# -*- coding: utf-8 -*-
"""Copyright 2019 DScribe developers
Licensed under the Apache License, Version 2.0 (the "License");
you may not use this file except in compliance with the License.
You may obtain a copy of the License at
http://www.apache.org/licenses/LICENSE-2.0
Unless required by applicable law or agreed to in writing, software
distributed under the License is distributed on an "AS IS" BASIS,
WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
See the License for the specific language governing permissions and
limitations under the License.
"""
import numpy as np
from scipy.sparse import coo_matrix
from scipy.special import gamma
from scipy.linalg import sqrtm, inv
from ase import Atoms
import ase.data
from dscribe.descriptors import Descriptor
from dscribe.core import System
from dscribe.utils.geometry import get_extended_system
import dscribe.ext
[docs]class SOAP(Descriptor):
"""Class for generating a partial power spectrum from Smooth Overlap of
Atomic Orbitals (SOAP). This implementation uses real (tesseral) spherical
harmonics as the angular basis set and provides two orthonormalized
alternatives for the radial basis functions: spherical primitive gaussian
type orbitals ("gto") or the polynomial basis set ("polynomial").
For reference, see:
"On representing chemical environments, Albert P. Bartók, Risi Kondor, and
Gábor Csányi, Phys. Rev. B 87, 184115, (2013),
https://doi.org/10.1103/PhysRevB.87.184115
"Comparing molecules and solids across structural and alchemical space",
Sandip De, Albert P. Bartók, Gábor Csányi and Michele Ceriotti, Phys.
Chem. Chem. Phys. 18, 13754 (2016), https://doi.org/10.1039/c6cp00415f
"Machine learning hydrogen adsorption on nanoclusters through structural
descriptors", Marc O. J. Jäger, Eiaki V. Morooka, Filippo Federici Canova,
Lauri Himanen & Adam S. Foster, npj Comput. Mater., 4, 37 (2018),
https://doi.org/10.1038/s41524-018-0096-5
"""
[docs] def __init__(
self,
rcut,
nmax,
lmax,
sigma=1.0,
rbf="gto",
species=None,
periodic=False,
crossover=True,
average="off",
sparse=False,
):
"""
Args:
rcut (float): A cutoff for local region in angstroms. Should be
bigger than 1 angstrom.
nmax (int): The number of radial basis functions.
lmax (int): The maximum degree of spherical harmonics.
species (iterable): The chemical species as a list of atomic
numbers or as a list of chemical symbols. Notice that this is not
the atomic numbers that are present for an individual system, but
should contain all the elements that are ever going to be
encountered when creating the descriptors for a set of systems.
Keeping the number of chemical species as low as possible is
preferable.
sigma (float): The standard deviation of the gaussians used to expand the
atomic density.
rbf (str): The radial basis functions to use. The available options are:
* "gto": Spherical gaussian type orbitals defined as :math:`g_{nl}(r) = \sum_{n'=1}^{n_\mathrm{max}}\,\\beta_{nn'l} r^l e^{-\\alpha_{n'l}r^2}`
* "polynomial": Polynomial basis defined as :math:`g_{n}(r) = \sum_{n'=1}^{n_\mathrm{max}}\,\\beta_{nn'} (r-r_\mathrm{cut})^{n'+2}`
periodic (bool): Determines whether the system is considered to be
periodic.
crossover (bool): Determines if crossover of atomic types should
be included in the power spectrum. If enabled, the power
spectrum is calculated over all unique species combinations Z
and Z'. If disabled, the power spectrum does not contain
cross-species information and is only run over each unique
species Z. Turned on by default to correspond to the original
definition
average (str): The averaging mode over the centers of interest.
Valid options are:
* "off": No averaging.
