Representing molecule-surface interactions with symmetry-adapted neural networks

Jorg Behler; Sonke Lorenz; Karsten Reuter

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Representing molecule-surface interactions with symmetry-adapted neural networks

Jorg Behler, Sonke Lorenz, Karsten Reuter

arXiv:cond-mat/0702522v1Feb 22, 2007cond-mat.mtrl-sci

TL;DR

Accurate molecule-surface dynamics require PES representations, yet first-principles sampling is expensive and prior symmetry functions were difficult to construct reliably. This paper introduces symmetry-adapted NNs using surface-symmetry descriptors and demonstrates their accuracy and efficiency on the six-dimensional O2/Al(111) PES.

Problem
Extended-surface PES calculations are computationally demanding, while earlier symmetry functions were system-specific and difficult to construct without artificial or missing symmetries.
Method
The paper constructs symmetry-adapted neural networks whose symmetry functions encode the full surface symmetry and includes the molecular bond distance among the descriptors.
Results
0.070 eV test-set RMSE was achieved for the optimized O2/Al(111) NN, while NN PES evaluation remained about 5-6 orders of magnitude faster than direct DFT-based molecular dynamics.
Takeaways & Limitations
The O2/Al(111) application demonstrates that symmetry-adapted NNs can accurately interpolate PES data while supporting substantially cheaper molecular-dynamics evaluation.
Takeaways & Limitations
The presented scheme has limited dimensionality and is consequently restricted to the frozen-surface approximation.

Abstract

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The accurate description of molecule-surface interactions requires a detailed knowledge of the underlying potential-energy surface (PES). Recently, neural networks (NNs) have been shown to be an efficient technique to accurately interpolate the PES information provided for a set of molecular configurations, e.g. by first-principles calculations. Here, we further develop this approach by building the NN on a new type of symmetry functions, which allows to take the symmetry of the surface exactly into account. The accuracy and efficiency of such symmetry-adapted NNs is illustrated by the application to a six-dimensional PES describing the interaction of oxygen molecules with the Al(111) surface.

I. INTRODUCTION

Molecule-surface dynamics depend on accurate PESs, but first-principles calculations are costly, motivating interpolation methods such as neural networks. The paper presents feed-forward NNs as flexible PES fits whose weights are optimized against known energies and whose derivatives provide forces.

Motivation: PESs describe molecular potential energy as a function of nuclear positions and are central to understanding molecule-surface processes.Relevant processes include physisorption, chemisorption, dissociation, diffusion, and desorption.
Motivation: DFT can provide accurate PES information, but calculating every point for an extended surface is computationally demanding.This restricts the number and duration of on-the-fly ab initio molecular-dynamics trajectories.
Interpolation methods: Analytical fits can reduce first-principles input but become increasingly difficult and inflexible as PES dimensionality grows.Their fixed functional forms depend on choosing a suitable physical ansatz.
Neural-network approach: Neural networks provide general function approximators that can fit PES data without prespecifying the underlying functional form.They had already been applied successfully to small gas-phase molecular PESs.
Neural-network architecture: A feed-forward NN maps molecular coordinates or descriptors at its input to the potential energy at its output through weighted hidden layers.Biases shift activation-function regimes, while nonlinear hidden activations support fitting nonlinear functions.
Training: NN weights are optimized iteratively by minimizing errors between predicted and known training energies, using methods such as backpropagation, conjugate gradient, or EKF.After optimization, the NN supplies a continuous PES and can predict energies and forces across the PES.

III. SYMMETRY-ADAPTED NEURAL NETWORKS

The symmetry-adapted approach addresses the inefficiency and inaccuracies that arise when ordinary NN fits must represent symmetry-related surface regions explicitly. It maps molecular coordinates to symmetry functions encoding the surface symmetry while retaining the molecular degrees of freedom needed for the PES.

III. SYMMETRY-ADAPTED NEURAL NETWORKS: The method targets molecule-surface PESs for diatomic molecules using the frozen-surface approximation.For diatomics, fixing substrate atoms reduces the problem to six molecular degrees of freedom.
Symmetry representation: Surface symmetry allows the PES to be mapped initially only within the irreducible wedge of a surface unit cell.The NN must nevertheless provide energies and forces outside that wedge for unrestricted molecular dynamics.
Symmetry representation: Copying data from the irreducible wedge into neighboring wedges would enlarge the training set and NN unnecessarily, making the fit inefficient.The approach instead seeks to encode symmetry directly in the representation.
Symmetry functions: Symmetry functions map molecular coordinates to descriptors that encode the surface symmetry and replace the original coordinates as NN inputs.The mapping separates symmetry assignment from the subsequent energy assignment by the NN.
Limitations of prior schemes: Earlier symmetry functions were system-specific and difficult to construct because they had to capture interdependence among periodic coordinates.Physical-intuition constructions could introduce artificial symmetries or omit real ones, affecting PES topology and molecular trajectories.

