Behler–Michele Parrinell symmetry functions

Formulation

Behler & Parrinello (2007):

The cutoff function $f_\mathrm{c}$ is given by

$f_\mathrm{c}(R_{ij}) = \left\{ \begin{aligned} & \tfrac{1}{2} \, \big[ \cos \big(\tfrac{\pi R_{ij}}{R_\mathrm{c}}\big) + 1 \big] & (R_{ij} \le R_\mathrm{c}) \\ & 0 & (R_{ij} > R_\mathrm{c}) \end{aligned} \right.$

The radial term $G^1_i$ is given by

$G^1_i = \sum_{j~(j\neq i)} e^{-\eta \, (R_{ij} - R_s)^2} f_\mathrm{c}(R_{ij})$

The angular term $G^2_i$ is given by

$G^2_i = \sum_{j,\,k~(j,\,k\neq i)} (1 + \lambda \cos \theta_{ijk})^\xi \cdot e^{-\eta \, (R_{ij}^2 + R_{ik}^2 + R_{jk}^2)^2} f_\mathrm{c}(R_{ij}) \, f_\mathrm{c}(R_{ik}) \, f_\mathrm{c}(R_{jk})$

where $\cos \theta_{ijk} = \frac{\mathbf{R}_{ij}\cdot \mathbf{R}_{ij}}{R_{ij} R_{ij}}$ .

Applications

ANI (e.g., TorchANI, Neurochem) uses a heavily modified version of the Behler and Parrinello symmetry functions (Smith et al., 2017).
ænet use the BP’s symmetry functions + smooth overlap of atomic positions (SOAP) approach (Artrith et al., 2017).

References

Artrith, N., Urban, A., & Ceder, G. (2017). Efficient and accurate machine-learning interpolation of atomic energies in compositions with many species. Physical Review B, 96(1), 014112. https://doi.org/10.1103/PhysRevB.96.014112
Behler, J., & Parrinello, M. (2007). Generalized Neural-Network Representation of High-Dimensional Potential-Energy Surfaces. Physical Review Letters, 98(14), 146401. https://doi.org/10.1103/PhysRevLett.98.146401
Smith, J. S., Isayev, O., & Roitberg, A. E. (2017). ANI-1: An extensible neural network potential with DFT accuracy at force field computational cost. Chemical Science, 8(4), 3192–3203. https://doi.org/10.1039/C6SC05720A