---
title: 'SOAP: Smooth Overlap of Atomic Positions'
url: https://www.emergentmind.com/topics/smooth-overlap-of-atomic-positions-soap
type: topic
---

# SOAP: Smooth Overlap of Atomic Positions

The Smooth Overlap of Atomic Positions (SOAP) is a formally rigorous, atom-centered structural descriptor that encodes the three-dimensional local environment around atoms as a high-dimensional, rotation-, translation-, and permutation-invariant feature vector suitable for quantitative comparison and efficient machine learning of atomic-scale properties of molecules and solids. SOAP has been established as a state-of-the-art representation for atomistic machine learning across chemistry, materials science, and molecular physics, offering a systematic, differentiable, and complete mapping from geometric configurations to feature space.

## 1. Mathematical Construction of SOAP

The foundational object in SOAP is the atom-centered local neighbor density:
\[
\rho_i(\mathbf{r}) = \sum_{j \in \mathrm{neigh}(i)} \exp\left(-\frac{|\mathbf{r} - \mathbf{r}_{ij}|^2}{2\sigma^2}\right) f_\mathrm{cut}(|\mathbf{r}_{ij}|)
\]
where $\mathbf{r}_{ij} = \mathbf{r}_j - \mathbf{r}_i$ is the vector from atom $i$ to neighbor $j$, $\sigma$ is the Gaussian width, and $f_\mathrm{cut}$ is a smooth cutoff function limiting the range to $r_\mathrm{cut}$ [2010.12857, 1601.04077, 1906.07816, 1904.08875].

This neighbor density is then expanded in an orthonormal basis formed by radial functions $g_n(r)$ and spherical harmonics $Y_{lm}(\hat{\mathbf{r}})$:
\[
\rho_i(\mathbf{r}) = \sum_{n=1}^{N} \sum_{l=0}^{L} \sum_{m=-l}^{l} c^i_{nlm}\;g_n(r)\;Y_{lm}(\hat{\mathbf{r}})
\]
with expansion coefficients obtained by the projection
\[
c^i_{nlm} = \int d^3r\,g_n(r)\,Y_{lm}^*(\hat{\mathbf{r}})\,\rho_i(\mathbf{r})
\]

To obtain rotational invariance, the so-called power spectrum is constructed:
\[
p^i_{nn'l} = \sum_{m=-l}^{l} (c^i_{nlm})^*\,c^i_{n'lm}
\]
This transformation eliminates explicit orientation dependence, producing a descriptor vector $p^i = \{p^i_{nn'l}\}$ that encodes up to three-body correlations of the atomic density around the central atom [2010.12857, 1601.04077, 1209.3140, 1807.00408].

Comparison between two atomic environments is performed using the dot product of their normalized power spectra, forming a positive-definite kernel:
\[
K_{ij} = \left( \frac{\sum_{n,n',l} p^i_{nn'l} p^j_{nn'l} }{ \sqrt{ \sum_{n,n',l} (p^i_{nn'l})^2 } \sqrt{ \sum_{n,n',l} (p^j_{nn'l})^2 } } \right)^{\zeta}
\]
where $\zeta$ (typically $\zeta=1$ or $2$) acts as a sharpening exponent [1906.07816, 1904.08875].

## 2. Hyperparameters, Basis Choices, and Implementation

SOAP's expressiveness and computational cost are governed by several key hyperparameters:

- **Gaussian width $\sigma$**: Controls spatial resolution/smoothness [2010.12857, 1601.04077]. Smaller values resolve fine structure but require higher angular cutoffs.
- **Cutoff radius $r_\mathrm{cut}$**: Sets environment size; critical for capturing relevant correlations (e.g., $r_\mathrm{cut}=3$–6 Å in typical applications; up to $10$–$12$ Å for ice polymorph discrimination) [2010.12857, 2409.19123].
- **Radial basis size $N$ and angular cutoff $L$**: Number of radial and angular functions limit the descriptor completeness. Common choices are $N=8$-$12$, $L=6$-$8$ [2010.12857, 1906.07816].
- **Choice of radial basis functions**: Either orthonormalized Gaussians or polynomials (implemented via analytical or numerical quadrature) [1904.08875].

