# coding: utf-8
# Copyright (c) Pymatgen Development Team.
# Distributed under the terms of the MIT License.
from __future__ import division, unicode_literals
import warnings
import six
import ruamel.yaml as yaml
import os
"""
This module provides classes to perform topological analyses of structures.
"""
__author__ = "Shyue Ping Ong, Geoffroy Hautier, Sai Jayaraman"
__copyright__ = "Copyright 2011, The Materials Project"
__version__ = "1.0"
__maintainer__ = "Shyue Ping Ong"
__email__ = "shyuep@gmail.com"
__status__ = "Production"
__date__ = "Sep 23, 2011"
import math
from math import pi, asin, atan, sqrt, exp, cos
import numpy as np
import itertools
import collections
from warnings import warn
from scipy.spatial import Voronoi
from pymatgen import PeriodicSite
from pymatgen import Element, Specie, Composition
from pymatgen.util.num import abs_cap
from pymatgen.symmetry.analyzer import SpacegroupAnalyzer
from pymatgen.core.surface import Slab, SlabGenerator
[docs]class VoronoiCoordFinder(object):
"""
Uses a Voronoi algorithm to determine the coordination for each site in a
structure.
Args:
structure (Structure): Input structure
target ([Element/Specie]): A list of target species to determine
coordination for.
cutoff (float): Radius in Angstrom cutoff to look for coordinating
atoms. Defaults to 10.0.
allow_pathological (bool): whether to allow infinite vertices in
determination of Voronoi coordination
"""
def __init__(self, structure, target=None, cutoff=10.0,
allow_pathological=False):
warnings.warn("VoronoiCoordFinder will be moved to local_env.py"
" with pymatgen >= 2018")
self._structure = structure
self.cutoff = cutoff
self.allow_pathological = allow_pathological
if target is None:
self._target = structure.composition.elements
else:
self._target = target
[docs] def get_voronoi_polyhedra(self, n):
"""
Gives a weighted polyhedra around a site. This uses the voronoi
construction with solid angle weights.
See ref: A Proposed Rigorous Definition of Coordination Number,
M. O'Keeffe, Acta Cryst. (1979). A35, 772-775
Args:
n (int): Site index
Returns:
A dict of sites sharing a common Voronoi facet with the site
n and their solid angle weights
"""
localtarget = self._target
center = self._structure[n]
neighbors = self._structure.get_sites_in_sphere(
center.coords, self.cutoff)
neighbors = [i[0] for i in sorted(neighbors, key=lambda s: s[1])]
qvoronoi_input = [s.coords for s in neighbors]
voro = Voronoi(qvoronoi_input)
all_vertices = voro.vertices
results = {}
for nn, vind in voro.ridge_dict.items():
if 0 in nn:
if -1 in vind:
if self.allow_pathological:
continue
else:
raise RuntimeError("This structure is pathological,"
" infinite vertex in the voronoi "
"construction")
facets = [all_vertices[i] for i in vind]
results[neighbors[sorted(nn)[1]]] = solid_angle(
center.coords, facets)
maxangle = max(results.values())
resultweighted = {}
for nn, angle in results.items():
# is nn site is ordered use "nn.specie" to get species, else use "nn.species_and_occu" to get species
if nn.is_ordered:
if nn.specie in localtarget:
resultweighted[nn] = angle / maxangle
else: # is nn site is disordered
for disordered_sp in nn.species_and_occu.keys():
if disordered_sp in localtarget:
resultweighted[nn] = angle / maxangle
return resultweighted
[docs] def get_coordination_number(self, n):
"""
Returns the coordination number of site with index n.
Args:
n (int): Site index
"""
return sum(self.get_voronoi_polyhedra(n).values())
[docs] def get_coordinated_sites(self, n, tol=0, target=None):
"""
Returns the sites that are in the coordination radius of site with
index n.
Args:
n (int): Site index.
tol (float): Weight tolerance to determine if a particular pair is
considered a neighbor.
target (Element): Target element
Returns:
Sites coordinating input site.
"""
coordinated_sites = []
for site, weight in self.get_voronoi_polyhedra(n).items():
if weight > tol and (target is None or site.specie == target):
coordinated_sites.append(site)
return coordinated_sites
[docs]class JMolCoordFinder:
"""
Determine coordinated sites and coordination number using an emulation of
JMol's default autoBond() algorithm. This version of the algorithm does not
take into account any information regarding known charge states.