A. Interaction of an atom with a fcc(111) surface

For an atom on fcc(111), the irreducible surface wedge is defined by top, fcc, and hcp sites. Fourier terms replace distance-based descriptors to preserve surface symmetry while avoiding derivative discontinuities.

The irreducible wedge is spanned by the top, fcc, and hcp sites, with mirror planes and threefold rotation axes defining the surface symmetry.
Distances to the nearest top, fcc, and hcp sites provide symmetry-aware coordinates for an atom within the surface unit-cell.
Fourier terms centered at high-symmetry sites replace atomic distances and provide a unique, smoothly varying representation of the atom’s position.Their continuous derivatives remove the bridge-site discontinuity while retaining the surface symmetry.
Distance-based descriptors can develop derivative discontinuities when an atom crosses the bridge site between neighboring top sites.
The Fourier terms are laterally periodic and include vertical-distance dependence, making fitted energies independent of lateral position at large atom-surface separation.

B. Interaction of a diatomic molecule with a fcc(111) surface

For a diatomic molecule, atomic Fourier terms describe each constituent atom relative to the fcc(111) surface. Explicit atomic heights and the bond distance complete the representation by distinguishing vertical separations and the atoms’ relative position.

Molecular symmetry functions are constructed from Fourier terms for the two constituent atoms, using their Cartesian coordinates relative to the surface.
Explicit Z1 and Z2 terms distinguish different separations of the two atoms from the surface that a center-of-mass height alone cannot resolve.
The resulting Fourier-based functions uniquely describe both atoms with respect to the top, fcc, and hcp sites.
Atomic surface descriptors alone cannot determine the molecule’s internal relative position because equivalent function values can correspond to different atom-atom separations.
Adding the interatomic distance r completes the symmetry-function set for a heteronuclear diatomic molecule.

C. Incorporation of internal molecular symmetries

The method incorporates molecular exchange symmetry by combining atomic Fourier terms, preserving structural information while preventing artificial symmetry breaking. The resulting functions reproduce surface-site and angular symmetries across configurations.

C. Incorporation of internal molecular symmetries: The additional symmetry may be numerically necessary to avoid artificial symmetry breaking and noise in fitted total energies and forces.
C. Incorporation of internal molecular symmetries: Homonuclear molecular exchange symmetry is incorporated by symmetrizing and antisymmetrizing Fourier terms from both atoms for each surface site.This yields molecular Fourier terms that retain structural information.
C. Incorporation of internal molecular symmetries: The molecular Fourier terms exhibit the correct lateral periodicities when plotted against center-of-mass coordinates at fixed molecular geometry.
C. Incorporation of internal molecular symmetries: At an fcc site with the molecule parallel to the surface, rotating by 60° produces an energetically equivalent configuration and 60° periodicity in the symmetry functions.
C. Incorporation of internal molecular symmetries: Tilting the molecule reduces the symmetry from sixfold to threefold, while bridge and low-symmetry sites produce the corresponding reduced or absent angular symmetries.

D. Adding redundant symmetry functions

The authors add redundant symmetry functions to assist neural-network fitting without violating the exact symmetry encoded by the core functions. These additions emphasize structurally relevant molecular separation and center-of-mass position.

D. Adding redundant symmetry functions: Seven symmetry functions contain exhaustive configuration information for diatomic molecules, although redundant functions may improve numerical fit accuracy.
D. Adding redundant symmetry functions: The added terms include one depending only on molecule-surface separation and three Fourier terms depending on the molecular center of mass.
D. Adding redundant symmetry functions: Figure 5 plots six symmetrized and antisymmetrized Fourier terms versus X and Y center-of-mass coordinates at fixed molecular geometry.
D. Adding redundant symmetry functions: The redundant terms preserve the symmetry of the core functions and assist the network in extracting energetically relevant structural information.

E. Calculation of forces

Forces are obtained analytically by differentiating the neural-network energy through the symmetry functions with respect to molecular coordinates. Continuous symmetry-function derivatives ensure consistency between calculated energies and forces.

E. Calculation of forces: The force along molecular coordinate α is obtained analytically from the neural network.
E. Calculation of forces: Figure 6 evaluates symmetry functions during molecular rotation about φ across fcc, bridge, and low-symmetry surface sites.
E. Calculation of forces: The network determines energy derivatives with respect to symmetry functions, while their coordinate derivatives follow analytically from the symmetry-function definitions.

IV. APPLICATION TO O2 DISSOCIATION AT AL(111)

The developed symmetry functions are applied to neural-network interpolation of the six-dimensional PES for triplet O2 interacting with Al(111). The PES serves as an example for the interpolation scheme and relates to the system’s experimentally low sticking coefficient.

IV. APPLICATION TO O2 DISSOCIATION AT AL(111): The application concerns the six-dimensional PES describing O2 dissociation at an Al(111) surface.
IV. APPLICATION TO O2 DISSOCIATION AT AL(111): The O2 molecule is constrained to its spin-triplet state, a system studied in connection with its experimentally measured low sticking coefficient.
IV. APPLICATION TO O2 DISSOCIATION AT AL(111): The particular PES is used as an example to illustrate the neural-network interpolation scheme.

