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 {(l₁,t₁),(l₂,t₂), …,(l_p,t_p)} is defined as a linear combination of products of p order parameters, expressed as

$d_{nl_1l_2\cdots l_p,t_1t_2\cdots t_p,(\sigma)}^{(i)} = \sum_{m_1,m_2,\cdots, m_p} c^{l_1l_2\cdots l_p,(\sigma)}_{m_1m_2\cdots m_p} a_{nl_1m_1,t_1}^{(i)} a_{nl_2m_2,t_2}^{(i)} \cdots a_{nl_pm_p,t_p}^{(i)}.$

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 $\sigma$ 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

$a_{nlm,\{s_i,s\}}^{(i)} = \sum_{\{j | r_{ij} \leq r_c,s_j = s\} } f_n(r_{ij}) Y_{lm}^* (\theta_{ij}, \phi_{ij}),$

where ( $r_{ij}$ , $\theta_{ij}$ , $\phi_{ij}$ ) 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

$F_1 \left(D\right) = \sum_i w_i d_i \\$

$F_{2,\rm pow} \left(D\right) = \sum_{i} w_{ii} d_i d_i \\$

$F_2 \left(D\right) = \sum_{\{i,j\}} w_{ij} d_i d_j \\$

$F_{3,\rm pow} \left(D\right) = \sum_{i} w_{iii} d_i d_i d_i \\$

$F_3 \left(D\right) = \sum_{\{i,j,k\}} w_{ijk} d_i d_j d_k$

where w denotes a coefficient.

Pairwise structural features

$D_{\rm pair}^{(i)} = \{d_{n0}^{(i)}\}$

Polynomial invariants

$D^{(i)} = D_{\rm pair}^{(i)} \cup D_2^{(i)} \cup D_3^{(i)} \cup D_4^{(i)} \cup \cdots$

$D_2^{(i)} = \{d_{nll}^{(i)}\}$

$D_3^{(i)} = \{d_{nl_1l_2l_3}^{(i)}\}$

$D_4^{(i)} = \{d_{nl_1l_2l_3l_4,(\sigma)}^{(i)}\}$

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

model type = 1, feature type = pair

$E^{(i)} = F_1\left(D_{\rm pair}^{(i)} \right) + F_{2,\rm pow} \left(D_{\rm pair}^{(i)} \right) + F_{3,\rm pow} \left(D_{\rm pair}^{(i)} \right) + \cdots$

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

$E^{(i)} = F_1\left(D_{\rm pair}^{(i)} \right) + F_{2} \left(D_{\rm pair}^{(i)} \right) + F_{3} \left(D_{\rm pair}^{(i)} \right) + \cdots$

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

$E^{(i)} = F_1 \left(D^{(i)} \right)$

model type = 1, feature type = polynomial invariants

$E^{(i)} = F_1 \left(D^{(i)} \right) + F_{2,\rm pow} \left(D^{(i)} \right) + F_{3,\rm pow} \left(D^{(i)} \right) + \cdots$

model type = 2, feature type = polynomial invariants

$E^{(i)} = F_1 \left(D^{(i)} \right) + F_{2} \left(D^{(i)} \right) + F_{3} \left(D^{(i)} \right) + \cdots$

model type = 3, feature type = polynomial invariants

$E^{(i)} = F_1 \left( D^{(i)} \right) + F_2 \left( D_{\rm pair}^{(i)} \right) + F_3 \left( D_{\rm pair}^{(i)} \right) + \cdots$

model type = 4, feature type = polynomial invariants

$E^{(i)} = F_1 \left( D^{(i)} \right) + F_2 \left( D_{\rm pair}^{(i)} \cup D_2^{(i)} \right) + \cdots$

Parameters for developing MLPs¶

Parameters used for developing MLPs are found in polymlp.in links at column “Files”.

Parameter example

# structural feature type (pair or gtinv)
feature_type gtinv

# cutoff radius in angstrom
cutoff 10.0

'''pairwise Gaussian function (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

# use derivatives in training or not: True or False
include_force True
include_stress True

# regularization parameter setting in linear ridge regression
reg_alpha_params -4 3 15

# 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 3

# degree of polynomials
max_p 2

# maximum order of polynomial invariants
gtinv_order 4
# maximum l values of polynomial invariants
gtinv_maxl 12 8 2

# number of atom species
n_type 2

# element types
elements Ag Au

# atomic energy values
atomic_energy -0.19820116 -0.18494148