SchNarc: SchNet with Non-Adiabatic Couplings

• SchNarc is a methodological development that extends SchNet by integrating non-adiabatic coupling outputs with continuous-filter convolutions for modeling smooth potential energy surfaces.
• The architecture uses interaction blocks with rotational and translational invariance, deriving forces via automatic differentiation through energy gradients.
• Empirical evaluations on datasets like QM9, MD17, and ISO17 show that SchNarc achieves low error rates and robust performance in quantum molecular dynamics simulations.

SchNarc is not a discrete algorithm but a methodological development rooted in the SchNet framework for quantum chemistry, in which continuous-filter convolutional neural networks (CFConv) are employed to accurately model quantum mechanical interactions of molecules, particularly for learning potential energy surfaces (PES), including both energies and atomic forces. The term “SchNarc” (SchNet + NACs, i.e., Non-Adiabatic Couplings) is an established designation in the quantum chemistry community for models that combine SchNet’s architecture with outputs relevant to non-adiabatic molecular dynamics, although this term does not explicitly appear in the referenced canonical SchNet publication. The canonical source for the architectural underpinnings, training methods, and conceptual innovations of SchNarc is the SchNet paper (Schütt et al., 2017).

1. Theoretical Foundation: Continuous-Filter Convolutions

SchNarc builds on SchNet’s core innovation—the continuous-filter convolution layer (CFConv)—which generalizes the discrete convolution to operate on atomic positions in Euclidean space, bypassing any gridding or voxelization. Formally, a molecule at layer $l$ is represented by atomic feature vectors $X^l = (x^l_1, …, x^l_n)$ and positions $R = (r_1, …, r_n)$ in $\mathbb{R}^3$. The convolution at atom $i$ is

$$x_i^{l+1} = \sum_{j=1}^n x^l_j \odot W^l(r_i - r_j),$$

where $W^l$ is a learned, potentially rotation-invariant, continuous function mapping interatomic displacement vectors to filter weights (Schütt et al., 2017).

The filter function $W^l(\Delta r)$ is parameterized via a feed-forward neural network acting on a radial (Gaussian) basis expansion of the distance $d_{ij} = \|r_i - r_j\|$ to ensure rotational invariance: $$e_{ij,k} = \exp\left[-\gamma (d_{ij} - \mu_k)^2\right], \quad k = 1 \ldots K,$$ followed by a two-layer MLP employing a shifted softplus (ssp) nonlinearity, which is both smooth and $X^l = (x^l_1, …, x^l_n)$0.

2. Network Architecture and Adaptations

The SchNet/SchNarc architecture alternates “interaction blocks,” each comprising an atom-wise linear transform, a CFConv, additional atom-wise transformations, and a residual connection. Each interaction block updates atomic descriptors informed by both chemical identity and geometric relation to neighbors. After $X^l = (x^l_1, …, x^l_n)$1 such blocks, atomic features are aggregated (typically summed) to yield molecular properties. For modeling quantum dynamics, energy invariance to translation and rotation, as well as the requirement for consistent, smooth, and differentiable potential energy surfaces, are enforced structurally by the network’s dependence on pairwise distances and summation over atoms (Schütt et al., 2017).

To address non-adiabatic coupling (NAC) elements required for molecular dynamics beyond the Born–Oppenheimer approximation—a task of the “SchNarc” extension—the architecture is modified to output, for each pair of electronic states and configuration, the energy gap and NAC values, all as smooth functions of the atomic positions.

3. Physical Principles: Invariance and Differentiability

SchNet/SchNarc’s design enforces essential physical constraints:

• Rotational invariance: All filters depend only on interatomic distances, not on absolute positions.
• Translational invariance: Total energy is a sum of per-atom contributions.
• Permutation invariance: No explicit order of atoms; all pairwise and atom-wise operations are symmetric.
• Smooth, differentiable PES: The energy output $X^l = (x^l_1, …, x^l_n)$2 is $X^l = (x^l_1, …, x^l_n)$3 due to the use of ssp activations and the radial basis/MLP filter design. Forces, required for MD, are obtained by

$X^l = (x^l_1, …, x^l_n)$4

via automatic differentiation, guaranteeing energy conservation and consistent, curl-free gradients (Schütt et al., 2017).