Implementations such as DScribe rely on analytical integrals for GTO radial bases and exploit recursive relations for efficient generation of both radial overlap integrals and angular functions [1904.08875, 1905.02142]. Caro's separable radial-angular approximation enables a tenfold speedup and improves numerical stability for distant neighbors without degrading regression power [1905.02142].

## 3. Multi-Species Extensions and Compression Strategies

SOAP straightforwardly generalizes to multi-component systems by constructing separate densities $\rho_i^\alpha$ for each element $\alpha$ and expanding in combined species-radial-angular bases. The resulting power spectrum includes cross-element terms $p_{nn'l}^{\alpha\beta}$ [1601.04077, 2112.13055]. The dimensionality grows as $O(N^2 S^2)$ ($S$ is the number of species), motivating compression schemes.

Recent work has demonstrated rank-deficient structure in the power spectrum, enabling lossless compression to $O(N S)$ scaling via Gram-matrix techniques and projection [2112.13055]. Further, reduced or agnostic variants combine element-specific and element-agnostic densities to lower scaling at small additional loss in fidelity, facilitating high-element-count applications (e.g., HEAs) [2112.13055].

## 4. Algorithms, Kernels, and Dataset-Level Similarity

SOAP kernels can be composed to compare whole molecules or periodic structures. The REMatch (Regularized Entropy Match) kernel provides a principled means of aggregating per-environment similarities using an optimal transport plan $P$ over all atomic environments in two structures $A$ and $B$:
\[
K(A,B) = \max_{P \in U(N,N)}\left\{ \sum_{i,j} P_{ij} C_{ij} - \gamma \sum_{i,j} P_{ij}\ln P_{ij} \right\}
\]
where $C_{ij}$ is the normalized environment-kernel matrix and $U(N,N)$ constrains $P$ to be doubly stochastic [1601.04077, 2010.12857]. The entropic regularization $\gamma$ tunes the assignment sharpness, and the Sinkhorn–Knopp algorithm is used for practical solution.

Kernel ridge regression (KRR) models based on SOAP-REMatch kernels have achieved sub-kcal/mol errors for molecular atomization energies—matching or exceeding previous Coulomb-matrix and deep neural network models [1601.04077].

## 5. Advanced and Generalized SOAP Variants

- **Tensorial (λ-SOAP)**: SOAP can be generalized to predict vector or tensor properties by constructing symmetry-adapted kernels that transform covariantly with the learning target under rotations. The $\lambda$-SOAP(2) formalism contracts Clebsch–Gordan coupled spherical harmonics and supports accurate regression of molecular polarizabilities, dipoles, and response tensors [1904.01623].
- **Time-Resolved and Structure-Dynamics Analysis**: TimeSOAP and combinations of SOAP with dynamical descriptors (e.g., LENS) quantify time-dependent changes in local environment, enabling detection of rare events, interface dynamics, and phase transitions [2302.09673, 2308.16829].
- **Anisotropic SOAP (AniSOAP)**: For coarse-grained or orientational degrees of freedom, SOAP can incorporate oriented multivariate Gaussians, yielding power spectra and kernels that encode both position and orientation (with analytic integration for ellipsoids, coarse-grained mesogens, etc.) [2403.19039].
- **Compression for Scalability**: Lossless and controlled-lossy compressions reduce SOAP power spectrum dimension from $O(N^2 S^2)$ to $O(N S)$, enabling fitting ML potentials for systems with up to 40 chemical species without significant loss in regression accuracy [2112.13055].

