# Theoretical background¶

## Structural features¶

The atomic energy is described by a function of polynomial invariants for the O(3) group. A pth-order polynomial invariant for a radial index n and a set of pairs composed of the angular number and the element unordered pair {(l_{1},t_{1}),(l_{2},t_{2}), …,(l_{p},t_{p})} is defined as a linear combination of products of p order parameters, expressed as

Linearly independent coefficient sets are obtained using the group-theoretical projector operation method for a given set of angular numbers, ensuring that the linear combinations are invariant for arbitrary rotation. In terms of fourth- and higher-order polynomial invariants, multiple invariants are linearly independent for most of the angular number sets, which are distinguished by index if necessary. The order parameters of atom i and element pair (s_{i},s) are approximately estimated from the neighboring atomic density of element s around atom i as

where (, , ) denotes the spherical coordinates of neighboring atom j centered at the position of atom i.

[1] A. Seko, A. Togo, and I. Tanaka, Group-theoretical high-order rotational invariants for structural representations: Application to linearized machine learning interatomic potential, Phys. Rev. B 99, 214108 (2019)

[2] A. Seko, Machine learning potentials for multicomponent systems: The Ti-Al binary system, Phys. Rev. B 102, 174104 (2020)

## Potential energy models¶

In the repository, the atomic energy is measured from the energy of the isolated state of the atom. A potential energy model is identified with a combination of polynomial functions and structural features. Given a set of structural features D, polynomial functions are written as

where w denotes a coefficient.

Pairwise structural features

Polynomial invariants

Machine learning potentials in the repository are developed from the following potential energy models.

model type = 1, feature type = pair

model type = 2, feature type = pair (An extension of EAM potentials)

model type = 1, feature type = polynomial invariants (Linear polynomial form)

model type = 1, feature type = polynomial invariants

model type = 2, feature type = polynomial invariants

model type = 3, feature type = polynomial invariants

model type = 4, feature type = polynomial invariants

## Parameters for developing MLPs¶

Parameters used for developing an MLP are found in **log** and **input** links at column “Files”.

Example of parameters

# number of atom species n_type 2 # use derivatives in training or not: True or False with_force True # regression method (ridge or lasso or normal) reg_method ridge # cutoff radius cutoff 10.0 # model type with respect to structural features # 1 (power of features) # 2 (polynomial of all features) # 3 (polynomial of pair features + invariants) # 4 (polynomial of pair features and invariants(order=2) + invariants) model_type 1 # degree of polynomials max_p 2 # structural feature type (pair or gtinv) des_type gtinv # pairwise function type (gaussian or sph_bessel) pair_type gaussian # sequence for a in exp(-a(r-b)^2) [min, max, n] gaussian_params1 1.0 1.0 1 # sequence for b in exp(-a(r-b)^2) [min, max, n] gaussian_params2 0 10.0 15 # maximum order of group-theoretical invariants gtinv_order 4 # maximum l values of group-theoretical invariants gtinv_maxl 7 7 2 0 0 # use only symmetric invariants or not gtinv_sym False False False False False