tests showed disappointing out-of-sample accuracy.
- The Fourier-domain representation is real and translation-invariant after multiplying the Fourier transform by its conjugate and simplifying with Euler’s formula.
- The resulting function can be interpreted as a sum over elements of a symmetric atom–atom pairwise matrix, making it reminiscent of the Coulomb matrix.
- At ω = 0 and a = (1/4)1/3, the matrix’s diagonal elements match those of a preliminary Coulomb-matrix version, 0.5 ZI^2.
- The three-dimensional Fourier representation still lacks rotational invariance.
- Preliminary atomization-energy models using aligned molecule pairs achieved disappointing predictive accuracy for out-of-sample molecules.
D. 1-D version
The one-dimensional version removes rotational dependence by projecting the Fourier transform onto interatomic distances, while preserving translation and nuclear-permutation invariance.
- Rotational dependence is removed by replacing the cosine argument with the scalar product of a frequency and an interatomic distance.
- The resulting double sum over atoms is invariant to nuclear permutation, translation, and rotation.
- Changes in interatomic distances alter oscillatory frequency, whereas elemental composition changes the overall amplitude and oscillatory-band width.
- The Gaussian prefactor dampens the descriptor toward zero at large frequencies.
E. Uniqueness
The descriptor addresses homometric molecules by replacing pairwise distance arguments with atomic radial distribution functions in a Fourier series, improving uniqueness while preserving a one-dimensional distance representation.
- The 1-D Fourier fingerprint is invariant to translation, rotation, and nuclear permutation but cannot distinguish all homometric molecules because cosine modification loses information.Homometric compounds can share identical interatomic-distance sets without being enantiomers.
- A continuous atomic radial distribution function replaces the scalar ω×rIJ argument in the Fourier series.This changes the representation from a frequency-domain variable to a real-space interatomic-distance variable.
- The Gaussian radial distribution function produced the best performance among the tested radial-distribution variants.Its prefactor was omitted to keep the descriptor complete, while global parameters l, m, n, and b were optimized by cross-validation.
- The Fourier summation makes atomic radial-distribution terms linearly independent unless the corresponding radial distributions are identical.Consequently, two molecules share FR only when all their atomic radial distributions match.
- FR distinguishes homometric geometries when their atomic radial distributions differ, as illustrated for rectangular and triangular compounds.The rectangular and triangular compounds have different distributions of short, medium, and long distances around specific atoms.
A. Organic molecules
For three seven-atom organic molecules, FR fingerprints reflect stoichiometric and constitutional differences, especially at larger interatomic distances, while smaller-distance features are more similar.
- FR fingerprints distinguish three iso-electronic organic molecules, including constitutional isomers and a molecule with different stoichiometry.The examples contain seven non-hydrogen atoms and were drawn randomly from the GDB-7 database.
- Larger interatomic distances show particularly clear fingerprint differences between the organic molecules.The fingerprints have a spectrum-like character for these more complex molecular structures.
- The three molecules have similar, though not identical, atomic radial distributions at smaller distances despite varied bonding patterns.The examples include single, double, cyclic, and triple bonds involving CH, NH, CC, CN, and NN pairs.
B. Machine Learning models
The study evaluates Fourier radial (FR) and sorted Coulomb matrix descriptors for machine-learning atomization enthalpies across training-set sizes. FR errors decrease with more data, but sorted CM learns faster at larger training sizes.
- Descriptor and metric: The FR descriptor compares molecules using the Euclidean norm between their radial-distribution fingerprints.Its distance is defined by integrating the squared difference between two FR descriptors over geometric distance.
- Dataset and task: The models use 134k organic molecules from the GDB database and predict atomization enthalpies for out-of-sample molecules.The dataset contains organic molecules with up to 17 heavy atoms and specified elements, with hydrogens saturating valencies.
- Model construction: FR uses a Gaussian kernel with Euclidean norm, whereas sorted CM uses a Laplacian kernel with Manhattan norm.The CM choice follows findings reported in earlier work.
- Performance: 1k training molecules: FR starts with slightly lower MAE and significantly lower RMSE than sorted CM.The comparison concerns out-of-sample atomization-enthalpy predictions shown as a function of training-set size.
- Performance: MAE decreases systematically as training-set size increases, reaching approximately 11 kcal/mol for both models by 2k molecules.For larger training sets, FR errors continue decreasing but at a substantially slower learning rate than CM errors.
- Performance: Beyond 2k molecules, sorted CM errors decay significantly faster than FR errors.The authors suggest that geometry-dominated energy differences may be captured less efficiently by FR because of its high-frequency oscillatory character.
C. Computational details
Hyperparameters are selected by five-fold cross-validation on randomly partitioned training sets, then used to predict every molecule excluded from training.
- Cross-validation: N training molecules are randomly divided into five bins, with each bin used once as the holdout set.The remaining four bins form the training portion for each cross-validation fold.
- Cross-validation: Hyperparameters are optimized by minimizing holdout-bin MAE, and globally optimal values are obtained from the median across folds.The resulting globally optimized kernel is then used for final prediction.
- Out-of-sample evaluation: The final model predicts atomization enthalpies for 134k−N out-of-sample molecules that never contributed to training.This procedure evaluates predictions on the remaining molecules in the full dataset.
IV. CONCLUSIONS
The conclusions present FR as a Fourier-series fingerprint of atomic radial distribution functions with broad invariance, uniqueness, and differentiability properties. These properties are intended to support machine learning of quantum-chemical molecular observables.
- Descriptor: FR is an intramolecular distance-based fingerprint constructed from Fourier series of atomic radial distribution functions.The descriptor is derived using physical arguments involving the external potential, Fourier transforms, and radial distribution functions.
- Descriptor properties: FR is invariant to translation, rotation, and atom indexing.These invariances are listed among the descriptor properties required for molecular representations.
- Descriptor properties: FR is differentiable with respect to nuclear displacement and nuclear-charge changes.The latter are described as alchemical changes and are noted as potentially useful for computational materials design.