A. Mapping of the six-dimensional PES

The six-dimensional spin-triplet PES for O2 on Al(111) was mapped with DFT, using elbow plots to sample molecular bond length and surface distance across orientations and sites. High-energy repulsive regions were transformed before fitting because they hindered accuracy but are not accessed in the targeted MD regime.

DFT mapping: The mapped PES describes the spin-triplet oxygen molecule interacting with Al(111) using DFT implemented in DMol3 with the RPBE functional.The spin-triplet state was enforced with locally constrained DFT.
DFT mapping: 3768 DFT data points mapped the six-dimensional PES, mostly in 38 elbow plots varying bond length r and surface distance Z for fixed orientation and site.The energy zero corresponds to infinite molecule-surface separation.
High-energy treatment: For small r and small Z, the potential becomes highly repulsive, creating a steep energy range that degrades NN fitting accuracy.These high-energy regions are typically inaccessible in MD simulations focused on maximum molecular kinetic energies of 1 eV.
High-energy treatment: Constraining the highest energies reduced the fitted energy range and improved NN accuracy in the remaining relevant energy region.

B. NN optimization procedure

The NN optimization emphasized physically important low-energy regions, compared minimal and expanded symmetry-function sets, and assessed both numerical accuracy and symmetry preservation. Including all 11 functions substantially improved errors and produced a PES with the correct surface-site symmetries.

Optimization design: 96 of 3768 DFT points were reserved as an independent test set, while fitting weights emphasized low-energy regions near minimum energy paths.
Optimization design: The study compared fits using functions G1−G7 with fits using all 11 symmetry functions G1−G11 to evaluate redundant-function effects.
Quantitative accuracy: Using G1−G7, training and test RMSEs were 0.472 eV and 0.329 eV, while the MAD for E < 1.0 eV was 0.070 eV.The entrance-channel error at r = 1.22 Å was 0.019 eV.
Quantitative accuracy: Using G1−G11 reduced training and test RMSEs by one order of magnitude, to 0.049 eV and 0.070 eV, respectively.The low-energy MAD became 0.012 eV, and the entrance-channel error at r = 1.22 Å became 1.4 meV.
Validation: The optimized NN reproduced corresponding ab initio MD trajectories with positional differences below 0.1 Å after 0.2 ps.The comparison followed molecules approaching the barrier region from Z = 5 Å toward Z = 2.5 Å.
Symmetry validation: The G1−G11 potential exhibited the correct symmetry properties for the different surface sites, supporting the paper’s central symmetry-adaptation objective.
Validation: Figure 7 compares DFT-splined and optimized-NN elbow plots across Z and r for three molecular orientations, with contours spaced by 0.2 eV.

V. DISCUSSION

The symmetry-adapted NN accurately interpolates the O2/Al(111) PES while preserving surface symmetry and remaining much faster than direct DFT-based dynamics. Its main limitations are the extra computational complexity of symmetry functions and the need for new optimization when adding degrees of freedom.

The O2/Al(111) model achieved an excellent representation of the supplied PES data, demonstrating accurate interpolation by symmetry-adapted NNs.The model concerns a six-dimensional molecule-surface PES.
5-6 orders of magnitude faster than direct DFT-based MD simulations, NN PES evaluation remains less susceptible to representation errors than less flexible analytical potentials.The NN approach is slightly more demanding than simple analytical potentials because it evaluates symmetry functions and their derivatives on-the-fly.
Adding further degrees of freedom requires a completely new NN optimization and a higher-dimensional PES mapping and interpolation.This limits the approach mainly to the frozen-surface approximation, although other surface unit-cell shapes and sizes are generally accessible.
The NN potential-energy symmetry follows the surface site: six-fold at one fcc configuration, three-fold at another, bridge-site symmetry, and no symmetry off-site.These rotational symmetries correspond to the symmetry-function values used by the network.
Systematic atomic-Fourier symmetry functions replace cumbersome empirical functions depending simultaneously on many degrees of freedom.The correct symmetry functions ensure equivalent structures share values while inequivalent configurations remain distinguishable, preventing numerical symmetry breaking in energies and forces.
The coordinate mapping is one-way, from six molecular coordinates to 11 symmetry functions, so reconstructing the original coordinates is unnecessary.This permits the use of symmetry functions that are difficult to invert.

VI. SUMMARY

The paper presents a symmetry-adapted neural-network representation of molecule-surface PESs using symmetry functions that fully encode surface symmetry. Its accuracy is demonstrated on the six-dimensional PES of triplet O2 interacting with Al(111).

The method represents molecule-surface potential-energy surfaces with neural networks whose inputs are systematically constructed symmetry functions rather than molecular coordinates.The symmetry functions are built from atomic Fourier terms and are intended for varied surface unit-cell sizes and shapes.