* "inner": Averaging over sites before summing up the magnetic quantum numbers: :math:`p_{nn'l}^{Z_1,Z_2} \sim \sum_m (\\frac{1}{n} \sum_i c_{nlm}^{i, Z_1})^{*} (\\frac{1}{n} \sum_i c_{n'lm}^{i, Z_2})`
* "outer": Averaging over the power spectrum of different sites: :math:`p_{nn'l}^{Z_1,Z_2} \sim \\frac{1}{n} \sum_i \sum_m (c_{nlm}^{i, Z_1})^{*} (c_{n'lm}^{i, Z_2})`
sparse (bool): Whether the output should be a sparse matrix or a
dense numpy array.
"""
super().__init__(periodic=periodic, flatten=True, sparse=sparse)
# Setup the involved chemical species
self.species = species
# Test that general settings are valid
if (sigma <= 0):
raise ValueError(
"Only positive gaussian width parameters 'sigma' are allowed."
)
self._eta = 1/(2*sigma**2)
self._sigma = sigma
if lmax < 0:
raise ValueError(
"lmax cannot be negative. lmax={}".format(lmax)
)
supported_rbf = set(("gto", "polynomial"))
if rbf not in supported_rbf:
raise ValueError(
"Invalid radial basis function of type '{}' given. Please use "
"one of the following: {}".format(rbf, supported_rbf)
)
if nmax < 1:
raise ValueError(
"Must have at least one radial basis function."
"nmax={}".format(nmax)
)
supported_average = set(("off", "inner", "outer"))
if average not in supported_average:
raise ValueError(
"Invalid average mode '{}' given. Please use "
"one of the following: {}".format(average, supported_average)
)
# Test that radial basis set specific settings are valid
if rbf == "gto":
if rcut <= 1:
raise ValueError(
"When using the gaussian radial basis set (gto), the radial "
"cutoff should be bigger than 1 angstrom."
)
if lmax > 9:
raise ValueError(
"When using the gaussian radial basis set (gto), lmax "
"cannot currently exceed 9. lmax={}".format(lmax)
)
# Precalculate the alpha and beta constants for the GTO basis
self._alphas, self._betas = self.get_basis_gto(rcut, nmax)
elif rbf == "polynomial":
if lmax > 20:
raise ValueError(
"When using the polynomial radial basis set, lmax "
"cannot currently exceed 20. lmax={}".format(lmax)
)
self._rcut = rcut
self._nmax = nmax
self._lmax = lmax
self._rbf = rbf
self._average = average
self.crossover = crossover
[docs] def create(self, system, positions=None, n_jobs=1, verbose=False):
"""Return the SOAP output for the given systems and given positions.
Args:
system (:class:`ase.Atoms` or list of :class:`ase.Atoms`): One or
many atomic structures.
positions (list): Positions where to calculate SOAP. Can be
provided as cartesian positions or atomic indices. If no
positions are defined, the SOAP output will be created for all
atoms in the system. When calculating SOAP for multiple
systems, provide the positions as a list for each system.
n_jobs (int): Number of parallel jobs to instantiate. Parallellizes
the calculation across samples. Defaults to serial calculation
with n_jobs=1.
verbose(bool): Controls whether to print the progress of each job
into to the console.
Returns:
np.ndarray | scipy.sparse.csr_matrix: The SOAP output for the given
systems and positions. The return type depends on the
'sparse'-attribute. The first dimension is determined by the amount
of positions and systems and the second dimension is determined by
the get_number_of_features()-function. When multiple systems are
provided the results are ordered by the input order of systems and
their positions.
"""
# If single system given, skip the parallelization
if isinstance(system, (Atoms, System)):
return self.create_single(system, positions)
else:
self._check_system_list(system)
# Combine input arguments
n_samples = len(system)
if positions is None:
inp = [(i_sys,) for i_sys in system]
else:
n_pos = len(positions)
if n_pos != n_samples:
raise ValueError(
"The given number of positions does not match the given"
"number os systems."
