PLIP model

The total potential energy of a configuration made of N atomic positions is first given by \(E_{\textrm{tot}} = \sum_{i = 0}^N E_{i}\) where Ei is the atomic energy. For Ei , we considered a weighted linear combination of descriptors indexed by n:

\[E^{(i)} = \sum_n \omega_n X_n^{(i)}\]

where :math:` omega_n` is the linear coefficient associated with the descriptor \(X_n^{(i)}\).

Descriptors

The descriptor space is divided into two-body, three-body and N-Body components with following equations:

\[\begin{split}&[2B]_n^i = \sum_j f_n(r_{ij}) \times f_c(r_{ij}), \\ &[3B]_{n,l}^i = \sum_j \sum_k f_n(r_{ij})f_c(r_{ij}) f_n(r_{ik})f_c(r_{ik})cos^l(\theta_{ijk}), \\ &[NB]_{n,m}^i = \left( \sum_j f_n(r_{ij}) \times f_c(r_{ij}) \times f_s(r_{ij}) \right)^m,\end{split}\]

where \(r_{ij}\) is the distance between atoms \(i\) and \(j\), \(\theta_{ijk}\) is the angle centered around the atom \(i\). The cut-off function is as follows:

\[f_c = \frac{1}{2}\left(1+\cos(\pi(R_{ij}/R_{\textrm{cut}})) \right)\]

Switch function:

\[\begin{split}u = (R_{ij}-r_1)/(r_2-r_1)\\ f_s(u) = 6{u}^5-15{u}^4+11^3\end{split}\]

Commonly, gaussian descriptors are used with the following equation:

\[f_n(R_{ij}) = \exp(-a_n(R_{ij}-b_n)^2)\]

where \(a_{n}\) and \(b_{n}\) are parameters to optimise.

Lasso Model

The set of descriptors are optimised using the Lasso regression method as follows:

\[\chi^2 = \chi^2_{\textrm{OLS}}+\alpha\sum_{n} \vert\omega_{(n)} \vert\]

where \(\alpha\) is a parameter to control regularization and \(\chi^2_{\textrm{OLS}}\) is ordinary least square objective function