"""
def __init__(self, el_radius_updates=None):
"""
Initialize coordination finder parameters (atomic radii)
Args:
el_radius_updates: (dict) symbol->float to override default atomic
radii table values
"""
warnings.warn("JMolCoordFinder will be moved to local_env.py"
" with pymatgen >= 2018")
# load elemental radii table
bonds_file = os.path.join(os.path.dirname(os.path.abspath(__file__)),
"bonds_jmol_ob.yaml")
with open(bonds_file, 'r') as f:
self.el_radius = yaml.safe_load(f)
# update any user preference elemental radii
if el_radius_updates:
self.el_radius.update(el_radius_updates)
[docs] def get_max_bond_distance(self, el1_sym, el2_sym, constant=0.56):
"""
Use JMol algorithm to determine bond length from atomic parameters
Args:
el1_sym: (str) symbol of atom 1
el2_sym: (str) symbol of atom 2
constant: (float) factor to tune model
Returns: (float) max bond length
"""
return math.sqrt(
(self.el_radius[el1_sym] + self.el_radius[el2_sym] + constant) ** 2)
[docs] def get_coordination_number(self, structure, n, tol=1E-3):
"""
Get the coordination number of a site
Args:
structure: (Structure)
n: (int) index of site in the structure to get CN for
tol: (float) a numerical tolerance to extend search
Returns: (int) the coordination number
"""
return len(self.get_coordinated_sites(structure, n, tol))
[docs] def get_coordinated_sites(self, structure, n, tol=1E-3):
"""
Get the coordinated sites for a site
Args:
structure: (Structure)
n: (int) index of site in the structure to analyze
tol: (float) a numerical tolerance to extend search
Returns: ([sites]) a list of coordinated sites
"""
site = structure[n]
# determine relevant bond lengths based on atomic radii table
bonds = {}
for el in structure.composition.elements:
bonds[site.specie, el] = self.get_max_bond_distance(
site.specie.symbol, el.symbol)
# search for neighbors up to max bond length + tolerance
max_rad = max(bonds.values()) + tol
all_neighbors = []
for neighb, dist in structure.get_neighbors(site, max_rad):
# confirm neighbor based on bond length specific to atom pair
if dist <= bonds[(site.specie, neighb.specie)] + tol:
all_neighbors.append(neighb)
return all_neighbors
[docs]def average_coordination_number(structures, freq=10):
"""
Calculates the ensemble averaged Voronoi coordination numbers
of a list of Structures using VoronoiCoordFinder.
Typically used for analyzing the output of a Molecular Dynamics run.
Args:
structures (list): list of Structures.
freq (int): sampling frequency of coordination number [every freq steps].
Returns:
Dictionary of elements as keys and average coordination numbers as values.
"""
coordination_numbers = {}
for el in structures[0].composition.elements:
coordination_numbers[el.name] = 0.0
count = 0
for t in range(len(structures)):
if t % freq != 0:
continue
count += 1
vor = VoronoiCoordFinder(structures[t])
for atom in range(len(structures[0])):
cn = vor.get_coordination_number(atom)
coordination_numbers[structures[t][atom].species_string] += cn
elements = structures[0].composition.as_dict()
for el in coordination_numbers:
coordination_numbers[el] = coordination_numbers[el] / elements[
el] / count
return coordination_numbers
[docs]class VoronoiAnalyzer(object):
"""
Performs a statistical analysis of Voronoi polyhedra around each site.
Each Voronoi polyhedron is described using Schaefli notation.
That is a set of indices {c_i} where c_i is the number of faces with i
number of vertices. E.g. for a bcc crystal, there is only one polyhedron
notation of which is [0,6,0,8,0,0,...].
In perfect crystals, these also corresponds to the Wigner-Seitz cells.
For distorted-crystals, liquids or amorphous structures, rather than one-type,
there is a statistical distribution of polyhedra.
See ref: Microstructure and its relaxation in Fe-B amorphous system
simulated by molecular dynamics,
Stepanyuk et al., J. Non-cryst. Solids (1993), 159, 80-87.
Args:
cutoff (float): cutoff distance to search for neighbors of a given atom
(default = 5.0)
qhull_options (str): options to pass to qhull (optional)
"""
def __init__(self, cutoff=5.0, qhull_options="Qbb Qc Qz"):
self.cutoff = cutoff
self.qhull_options = qhull_options
[docs] def analyze(self, structure, n=0):
"""
Performs Voronoi analysis and returns the polyhedra around atom n
in Schlaefli notation.
Args:
structure (Structure): structure to analyze
n (int): index of the center atom in structure
Returns:
voronoi index of n: <c3,c4,c6,c6,c7,c8,c9,c10>
where c_i denotes number of facets with i vertices.
"""
center = structure[n]
neighbors = structure.get_sites_in_sphere(center.coords, self.cutoff)
neighbors = [i[0] for i in sorted(neighbors, key=lambda s: s[1])]
qvoronoi_input = np.array([s.coords for s in neighbors])
voro = Voronoi(qvoronoi_input, qhull_options=self.qhull_options)
vor_index = np.array([0, 0, 0, 0, 0, 0, 0, 0])
for key in voro.ridge_dict:
if 0 in key: # This means if the center atom is in key
if -1 in key: # This means if an infinity point is in key
raise ValueError("Cutoff too short.")
else:
try:
vor_index[len(voro.ridge_dict[key]) - 3] += 1
except IndexError:
# If a facet has more than 10 edges, it's skipped here.
pass
return vor_index
[docs] def analyze_structures(self, structures, step_freq=10,
most_frequent_polyhedra=15):
"""
Perform Voronoi analysis on a list of Structures.
Note that this might take a significant amount of time depending on the
size and number of structures.
Args:
structures (list): list of Structures
cutoff (float: cutoff distance around an atom to search for
neighbors
step_freq (int): perform analysis every step_freq steps
qhull_options (str): options to pass to qhull
most_frequent_polyhedra (int): this many unique polyhedra with
highest frequences is stored.
Returns:
A list of tuples in the form (voronoi_index,frequency)
"""
voro_dict = {}
step = 0
for structure in structures:
step += 1
if step % step_freq != 0:
continue
v = []
for n in range(len(structure)):
v.append(str(self.analyze(structure, n=n).view()))
for voro in v:
if voro in voro_dict:
voro_dict[voro] += 1
else:
voro_dict[voro] = 1
return sorted(voro_dict.items(),
key=lambda x: (x[1], x[0]),
reverse=True)[:most_frequent_polyhedra]
[docs] @staticmethod
def plot_vor_analysis(voronoi_ensemble):
t = zip(*voronoi_ensemble)
labels = t[0]
val = list(t[1])
tot = np.sum(val)
val = [float(j) / tot for j in val]
pos = np.arange(len(val)) + .5 # the bar centers on the y axis
import matplotlib.pyplot as plt
plt.figure()
plt.barh(pos, val, align='center', alpha=0.5)
plt.yticks(pos, labels)
plt.xlabel('Count')
plt.title('Voronoi Spectra')
plt.grid(True)
return plt
[docs]class RelaxationAnalyzer(object):
"""
This class analyzes the relaxation in a calculation.