)
inp = list(zip(system, positions))
# For SOAP the output size for each job depends on the exact arguments.
# Here we precalculate the size for each job to preallocate memory and
# make the process faster.
k, m = divmod(n_samples, n_jobs)
jobs = (inp[i * k + min(i, m):(i + 1) * k + min(i + 1, m)] for i in range(n_jobs))
output_sizes = []
for i_job in jobs:
n_desc = 0
if self._average == "outer" or self._average == "inner":
n_desc = len(i_job)
elif positions is None:
n_desc = 0
for job in i_job:
n_desc += len(job[0])
else:
n_desc = 0
for i_sample, i_pos in i_job:
if i_pos is not None:
n_desc += len(i_pos)
else:
n_desc += len(i_sample)
output_sizes.append(n_desc)
# Create in parallel
output = self.create_parallel(inp, self.create_single, n_jobs, output_sizes, verbose=verbose)
return output
[docs] def create_single(self, system, positions=None):
"""Return the SOAP output for the given system and given positions.
Args:
system (:class:`ase.Atoms` | :class:`.System`): Input system.
positions (list): Cartesian positions or atomic indices. If
specified, the SOAP spectrum will be created for these points.
If no positions are defined, the SOAP output will be created
for all atoms in the system.
Returns:
np.ndarray | scipy.sparse.coo_matrix: The SOAP output for the
given system and positions. The return type depends on the
'sparse'-attribute. The first dimension is given by the number of
positions and the second dimension is determined by the
get_number_of_features()-function.
"""
# Transform the input system into the internal System-object
system = self.get_system(system)
# Check that the system does not have elements that are not in the list
# of atomic numbers
self.check_atomic_numbers(system.get_atomic_numbers())
# Check if periodic is valid
if self.periodic:
cell = system.get_cell()
if np.cross(cell[0], cell[1]).dot(cell[2]) == 0:
raise ValueError(
"System doesn't have cell to justify periodicity."
)
# Setup the local positions
if positions is None:
list_positions = system.get_positions()
else:
# Check validity of position definitions and create final cartesian
# position list
list_positions = []
if len(positions) == 0:
raise ValueError(
"The argument 'positions' should contain a non-empty set of"
" atomic indices or cartesian coordinates with x, y and z "
"components."
)
for i in positions:
if np.issubdtype(type(i), np.integer):
list_positions.append(system.get_positions()[i])
elif isinstance(i, (list, tuple, np.ndarray)):
if len(i) != 3:
raise ValueError(
"The argument 'positions' should contain a "
"non-empty set of atomic indices or cartesian "
"coordinates with x, y and z components."
)
list_positions.append(i)
else:
raise ValueError(
"Create method requires the argument 'positions', a "
"list of atom indices and/or positions."
)
# The radial cutoff is extended by adding a padding that depends on
# the used used sigma value. The padding is chosen so that the
# gaussians decay to the specified threshold value at the cutoff
# distance.
threshold = 0.001
cutoff_padding = self._sigma*np.sqrt(-2*np.log(threshold))
# Create the extended system if periodicity is requested
if self.periodic:
system = get_extended_system(system, self._rcut+cutoff_padding, return_cell_indices=False)
# Determine the SOAPLite function to call based on periodicity and
# rbf
if self._rbf == "gto":
soap_mat = self.get_soap_locals_gto(
system,
list_positions,
self._alphas,
self._betas,
rcut=self._rcut,
cutoff_padding=cutoff_padding,
nmax=self._nmax,
lmax=self._lmax,
eta=self._eta,
crossover=self.crossover,
average=self._average,
atomic_numbers=None,
)
elif self._rbf == "polynomial":
soap_mat = self.get_soap_locals_poly(
system,
list_positions,
rcut=self._rcut,
cutoff_padding=cutoff_padding,
nmax=self._nmax,
lmax=self._lmax,
eta=self._eta,
crossover=self.crossover,
average=self._average,
atomic_numbers=None,
)
# Make into a sparse array if requested
if self._sparse:
soap_mat = coo_matrix(soap_mat)
return soap_mat
@property
def species(self):
return self._species
@species.setter
def species(self, value):
"""Used to check the validity of given atomic numbers and to initialize
the C-memory layout for them.