## 6. Practical Applications and Empirical Performance

SOAP descriptors serve as input features for a variety of atomic-scale regression tasks:

- **Potential energy surfaces (PES):** SOAP-driven Gaussian Approximation Potentials (GAP) achieve near-DFT accuracy for energies, forces, and virials of alloys and complex bulk systems [1906.07816].
- **Molecular property prediction:** REMatch–SOAP kernels deliver chemical-accuracy (<1 kcal/mol MAE) in atomization energies across organic molecule datasets and generalize better in scaffold splits compared to both 2D graph and 3D radial-fingerprint baselines [2010.12857, 1601.04077].
- **Phase mapping and polymorph discrimination:** High-dimensional SOAP vectors combined with dimensionality reduction (e.g., variational autoencoders) unambiguously resolve all known ice polymorphs and liquids, outperforming linear PCA and conventional order parameters [2409.19123].
- **Zeolites and complex frameworks:** Kernel PCA of SOAP descriptors maps the structural landscape to correlatively highlight motif contributions to density and lattice energy, outperforming classical order-parameter sets [1907.03517].
- **Structure-dynamics coupling:** SOAP+LENS and TimeSOAP approaches reveal dynamical domains, transition pathways, and rare-events in molecular systems at single-atom resolution [2302.09673, 2308.16829].

## 7. Limitations, Information Content, and Best-Practice Recommendations

SOAP features are maximally sensitive to up to three-body correlations by design; finite “quasi-constant” manifolds in SOAP feature space make some four-body (e.g., torsional) interactions invisible—manifesting as failures in machine-learned force fields for torsional energetics [2102.06915]. Introducing explicit four-body terms or switching to many-body (Overlap Matrix) descriptors is required for rigorous learning in torsion-sensitive systems.

Systematic tests reveal that SOAP similarity is not strictly equivalent to traditional geometric order parameters (bond angles, coordination, hydrogen bonding) in water and other disordered systems; these descriptors each encode complementary aspects, and reducing the Gaussian width in SOAP increases information content but at the computational cost of larger feature vectors [2211.16196]. Including angular information (non-zero $L$) is essential; purely radial (E3FP-like) descriptors are sub-optimal [2010.12857].

For optimal accuracy and efficiency, recommended practice is to (1) tune (or cross-validate) $r_\mathrm{cut}$, $\sigma$, $N$, $L$ to the complexity of the target environments, (2) use standard ranges for molecular and condensed-phase systems (e.g., $N=8$–$12$, $L=6$–$8$, $\sigma=0.2$–$0.5$ Å, $r_\mathrm{cut}=3$–$10$ Å), (3) exploit lossless compression for multi-element systems, and (4) normalize feature vectors prior to any regression or classification [2010.12857, 2112.13055, 2409.19123].

---

**References**  
- "Investigating 3D Atomic Environments for Enhanced QSAR" [2010.12857]  
- "Comparing molecules and solids across structural and alchemical space" [1601.04077]  
- "Machine-learned Interatomic Potentials for Alloys and Alloy Phase Diagrams" [1906.07816]  
- "TimeSOAP: Tracking high-dimensional fluctuations in complex molecular systems via time-variations of SOAP spectra" [2302.09673]  
- "IceCoder: Identification of Ice phases in molecular simulation using variational autoencoder" [2409.19123]  
- "On representing chemical environments" [1209.3140]  
- "A New Kind of Atlas of Zeolite Building Blocks" [1907.03517]  
- "Quantum chemical roots of machine-learning molecular similarity descriptors" [2207.03599]  
- "Manifolds of quasi-constant SOAP and ACSF fingerprints and the resulting failure to machine learn four-body interactions" [2102.06915]  
- "Do Machine-Learning Atomic Descriptors and Order Parameters Tell the Same Story? The Case of Liquid Water" [2211.16196]  
- "Atom-Density Representations for Machine Learning" [1807.00408]  
- "Compressing local atomic neighbourhood descriptors" [2112.13055]  
- "Machine-learning of atomic-scale properties based on physical principles" [1901.10971]  
- "Atomic-scale representation and statistical learning of tensorial properties" [1904.01623]  
- "Machine learning of microscopic structure-dynamics relationships in complex molecular systems" [2308.16829]  
- "DScribe: Library of Descriptors for Machine Learning in Materials Science" [1904.08875]  
- "Optimizing many-body atomic descriptors for enhanced computational performance of machine learning based interatomic potentials" [1905.02142]  
- "Expanding Density-Correlation Machine Learning Representations for Anisotropic Coarse-Grained Particles" [2403.19039]

Source: https://www.emergentmind.com/topics/smooth-overlap-of-atomic-positions-soap