"""
def __init__(self, initial_structure, final_structure):
"""
Please note that the input and final structures should have the same
ordering of sites. This is typically the case for most computational
codes.
Args:
initial_structure (Structure): Initial input structure to
calculation.
final_structure (Structure): Final output structure from
calculation.
"""
if final_structure.formula != initial_structure.formula:
raise ValueError("Initial and final structures have different " +
"formulas!")
self.initial = initial_structure
self.final = final_structure
[docs] def get_percentage_volume_change(self):
"""
Returns the percentage volume change.
Returns:
Volume change in percentage, e.g., 0.055 implies a 5.5% increase.
"""
initial_vol = self.initial.lattice.volume
final_vol = self.final.lattice.volume
return final_vol / initial_vol - 1
[docs] def get_percentage_lattice_parameter_changes(self):
"""
Returns the percentage lattice parameter changes.
Returns:
A dict of the percentage change in lattice parameter, e.g.,
{'a': 0.012, 'b': 0.021, 'c': -0.031} implies a change of 1.2%,
2.1% and -3.1% in the a, b and c lattice parameters respectively.
"""
initial_latt = self.initial.lattice
final_latt = self.final.lattice
d = {l: getattr(final_latt, l) / getattr(initial_latt, l) - 1
for l in ["a", "b", "c"]}
return d
[docs] def get_percentage_bond_dist_changes(self, max_radius=3.0):
"""
Returns the percentage bond distance changes for each site up to a
maximum radius for nearest neighbors.
Args:
max_radius (float): Maximum radius to search for nearest
neighbors. This radius is applied to the initial structure,
not the final structure.
Returns:
Bond distance changes as a dict of dicts. E.g.,
{index1: {index2: 0.011, ...}}. For economy of representation, the
index1 is always less than index2, i.e., since bonding between
site1 and siten is the same as bonding between siten and site1,
there is no reason to duplicate the information or computation.
"""
data = collections.defaultdict(dict)
for inds in itertools.combinations(list(range(len(self.initial))), 2):
(i, j) = sorted(inds)
initial_dist = self.initial[i].distance(self.initial[j])
if initial_dist < max_radius:
final_dist = self.final[i].distance(self.final[j])
data[i][j] = final_dist / initial_dist - 1
return data
[docs]class VoronoiConnectivity(object):
"""
Computes the solid angles swept out by the shared face of the voronoi
polyhedron between two sites.
Args:
structure (Structure): Input structure
cutoff (float) Cutoff distance.
"""
# Radius in Angstrom cutoff to look for coordinating atoms
def __init__(self, structure, cutoff=10):
self.cutoff = cutoff
self.s = structure
recp_len = np.array(self.s.lattice.reciprocal_lattice.abc)
i = np.ceil(cutoff * recp_len / (2 * math.pi))
offsets = np.mgrid[-i[0]:i[0] + 1, -i[1]:i[1] + 1, -i[2]:i[2] + 1].T
self.offsets = np.reshape(offsets, (-1, 3))
# shape = [image, axis]
self.cart_offsets = self.s.lattice.get_cartesian_coords(self.offsets)
@property
def connectivity_array(self):
"""
Provides connectivity array.
Returns:
connectivity: An array of shape [atomi, atomj, imagej]. atomi is
the index of the atom in the input structure. Since the second
atom can be outside of the unit cell, it must be described
by both an atom index and an image index. Array data is the
solid angle of polygon between atomi and imagej of atomj
"""
# shape = [site, axis]
cart_coords = np.array(self.s.cart_coords)
# shape = [site, image, axis]
all_sites = cart_coords[:, None, :] + self.cart_offsets[None, :, :]
vt = Voronoi(all_sites.reshape((-1, 3)))
n_images = all_sites.shape[1]
cs = (len(self.s), len(self.s), len(self.cart_offsets))
connectivity = np.zeros(cs)
vts = np.array(vt.vertices)
for (ki, kj), v in vt.ridge_dict.items():
atomi = ki // n_images
atomj = kj // n_images
imagei = ki % n_images
imagej = kj % n_images
if imagei != n_images // 2 and imagej != n_images // 2:
continue
if imagei == n_images // 2:
# atomi is in original cell
val = solid_angle(vt.points[ki], vts[v])
connectivity[atomi, atomj, imagej] = val
if imagej == n_images // 2:
# atomj is in original cell
val = solid_angle(vt.points[kj], vts[v])
connectivity[atomj, atomi, imagei] = val
if -10.101 in vts[v]:
warn('Found connectivity with infinite vertex. '
'Cutoff is too low, and results may be '
'incorrect')
return connectivity
@property
def max_connectivity(self):
"""
returns the 2d array [sitei, sitej] that represents
the maximum connectivity of site i to any periodic
image of site j
"""
return np.max(self.connectivity_array, axis=2)
[docs] def get_connections(self):
"""
Returns a list of site pairs that are Voronoi Neighbors, along
with their real-space distances.