Args:
value(iterable): Chemical species either as a list of atomic
numbers or list of chemical symbols.
"""
# The species are stored as atomic numbers for internal use.
self._set_species(value)
# Setup mappings between atom indices and types
self.atomic_number_to_index = {}
self.index_to_atomic_number = {}
for i_atom, atomic_number in enumerate(self._atomic_numbers):
self.atomic_number_to_index[atomic_number] = i_atom
self.index_to_atomic_number[i_atom] = atomic_number
self.n_elements = len(self._atomic_numbers)
[docs] def get_number_of_features(self):
"""Used to inquire the final number of features that this descriptor
will have.
Returns:
int: Number of features for this descriptor.
"""
n_elem = len(self._atomic_numbers)
if self.crossover:
n_elem_radial = n_elem * self._nmax
return int((n_elem_radial) * (n_elem_radial + 1) / 2 * (self._lmax + 1))
else:
return int(n_elem * self._nmax * (self._nmax + 1) / 2 * (self._lmax + 1))
[docs] def get_location(self, species):
"""Can be used to query the location of a species combination in the
the flattened output.
Args:
species(tuple): A tuple containing a pair of species as chemical
symbols or atomic numbers. The tuple can be for example ("H", "O").
Returns:
slice: slice containing the location of the specified species
combination. The location is given as a python slice-object, that
can be directly used to target ranges in the output.
Raises:
ValueError: If the requested species combination is not in the
output or if invalid species defined.
"""
# Check that the corresponding part is calculated
if len(species) != 2:
raise ValueError(
"Please use a pair of atomic numbers or chemical symbols."
)
# Change chemical elements into atomic numbers
numbers = []
for specie in species:
if isinstance(specie, str):
try:
specie = ase.data.atomic_numbers[specie]
except KeyError:
raise ValueError("Invalid chemical species: {}".format(specie))
numbers.append(specie)
# See if species defined
for number in numbers:
if number not in self._atomic_number_set:
raise ValueError(
"Atomic number {} was not specified in the species."
.format(number)
)
# Change into internal indexing
numbers = [self.atomic_number_to_index[x] for x in numbers]
n_elem = self.n_elements
if numbers[0] > numbers[1]:
numbers = list(reversed(numbers))
i = numbers[0]
j = numbers[1]
n_elem_feat_symm = self._nmax*(self._nmax+1)/2*(self._lmax + 1)
if self.crossover:
n_elem_feat_unsymm = self._nmax*self._nmax*(self._lmax + 1)
n_elem_feat = n_elem_feat_symm if i == j else n_elem_feat_unsymm
# The diagonal terms are symmetric and off-diagonal terms are
# unsymmetric
m_symm = i + int(j > i)
m_unsymm = j + i*n_elem - i*(i+1)/2 - m_symm
start = int(m_symm*n_elem_feat_symm+m_unsymm*n_elem_feat_unsymm)
end = int(start+n_elem_feat)
else:
if i != j:
raise ValueError(
"Crossover is set to False. No cross-species output "
"available"
)
start = int(i*n_elem_feat_symm)
end = int(start+n_elem_feat_symm)
return slice(start, end)
[docs] def flatten_positions(self, system, atomic_numbers=None):
"""Takes an ase Atoms object and returns flattened numpy arrays for the
C-extension to use.
Args:
system (ase.atoms): The system to convert.
atomic_numbers(): The atomic numbers to consider. Atoms that do not
have these atomic numbers are ignored.
Returns:
(np.ndarray, list, int, np.ndarray): Returns the positions
flattened and sorted by atomic number, atomic numbers flattened and
sorted by atomic number, number of different species and the sorted
set of atomic numbers.
