Symmetry Functions

Below, an overview of the symmetry functions are provided. These symmetry functions are largerly based on the work of Behler.

One-body functions

The function \(G_{0}\) encodes the atomic identity. Canonically, this function is evaluated 6 times with values for \(\beta\) corresponding to \((0.4, 0.5, 0.6, 0.8, 0.9, 1.0)\).

\[G^{(0)}_{i} = \frac{\min(|\beta| , 1) \cdot Z_{i}}{118}\]

\(Z\) corresponds to the atomic number, i.e. \(Z=1\) for H and \(Z=6\) for carbon.

Two-body functions

The two-body functions, also called the radial symmetry functions, probe the radial environment of every atom \(i\).

\[G^{(1)}_{i} = \sum_{j \neq i} f_{c} \left(R_{ij} \right)\]

where \(f_{c}\) is a smooth cut-off function given by

\[\begin{split}f_{c}(R_{ij}) = \begin{cases} 0.5 \cdot \left[ \cos \left( \frac{\pi R_{ij}}{R_{c}} \right) + 1 \right],\;&\text{for}\; R_{ij} \leq R_{c} \\ 0&\text{otherwise} \end{cases}\end{split}\]

In the above equation, \(R_{c}\) acts as the cut-off radius, which is predefined at the start of the study. A typical value is \(R_{c} = 6.5\).

\[G^{(2)}_{i} = \sum_{j \neq i} \exp \left(- \eta \left(R_{ij} - R_{s} \right)^{2} \right) f_{c} \left(R_{ij} \right)\]

In the above equation, the atomic environment is sampled using Gaussians. These Gaussians can be shifted using \(R_{s}\). When using atom-centered Gaussians, the value used corresponds to \(R_{s} = 0\). The parameter \(\eta\) is used to tune the width of the Gaussians, its values are typically set to \(\eta = (0, 0.01, 0.025, 0.05, 0.1, 0.15, 0.25, 0.5, 1)\).