"""
con = []
maxconn = self.max_connectivity
for ii in range(0, maxconn.shape[0]):
for jj in range(0, maxconn.shape[1]):
if maxconn[ii][jj] != 0:
dist = self.s.get_distance(ii, jj)
con.append([ii, jj, dist])
return con
[docs] def get_sitej(self, site_index, image_index):
"""
Assuming there is some value in the connectivity array at indices
(1, 3, 12). sitei can be obtained directly from the input structure
(structure[1]). sitej can be obtained by passing 3, 12 to this function
Args:
site_index (int): index of the site (3 in the example)
image_index (int): index of the image (12 in the example)
"""
atoms_n_occu = self.s[site_index].species_and_occu
lattice = self.s.lattice
coords = self.s[site_index].frac_coords + self.offsets[image_index]
return PeriodicSite(atoms_n_occu, coords, lattice)
[docs]def solid_angle(center, coords):
"""
Helper method to calculate the solid angle of a set of coords from the
center.
Args:
center (3x1 array): Center to measure solid angle from.
coords (Nx3 array): List of coords to determine solid angle.
Returns:
The solid angle.
"""
o = np.array(center)
r = [np.array(c) - o for c in coords]
r.append(r[0])
n = [np.cross(r[i + 1], r[i]) for i in range(len(r) - 1)]
n.append(np.cross(r[1], r[0]))
vals = []
for i in range(len(n) - 1):
v = -np.dot(n[i], n[i + 1]) \
/ (np.linalg.norm(n[i]) * np.linalg.norm(n[i + 1]))
vals.append(math.acos(abs_cap(v)))
phi = sum(vals)
return phi + (3 - len(r)) * math.pi
[docs]def get_max_bond_lengths(structure, el_radius_updates=None):
"""
Provides max bond length estimates for a structure based on the JMol
table and algorithms.
Args:
structure: (structure)
el_radius_updates: (dict) symbol->float to update atomic radii
Returns: (dict) - (Element1, Element2) -> float. The two elements are
ordered by Z.
"""
jmc = JMolCoordFinder(el_radius_updates)
bonds_lens = {}
els = sorted(structure.composition.elements, key=lambda x: x.Z)
for i1 in range(len(els)):
for i2 in range(len(els) - i1):
bonds_lens[els[i1], els[i1 + i2]] = jmc.get_max_bond_distance(
els[i1].symbol, els[i1 + i2].symbol)
return bonds_lens
[docs]def get_dimensionality(structure, max_hkl=2, el_radius_updates=None,
min_slab_size=5, min_vacuum_size=5,
standardize=True, bonds=None):
"""
This method returns whether a structure is 3D, 2D (layered), or 1D (linear
chains or molecules) according to the algorithm published in Gorai, P.,
Toberer, E. & Stevanovic, V. Computational Identification of Promising
Thermoelectric Materials Among Known Quasi-2D Binary Compounds. J. Mater.
Chem. A 2, 4136 (2016).
Note that a 1D structure detection might indicate problems in the bonding
algorithm, particularly for ionic crystals (e.g., NaCl)
Users can change the behavior of bonds detection by passing either
el_radius_updates to update atomic radii for auto-detection of max bond
distances, or bonds to explicitly specify max bond distances for atom pairs.
Note that if you pass both, el_radius_updates are ignored.
Args:
structure: (Structure) structure to analyze dimensionality for
max_hkl: (int) max index of planes to look for layers
el_radius_updates: (dict) symbol->float to update atomic radii
min_slab_size: (float) internal surface construction parameter
min_vacuum_size: (float) internal surface construction parameter
standardize (bool): whether to standardize the structure before
analysis. Set to False only if you already have the structure in a
convention where layers / chains will be along low <hkl> indexes.
bonds ({(specie1, specie2): max_bond_dist}: bonds are
specified as a dict of tuples: float of specie1, specie2
and the max bonding distance. For example, PO4 groups may be
defined as {("P", "O"): 3}.
Returns: (int) the dimensionality of the structure - 1 (molecules/chains),
2 (layered), or 3 (3D)
"""
if standardize:
structure = SpacegroupAnalyzer(structure).\
get_conventional_standard_structure()
if not bonds:
bonds = get_max_bond_lengths(structure, el_radius_updates)
num_surfaces = 0
for h in range(max_hkl):
for k in range(max_hkl):
for l in range(max_hkl):
if max([h, k, l]) > 0 and num_surfaces < 2:
sg = SlabGenerator(structure, (h, k, l),
min_slab_size=min_slab_size,
min_vacuum_size=min_vacuum_size)
slabs = sg.get_slabs(bonds)
for _ in slabs:
num_surfaces += 1
return 3 - min(num_surfaces, 2)
[docs]def contains_peroxide(structure, relative_cutoff=1.1):
"""
Determines if a structure contains peroxide anions.
Args:
structure (Structure): Input structure.
relative_cutoff: The peroxide bond distance is 1.49 Angstrom.
Relative_cutoff * 1.49 stipulates the maximum distance two O
atoms must be to each other to be considered a peroxide.
Returns:
Boolean indicating if structure contains a peroxide anion.
"""
ox_type = oxide_type(structure, relative_cutoff)
if ox_type == "peroxide":
return True
else:
return False
[docs]class OxideType(object):
"""
Separate class for determining oxide type.
Args:
structure: Input structure.
relative_cutoff: Relative_cutoff * act. cutoff stipulates the max.
distance two O atoms must be from each other. Default value is
1.1. At most 1.1 is recommended, nothing larger, otherwise the
script cannot distinguish between superoxides and peroxides.