"""
Z = system.get_atomic_numbers()
pos = system.get_positions()
# Get a sorted list of atom types
if atomic_numbers is not None:
atomtype_set = set(atomic_numbers)
else:
atomtype_set = set(Z)
atomic_numbers_sorted = np.sort(list(atomtype_set))
# Form a flattened list of atomic positions, sorted by atomic type
pos_lst = []
z_lst = []
for atomtype in atomic_numbers_sorted:
condition = Z == atomtype
pos_onetype = pos[condition]
z_onetype = Z[condition]
pos_lst.append(pos_onetype)
z_lst.append(z_onetype)
positions_sorted = np.concatenate(pos_lst, axis=0)
atomic_numbers_sorted = np.concatenate(z_lst).ravel()
return positions_sorted, atomic_numbers_sorted
[docs] def get_soap_locals_gto(self, system, centers, alphas, betas, rcut, cutoff_padding, nmax, lmax, eta, crossover, average, atomic_numbers=None):
"""Get the SOAP output for the given positions using the gto radial
basis.
Args:
system (ase.Atoms): Atomic structure for which the SOAP output is
calculated.
centers (np.ndarray): Positions at which to calculate SOAP.
alphas (np.ndarray): The alpha coeffients for the gto-basis.
betas (np.ndarray): The beta coeffients for the gto-basis.
rcut (float): Radial cutoff.
cutoff_padding (float): The padding that is added for including
atoms beyond the cutoff.
nmax (int): Maximum number of radial basis functions.
lmax (int): Maximum spherical harmonics degree.
eta (float): The gaussian smearing width.
crossover (bool): Whether to include species crossover in output.
atomic_numbers (np.ndarray): Can be used to specify the species for
which to calculate the output. If None, all species are included.
If given the output is calculated only for the given species and is
ordered by atomic number.
Returns:
np.ndarray: SOAP output with the gto radial basis for the given positions.
"""
n_atoms = len(system)
positions, Z_sorted = self.flatten_positions(system, atomic_numbers)
sorted_species = self._atomic_numbers
n_species = len(sorted_species)
centers = np.array(centers)
n_centers = centers.shape[0]
centers = centers.flatten()
alphas = alphas.flatten()
betas = betas.flatten()
# Determine shape
n_features = self.get_number_of_features()
if average == "inner" or average == "outer":
c = np.zeros((1, n_features), dtype=np.float64)
else:
c = np.zeros((n_centers, n_features), dtype=np.float64)
# Calculate with extension
dscribe.ext.soap_gto(
c,
positions,
centers,
alphas,
betas,
Z_sorted,
sorted_species,
rcut,
cutoff_padding,
n_atoms,
n_species,
nmax,
lmax,
n_centers,
eta,
crossover,
average,
)
return c
[docs] def get_soap_locals_poly(self, system, centers, rcut, cutoff_padding, nmax, lmax, eta, crossover, average, atomic_numbers=None):
"""Get the SOAP output using polynomial radial basis for the given
positions.
Args:
system(ase.Atoms): Atomic structure for which the SOAP output is
calculated.
centers(np.ndarray): Positions at which to calculate SOAP.
alphas (np.ndarray): The alpha coeffients for the gto-basis.
betas (np.ndarray): The beta coeffients for the gto-basis.
rCut (float): Radial cutoff.
cutoff_padding (float): The padding that is added for including
atoms beyond the cutoff.
nmax (int): Maximum number of radial basis functions.
lmax (int): Maximum spherical harmonics degree.
eta (float): The gaussian smearing width.
crossover (bool): Whether to include species crossover in output.
atomic_numbers (np.ndarray): Can be used to specify the species for
which to calculate the output. If None, all species are included.
If given the output is calculated only for the given species and is
ordered by atomic number.
Returns:
np.ndarray: SOAP output with the polynomial radial basis for the
given positions.