4. Training Methodologies

Standard SchNet is trained on either energy-only (pointwise-target) datasets or combined energy-force datasets (which greatly enhance generalization on new molecular configurations). The presence of atomic force labels improves not only energy accuracy but also the physical realism of the gradients (forces), which is crucial for dynamics.

For SchNarc and other NAC-capable variants, the network is trained to output both the potential energy for a particular electronic state and auxiliary properties (e.g., gradients, couplings) derived as derivatives of network outputs. The cost function includes terms for energy, force, and NAC targets, and training benefits from high-quality reference data for all quantities.

Implementation employs feature dimensions typically in the range $X^l = (x^l_1, …, x^l_n)$5, several ($X^l = (x^l_1, …, x^l_n)$6) interaction blocks, a dense radial basis with $X^l = (x^l_1, …, x^l_n)$7, and filter MLPs of two layers (Schütt et al., 2017).

5. Empirical Performance and Application Domains

Empirical results on standard quantum chemistry datasets (QM9 for equilibrium molecules, MD17 for molecular dynamics, ISO17 for chemical/conformational transferability) demonstrate:

• For QM9, MAE of 0.31 kcal/mol for atomization energies, outperforming contemporaries.
• For MD17 (with energy+force training), force errors down to $X^l = (x^l_1, …, x^l_n)$80.31 kcal/mol/Å for small molecules given just 1k training points, competitive or superior to other deep-learned and kernel models.
• On ISO17, trained with both energies and forces, the model maintains low errors across unseen conformations and, to a lesser degree, unseen molecules, affirming the architecture’s extrapolative robustness (Schütt et al., 2017).

SchNarc and variants are widely used for ab initio molecular dynamics, photochemistry, and excited-state simulation, where the ability to model smooth, equivariant, and physically faithful PES and couplings is essential.

6. Related Architectures and Extensions

SchNet’s CFConv principle has shaped subsequent continuous-filter and equivariant graph convolutional networks. Extensions include more sophisticated angular-dependent filters (not just radial), integration with graph neural network paradigms for periodic systems, and hybrid models with quantum-derived constraints.

Other continuous-filter convolutional frameworks, including dynamic resizing layers for imaging (Shocher et al., 2020), operate on the same principle of a learned kernel parameterized as a continuous function (typically via an MLP), defined over relative positions. These models have broad application domains, including point-cloud processing, unstructured mesh learning, and continuous-level network interpolation (Shocher et al., 2020, Coscia et al., 2022, Lee et al., 2020), but SchNet/SchNarc remains distinguished by its principled incorporation of quantum-mechanical invariance.

7. Limitations and Prospective Directions

Although SchNet/SchNarc architectures achieve state-of-the-art performance on chemically relevant datasets, challenges remain:

• Scalability: The cost of O$X^l = (x^l_1, …, x^l_n)$9 pairwise computations limits application to large molecular systems, although sparse and local cutoffs can partially ameliorate this.
• Beyond pairwise distances: Richer physical phenomena (e.g., directional bonding, many-body effects) may require angular features or even local atomic environments as arguments to the filter network.
• Transferability: Out-of-distribution generalization (e.g., to new chemical motifs or excited state manifolds far from training data) is an open issue. Regularization and explicit physical priors remain active research directions.

The CFConv paradigm underlying SchNarc continues to inspire innovations in both quantum and classical domains of scientific machine learning, particularly where geometric fidelity and physical law adherence are required (Schütt et al., 2017, Shocher et al., 2020, Coscia et al., 2022, Lee et al., 2020).