"""
def __init__(self, structure, relative_cutoff=1.1):
self.structure = structure
self.relative_cutoff = relative_cutoff
self.oxide_type, self.nbonds = self.parse_oxide()
[docs] def parse_oxide(self):
"""
Determines if an oxide is a peroxide/superoxide/ozonide/normal oxide.
Returns:
oxide_type (str): Type of oxide
ozonide/peroxide/superoxide/hydroxide/None.
nbonds (int): Number of peroxide/superoxide/hydroxide bonds in
structure.
"""
structure = self.structure
relative_cutoff = self.relative_cutoff
o_sites_frac_coords = []
h_sites_frac_coords = []
lattice = structure.lattice
if isinstance(structure.composition.elements[0], Element):
comp = structure.composition
elif isinstance(structure.composition.elements[0], Specie):
elmap = collections.defaultdict(float)
for site in structure:
for species, occu in site.species_and_occu.items():
elmap[species.element] += occu
comp = Composition(elmap)
if Element("O") not in comp or comp.is_element:
return "None", 0
for site in structure:
syms = [sp.symbol for sp in site.species_and_occu.keys()]
if "O" in syms:
o_sites_frac_coords.append(site.frac_coords)
if "H" in syms:
h_sites_frac_coords.append(site.frac_coords)
if h_sites_frac_coords:
dist_matrix = lattice.get_all_distances(o_sites_frac_coords,
h_sites_frac_coords)
if np.any(dist_matrix < relative_cutoff * 0.93):
return "hydroxide", len(
np.where(dist_matrix < relative_cutoff * 0.93)[0]) / 2.0
dist_matrix = lattice.get_all_distances(o_sites_frac_coords,
o_sites_frac_coords)
np.fill_diagonal(dist_matrix, 1000)
is_superoxide = False
is_peroxide = False
is_ozonide = False
if np.any(dist_matrix < relative_cutoff * 1.35):
bond_atoms = np.where(dist_matrix < relative_cutoff * 1.35)[0]
is_superoxide = True
elif np.any(dist_matrix < relative_cutoff * 1.49):
is_peroxide = True
bond_atoms = np.where(dist_matrix < relative_cutoff * 1.49)[0]
if is_superoxide:
if len(bond_atoms) > len(set(bond_atoms)):
is_superoxide = False
is_ozonide = True
try:
nbonds = len(set(bond_atoms))
except UnboundLocalError:
nbonds = 0.0
if is_ozonide:
str_oxide = "ozonide"
elif is_superoxide:
str_oxide = "superoxide"
elif is_peroxide:
str_oxide = "peroxide"
else:
str_oxide = "oxide"
if str_oxide == "oxide":
nbonds = comp["O"]
return str_oxide, nbonds
[docs]def oxide_type(structure, relative_cutoff=1.1, return_nbonds=False):
"""
Determines if an oxide is a peroxide/superoxide/ozonide/normal oxide
Args:
structure (Structure): Input structure.
relative_cutoff (float): Relative_cutoff * act. cutoff stipulates the
max distance two O atoms must be from each other.
return_nbonds (bool): Should number of bonds be requested?
"""
ox_obj = OxideType(structure, relative_cutoff)
if return_nbonds:
return ox_obj.oxide_type, ox_obj.nbonds
else:
return ox_obj.oxide_type
[docs]def sulfide_type(structure):
"""
Determines if a structure is a sulfide/polysulfide
Args:
structure (Structure): Input structure.
Returns:
(str) sulfide/polysulfide/sulfate
"""
structure = structure.copy()
structure.remove_oxidation_states()
s = Element("S")
comp = structure.composition
if comp.is_element or s not in comp:
return None
finder = SpacegroupAnalyzer(structure, symprec=0.1)
symm_structure = finder.get_symmetrized_structure()
s_sites = [sites[0] for sites in symm_structure.equivalent_sites if sites[0].specie == s]
def process_site(site):
neighbors = structure.get_neighbors(site, 4)
neighbors = sorted(neighbors, key=lambda n: n[1])
nn, dist = neighbors[0]
coord_elements = [site.specie for site, d in neighbors
if d < dist + 0.4][:4]
avg_electroneg = np.mean([e.X for e in coord_elements])
if avg_electroneg > s.X:
return "sulfate"
elif avg_electroneg == s.X and s in coord_elements:
return "polysulfide"
else:
return "sulfide"
types = set([process_site(site) for site in s_sites])
if "sulfate" in types:
return None
elif "polysulfide" in types:
return "polysulfide"
else:
return "sulfide"
[docs]def gramschmidt(vin, uin):
"""
Returns that part of the first input vector
that is orthogonal to the second input vector.
The output vector is not normalized.
Args:
vin (numpy array):
first input vector
uin (numpy array):
second input vector
"""
vin_uin = np.inner(vin, uin)
uin_uin = np.inner(uin, uin)
if uin_uin <= 0.0:
raise ValueError("Zero or negative inner product!")
return vin - (vin_uin / uin_uin) * uin
[docs]class OrderParameters(object):
"""
This class permits the calculation of various types of local order
parameters.