"""
# Get the discretized and orthogonalized polynomial radial basis
# function values
rx, gss = self.get_basis_poly(rcut, nmax)
gss = gss.flatten()
# Get atoms positions and species in sorted and flattened format
n_atoms = len(system)
positions, Z_sorted = self.flatten_positions(system, atomic_numbers)
sorted_species = self._atomic_numbers
n_species = len(sorted_species)
centers = np.array(centers)
n_centers = centers.shape[0]
# Determine shape
n_features = self.get_number_of_features()
if average == "inner" or average == "outer":
c = np.zeros((1, n_features), dtype=np.float64)
else:
c = np.zeros((n_centers, n_features), dtype=np.float64)
# Calculate with extension
dscribe.ext.soap_general(
c,
positions,
centers,
Z_sorted,
sorted_species,
rcut,
cutoff_padding,
n_atoms,
n_species,
nmax,
lmax,
n_centers,
eta,
rx,
gss,
crossover,
average
)
return c
[docs] def get_basis_gto(self, rcut, nmax):
"""Used to calculate the alpha and beta prefactors for the gto-radial
basis.
Args:
rcut(float): Radial cutoff.
nmax(int): Number of gto radial bases.
Returns:
(np.ndarray, np.ndarray): The alpha and beta prefactors for all bases
up to a fixed size of l=10.
"""
# These are the values for where the different basis functions should decay
# to: evenly space between 1 angstrom and rcut.
a = np.linspace(1, rcut, nmax)
threshold = 1e-3 # This is the fixed gaussian decay threshold
alphas_full = np.zeros((10, nmax))
betas_full = np.zeros((10, nmax, nmax))
for l in range(0, 10):
# The alphas are calculated so that the GTOs will decay to the set
# threshold value at their respective cutoffs
alphas = -np.log(threshold/np.power(a, l))/a**2
# Calculate the overlap matrix
m = np.zeros((alphas.shape[0], alphas.shape[0]))
m[:, :] = alphas
m = m + m.transpose()
S = 0.5*gamma(l + 3.0/2.0)*m**(-l-3.0/2.0)
# Get the beta factors that orthonormalize the set with Löwdin
# orthonormalization
betas = sqrtm(inv(S))
# If the result is complex, the calculation is currently halted.
if (betas.dtype == np.complex128):
raise ValueError(
"Could not calculate normalization factors for the radial "
"basis in the domain of real numbers. Lowering the number of "
"radial basis functions (nmax) or increasing the radial "
"cutoff (rcut) is advised."
)
alphas_full[l, :] = alphas
betas_full[l, :, :] = betas
return alphas_full, betas_full
[docs] def get_basis_poly(self, rcut, nmax):
"""Used to calculate discrete vectors for the polynomial basis functions.
Args:
rcut(float): Radial cutoff.
nmax(int): Number of polynomial radial bases.
Returns:
(np.ndarray, np.ndarray): Tuple containing the evaluation points in
radial direction as the first item, and the corresponding
orthonormalized polynomial radial basis set as the second item.
"""
# Calculate the overlap of the different polynomial functions in a
# matrix S. These overlaps defined through the dot product over the
# radial coordinate are analytically calculable: Integrate[(rc - r)^(a
# + 2) (rc - r)^(b + 2) r^2, {r, 0, rc}]. Then the weights B that make
# the basis orthonormal are given by B=S^{-1/2}
S = np.zeros((nmax, nmax), dtype=np.float64)
for i in range(1, nmax+1):
for j in range(1, nmax+1):
S[i-1, j-1] = (2*(rcut)**(7+i+j))/((5+i+j)*(6+i+j)*(7+i+j))
# Get the beta factors that orthonormalize the set with Löwdin
# orthonormalization
betas = sqrtm(np.linalg.inv(S))
# If the result is complex, the calculation is currently halted.
if (betas.dtype == np.complex128):
raise ValueError(
"Could not calculate normalization factors for the radial "
"basis in the domain of real numbers. Lowering the number of "
"radial basis functions (nmax) or increasing the radial "
"cutoff (rcut) is advised."
