SOAP polynomial RBF error about dscribe HOT 3 CLOSED

Hi @pujaltes!

Thank you for your note and keen observations on the code.

I had a quick look on the subject, and found a probable explanation. The equation by Bartok et. al. in the original paper has a small problem in that the integrals over the radial component do not contain the spherical coordinate Jacobian (r^2). This is mentioned in the errata that the authors have posted, and you can find it here. After you include it, you should find that our implementation is correct (when checking orthogonality, you also have to include the r^2 in the integration).

I'm not sure about the figures in our paper: I will have to take a deeper look at whether we have plotted them with the incorrect data.

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Here is an example that demonstrates that the radial basis functions are orthonormal (within numerical accuracy) when integrated under spherical coordinates:

import numpy as np
import matplotlib.pyplot as plt
from dscribe.descriptors import SOAP

n_max = 4
soap = SOAP(
    species=["H"],
    r_cut=3,
    n_max=n_max,
    l_max=2,
    rbf="polynomial",
)
r, g = soap.get_basis_poly(3, n_max)

integrals = np.empty((n_max, n_max))
for i in range(n_max):
    for j in range(n_max):
        # note the r^2 coming from the Jacobian of the spherical coordinates
        integrals[i, j] = np.trapz(g[i, :]*g[j, :]*r**2, r) 
print(integrals)

This means that the plot in our original paper is done using the incorrect basis set, but our current implementation should be correct.

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@lauri-codes thank you very much for taking a look at this and for your detailed response! This makes a lot of sense, thanks for clearing it up.

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