"""
__supported_types = (
"cn", "lin", "bent", "tet", "tet_legacy", "oct", "oct_legacy",
"bcc", "q2", "q4", "q6", "reg_tri", "sq", "sq_pyr", "tri_bipyr")
def __init__(self, types, parameters=None, cutoff=-10.0):
"""
Args:
types ([string]): list of strings representing the types of
order parameters to be calculated. Note that multiple
mentions of the same type may occur. Currently available
types are:
"cn": simple coordination number---normalized
if desired;
"lin": Peters-style OP recognizing linear coordination
(Zimmermann & Jain, in progress, 2017);
"bent": Peters-style OP recognizing bent coordination
(Zimmermann & Jain, in progress, 2017);
"tet": Peters-style OP recognizing tetrahedral
environments (Zimmermann et al., submitted,
2017);
"oct": Peters-style OP recognizing octahedral
environments (Zimmermann et al., submitted,
2017);
"bcc": Peters-style OP recognizing local
body-centered cubic environment (Peters,
J. Chem. Phys., 131, 244103, 2009);
"reg_tri": OP recognizing coordination with a regular triangle;
"sq": OP recognizing square coordination;
"sq_pyr": OP recognizing square pyramidal coordination;
"tri_bipyr": OP recognizing trigonal bipyramidal coord.;
"q2": bond orientational order parameter (BOOP)
of weight l=2 (Steinhardt et al., Phys. Rev. B,
28, 784-805, 1983);
"q4": BOOP of weight l=4;
"q6": BOOP of weight l=6.
"tet_legacy": original Peters-style OP recognizing
tetrahedral coordination environments
(Zimmermann et al., J. Am. Chem. Soc.,
137, 13352-13361, 2015) that can, however,
produce small negative values sometimes;
"oct_legacy": original Peters-style OP recognizing
octahedral coordination environments
(Zimmermann et al., J. Am. Chem. Soc.,
137, 13352-13361, 2015) that can, however,
produce small negative values sometimes.
parameters ([[float]]): 2D list of floating point numbers that store
parameters associated with the different order parameters
that are to be calculated (1st dimension = length of
types tuple; any 2nd dimension may be zero, in which case
default values are used). In the following, those order
parameters q_i are listed that require further parameters
for their computation (values in brackets denote default
values):
"cn": normalizing constant (1);
"lin": Gaussian width in fractions of pi (180 degrees)
reflecting the "speed of penalizing" deviations
away from 180 degrees of any individual
neighbor1-center-neighbor2 configuration (0.0667);
"bent": target angle in degrees (180);
Gaussian width for penalizing deviations away
from perfect target angle in fractions of pi
(0.0667);
"tet": Gaussian width for penalizing deviations
away perfect tetrahedral angle (0.0667);
"oct": threshold angle in degrees distinguishing
a second neighbor to be either close to the
south pole or close to the equator (160.0);
Gaussian width for penalizing deviations
away from south pole (0.0667); Gaussian
width for penalizing deviations away from
equator (0.0556);
"bcc": south-pole threshold angle as for "oct" (160.0);
south-pole Gaussian width as for "oct" (0.0667);
"reg_tri": Gaussian width for penalizing angles away from
the expected angles, given the estimated
height-to-side ratio of the trigonal pyramid
in which the central atom is located at the
tip (0.0222);
"sq": Gaussian width for penalizing angles away from
the expected angles, given the estimated
height-to-diagonal ratio of the pyramid in which
the central atom is located at the tip
(0.0333);
"sq_pyr": Gaussian width in fractions of pi
for penalizing angles away from 90 degrees
(0.0333);
Gaussian width in Angstrom for penalizing
variations in neighbor distances (0.1);
"tri_bipyr": threshold angle to identify close to
South pole positions (160.0, cf., oct).
Gaussian width for penalizing deviations away
from south pole (0.0667);
Gaussian width for penalizing deviations away
from equator (0.0556).
"tet_legacy": Gaussian width for penalizing deviations
away perfect tetrahedral angle (0.0667);
"oct_legacy": threshold angle in degrees distinguishing
a second neighbor to be either close to the
south pole or close to the equator (160.0);
Gaussian width for penalizing deviations
away from south pole (0.0667); Gaussian
width for penalizing deviations away from
equator (0.0556); constant for shifting
q_oct_legacy toward smaller values, which
can be helpful when trying to fine-tune
the capabilities of distinguishing between
different environments (e.g., tet vs oct)
given a single mutual threshold q_thresh;
cutoff (float): Cutoff radius to determine which nearest neighbors are
supposed to contribute to the order parameters.
If the value is negative the neighboring sites found by
distance and cutoff radius are further
pruned using the get_coordinated_sites method from the
VoronoiCoordFinder class.
"""
if len(types) == 0:
raise ValueError("Empty types list!")
for t in types:
if t not in OrderParameters.__supported_types:
raise ValueError("Unknown order parameter type (" + \
t + ")!")
if parameters is not None:
if len(types) != len(parameters):
raise ValueError("1st dimension of parameters array is not"
" consistent with types list!")
for lp in parameters:
if len(lp) > 0:
for p in lp:
if type(p) != float and type(p) != int:
raise AttributeError("Expected only float and"
" integer type parameters!")
loc_parameters = list(parameters)
else:
loc_parameters = [[] for t in types]
self._types = tuple(types)
tmpparas = []
self._computerijs = self._computerjks = self._geomops = False
self._geomops2 = self._boops = False
self._max_trig_order = -1
for i, t in enumerate(self._types):
# add here any additional parameter checking and
# default value assignment
tmpparas.append([])
if t == "cn":
if len(loc_parameters[i]) == 0:
tmpparas[i].append(1.0)
else:
if loc_parameters[i][0] == 0.0:
raise ValueError("Normalizing constant for"
" coordination-number based order"
" parameter is zero!")