)
# The radial basis is integrated in a very specific nonlinearly spaced
# grid given by rx
x = np.zeros(100)
x[0] = -0.999713726773441234
x[1] = -0.998491950639595818
x[2] = -0.996295134733125149
x[3] = -0.99312493703744346
x[4] = -0.98898439524299175
x[5] = -0.98387754070605702
x[6] = -0.97780935848691829
x[7] = -0.97078577576370633
x[8] = -0.962813654255815527
x[9] = -0.95390078292549174
x[10] = -0.94405587013625598
x[11] = -0.933288535043079546
x[12] = -0.921609298145333953
x[13] = -0.90902957098252969
x[14] = -0.895561644970726987
x[15] = -0.881218679385018416
x[16] = -0.86601468849716462
x[17] = -0.849964527879591284
x[18] = -0.833083879888400824
x[19] = -0.815389238339176254
x[20] = -0.79689789239031448
x[21] = -0.77762790964949548
x[22] = -0.757598118519707176
x[23] = -0.736828089802020706
x[24] = -0.715338117573056447
x[25] = -0.69314919935580197
x[26] = -0.670283015603141016
x[27] = -0.64676190851412928
x[28] = -0.622608860203707772
x[29] = -0.59784747024717872
x[30] = -0.57250193262138119
x[31] = -0.546597012065094168
x[32] = -0.520158019881763057
x[33] = -0.493210789208190934
x[34] = -0.465781649773358042
x[35] = -0.437897402172031513
x[36] = -0.409585291678301543
x[37] = -0.380872981624629957
x[38] = -0.351788526372421721
x[39] = -0.322360343900529152
x[40] = -0.292617188038471965
x[41] = -0.26258812037150348
x[42] = -0.23230248184497397
x[43] = -0.201789864095735997
x[44] = -0.171080080538603275
x[45] = -0.140203137236113973
x[46] = -0.109189203580061115
x[47] = -0.0780685828134366367
x[48] = -0.046871682421591632
x[49] = -0.015628984421543083
x[50] = 0.0156289844215430829
x[51] = 0.046871682421591632
x[52] = 0.078068582813436637
x[53] = 0.109189203580061115
x[54] = 0.140203137236113973
x[55] = 0.171080080538603275
x[56] = 0.201789864095735997
x[57] = 0.23230248184497397
x[58] = 0.262588120371503479
x[59] = 0.292617188038471965
x[60] = 0.322360343900529152
x[61] = 0.351788526372421721
x[62] = 0.380872981624629957
x[63] = 0.409585291678301543
x[64] = 0.437897402172031513
x[65] = 0.465781649773358042
x[66] = 0.49321078920819093
x[67] = 0.520158019881763057
x[68] = 0.546597012065094168
x[69] = 0.572501932621381191
x[70] = 0.59784747024717872
x[71] = 0.622608860203707772
x[72] = 0.64676190851412928
x[73] = 0.670283015603141016
x[74] = 0.693149199355801966
x[75] = 0.715338117573056447
x[76] = 0.736828089802020706
x[77] = 0.75759811851970718
x[78] = 0.77762790964949548
x[79] = 0.79689789239031448
x[80] = 0.81538923833917625
x[81] = 0.833083879888400824
x[82] = 0.849964527879591284
x[83] = 0.866014688497164623
x[84] = 0.881218679385018416
x[85] = 0.89556164497072699
x[86] = 0.90902957098252969
x[87] = 0.921609298145333953
x[88] = 0.933288535043079546
x[89] = 0.94405587013625598
x[90] = 0.953900782925491743
x[91] = 0.96281365425581553
x[92] = 0.970785775763706332
x[93] = 0.977809358486918289
x[94] = 0.983877540706057016
x[95] = 0.98898439524299175
x[96] = 0.99312493703744346
x[97] = 0.99629513473312515
x[98] = 0.998491950639595818
x[99] = 0.99971372677344123
rx = rcut*0.5*(x + 1)
# Calculate the value of the orthonormalized polynomial basis at the rx
# values
fs = np.zeros([nmax, len(x)])
for n in range(1, nmax+1):
fs[n-1, :] = (rcut-np.clip(rx, 0, rcut))**(n+2)
gss = np.dot(betas, fs)
return rx, gss