else:
tmpparas[i].append(loc_parameters[i][0])
elif t == "lin":
if len(loc_parameters[i]) == 0:
tmpparas[i] = [1.0 / 0.0667]
else:
if loc_parameters[i][0] == 0.0:
raise ValueError("Gaussian width for"
" linear order"
" parameter is zero!")
else:
tmpparas[i] = [1.0 / loc_parameters[i][0]]
elif t == "bent":
if len(loc_parameters[i]) == 0:
tmpparas[i] = [1.0, 1.0 / 0.0667]
else:
if loc_parameters[i][0] <= 0.0 or loc_parameters[i][
0] > 180.0:
warn("Target angle for bent order parameter is"
" not in ]0,180] interval.")
if loc_parameters[i][1] == 0.0:
raise ValueError("Gaussian width for"
" bent order"
" parameter is zero!")
else:
tmpparas[i] = [loc_parameters[i][0] / 180.0, \
1.0 / loc_parameters[i][1]]
elif t == "tet_legacy":
if len(loc_parameters[i]) == 0:
tmpparas[i].append(1.0 / 0.0667)
else:
if loc_parameters[i][0] == 0.0:
raise ValueError("Gaussian width for"
" original tetrahedral order"
" parameter is zero!")
else:
tmpparas[i].append(1.0 / loc_parameters[i][0])
elif t == "tet":
if len(loc_parameters[i]) == 0:
tmpparas[i].append(1.0 / 0.0667)
else:
if loc_parameters[i][0] == 0.0:
raise ValueError("Gaussian width for"
" tetrahedral order"
" parameter is zero!")
else:
tmpparas[i].append(1.0 / loc_parameters[i][0])
elif t == "oct_legacy":
if len(loc_parameters[i]) < 4:
tmpparas[i].append(8.0 * pi / 9.0)
tmpparas[i].append(1.0 / 0.0667)
tmpparas[i].append(1.0 / 0.0556)
tmpparas[i].append(0.25)
tmpparas[i].append(4.0 / 3.0)
else:
if loc_parameters[i][0] <= 0.0 or loc_parameters[i][
0] >= 180.0:
warn("Threshold value for south pole"
" configurations in original octahedral order"
" parameter outside ]0,180[")
tmpparas[i].append(loc_parameters[i][0] * pi / 180.0)
if loc_parameters[i][1] == 0.0:
raise ValueError("Gaussian width for south pole"
" configurations in original"
" octahedral order parameter is"
" zero!")
else:
tmpparas[i].append(1.0 / loc_parameters[i][1])
if loc_parameters[i][2] == 0.0:
raise ValueError("Gaussian width for equatorial"
" configurations in original"
" octahedral order parameter is"
" zero!")
else:
tmpparas[i].append(1.0 / loc_parameters[i][2])
if loc_parameters[i][3] - 1.0 == 0.0:
raise ValueError("Shift constant may not be"
" unity!")
if loc_parameters[i][3] < 0.0 or loc_parameters[i][3] > 1.0:
warn("Shift constant outside [0,1[.")
tmpparas[i].append(loc_parameters[i][3])
tmpparas[i].append(1.0 / (1.0 - loc_parameters[i][3]))
elif t == "oct":
if len(loc_parameters[i]) < 3:
tmpparas[i].append(8.0 * pi / 9.0)
tmpparas[i].append(1.0 / 0.0667)
tmpparas[i].append(1.0 / 0.0556)
else:
if loc_parameters[i][0] <= 0.0 or loc_parameters[i][
0] >= 180.0:
warn("Threshold value for south pole"
" configurations in octahedral order"
" parameter outside ]0,180[")
tmpparas[i].append(loc_parameters[i][0] * pi / 180.0)
if loc_parameters[i][1] == 0.0:
raise ValueError("Gaussian width for south pole"
" configurations in octahedral"
" order parameter is zero!")
else:
tmpparas[i].append(1.0 / loc_parameters[i][1])
if loc_parameters[i][2] == 0.0:
raise ValueError("Gaussian width for equatorial"
" configurations in octahedral"
" order parameter is zero!")
else:
tmpparas[i].append(1.0 / loc_parameters[i][2])
elif t == "bcc":
if len(loc_parameters[i]) < 2:
tmpparas[i].append(8.0 * pi / 9.0)
tmpparas[i].append(1.0 / 0.0667)
else:
if loc_parameters[i][0] <= 0.0 or loc_parameters[i][
0] >= 180.0:
warn("Threshold value for south pole"
" configurations in bcc order"
" parameter outside ]0,180[")
tmpparas[i].append(loc_parameters[i][0] * pi / 180.0)
if loc_parameters[i][1] == 0.0:
raise ValueError("Gaussian width for south pole"
" configurations in bcc"
" order parameter is zero!")
else:
tmpparas[i].append(1.0 / loc_parameters[i][1])
elif t == "reg_tri":
if len(loc_parameters[i]) == 0:
tmpparas[i] = [1.0 / 0.0222]
else:
if loc_parameters[i][0] == 0.0:
raise ValueError("Gaussian width for angles in"
" trigonal pyramid tip of regular triangle"
" order parameter is zero!")
tmpparas[i] = [1.0 / loc_parameters[i][0]]
elif t == "sq":
if len(loc_parameters[i]) == 0:
tmpparas[i] = [1.0 / 0.0333]
else:
if loc_parameters[i][0] == 0.0:
raise ValueError("Gaussian width for angles in"
" pyramid tip of square order parameter"
" is zero!")
tmpparas[i] = [1.0 / loc_parameters[i][0]]
elif t == "sq_pyr":
if len(loc_parameters[i]) == 0:
tmpparas[i] = [1.0 / 0.0333, 1.0 / 0.1]
else:
if loc_parameters[i][0] == 0.0:
raise ValueError("Gaussian width for angles in"
" square pyramid order parameter is zero!")
if loc_parameters[i][0] == 0.0:
raise ValueError("Gaussian width for lengths in"
" square pyramid order parameter is zero!")
tmpparas[i] = [1.0 / loc_parameters[i][0], \
1.0 / loc_parameters[i][1]]
elif t == "tri_bipyr":
if len(loc_parameters[i]) < 3:
tmpparas[i].append(8.0 * pi / 9.0)
tmpparas[i].append(1.0 / 0.0667)
tmpparas[i].append(1.0 / 0.0741)
else:
if loc_parameters[i][0] <= 0.0 or loc_parameters[i][
0] >= 180.0:
warn("Threshold value for south pole"
" configurations in trigonal bipyramidal"
" order parameter outside ]0,180[")
tmpparas[i].append(loc_parameters[i][0] * pi / 180.0)
if loc_parameters[i][1] == 0.0:
raise ValueError("Gaussian width for south pole"
" configurations in trigonal"
" bipyramidal order parameter is"
" zero!")
else:
tmpparas[i].append(1.0 / loc_parameters[i][1])
if loc_parameters[i][2] == 0.0:
raise ValueError("Gaussian width for equatorial"
" configurations in trigonal"
" bipyramidal order parameter"
" is zero!")
else:
tmpparas[i].append(1.0 / loc_parameters[i][2])
# All following types should be well-defined/-implemented,
# and they should not require parameters.
elif t != "q2" and t != "q4" and t != "q6":
raise ValueError("unknown order-parameter type \"" + t + "\"")
# Add here any additional flags to be used during calculation.
# self._computerijs: compute vectors from centeral atom i
# to any neighbor j.
# self._computerjks: compute vectors from non-centeral atom j
# to any non-central atom k.
if t == "tet" or t == "oct" or t == "bcc" or t == "sq_pyr" or \
t == "tri_bipyr" or t == "tet_legacy" or t == "oct_legacy":
self._computerijs = self._geomops = True
if t == "reg_tri" or t =="sq":
self._computerijs = self._computerjks = self._geomops2 = True
if t == "q2" or t == "q4" or t == "q6":
self._computerijs = self._boops = True
if t == "q2" and self._max_trig_order < 2:
self._max_trig_order = 2
if t == "q4" and self._max_trig_order < 4:
self._max_trig_order = 4
if t == "q6" and self._max_trig_order < 6:
self._max_trig_order = 6
# Finish parameter treatment.
self._paras = list(tmpparas)
if cutoff < 0.0:
self._cutoff = -cutoff
self._voroneigh = True
elif cutoff > 0.0:
self._cutoff = cutoff
self._voroneigh = False
else:
raise ValueError("Cutoff radius is zero!")
# Further variable definitions.
self._last_nneigh = -1
self._pow_sin_t = {}
self._pow_cos_t = {}
self._sin_n_p = {}
self._cos_n_p = {}
@property
def num_ops(self):
""""
Returns the number of different order parameters that are targeted
to be calculated.
"""
return len(self._types)
@property
def last_nneigh(self):
""""
Returns the number of neighbors encountered during the most
recent order-parameter calculation. A value of -1 indicates that
no such calculation has yet been performed for this instance.
"""
return len(self._last_nneigh)
[docs] def compute_trigonometric_terms(self, thetas, phis):
""""
Computes trigonometric terms that are required to
calculate bond orientational order parameters.
Args:
thetas ([float]):
polar angles of all neighbors in radians.
phis ([float]):
azimuth angles of all neighbors in radians. The list of
azimuth angles is expected to have the same size as the list
of polar angles; otherwise, a ValueError is raised. Also,
the two lists of angles have to be coherent in order. That
is, it is expected that the order in the list of azimuth
angles corresponds to a distinct sequence of neighbors.
And, this sequence has to equal the sequence
of neighbors in the list of polar angles.
"""
if len(thetas) != len(phis):
raise ValueError("List of polar and azimuthal angles have to be"
" equal!")
self._pow_sin_t.clear()
self._pow_cos_t.clear()
self._sin_n_p.clear()
self._cos_n_p.clear()
self._pow_sin_t[1] = [math.sin(float(t)) for t in thetas]
self._pow_cos_t[1] = [math.cos(float(t)) for t in thetas]
self._sin_n_p[1] = [math.sin(float(p)) for p in phis]
self._cos_n_p[1] = [math.cos(float(p)) for p in phis]
for i in range(2, self._max_trig_order + 1):
self._pow_sin_t[i] = [e[0] * e[1] for e in zip(
self._pow_sin_t[i - 1], self._pow_sin_t[1])]
self._pow_cos_t[i] = [e[0] * e[1] for e in zip(
self._pow_cos_t[i - 1], self._pow_cos_t[1])]
self._sin_n_p[i] = [math.sin(float(i) * float(p)) \
for p in phis]
self._cos_n_p[i] = [math.cos(float(i) * float(p)) \
for p in phis]
[docs] def get_q2(self, thetas=None, phis=None):
"""
Calculates the value of the bond orientational order parameter of
weight l=2. If the function is called with non-empty lists of
polar and azimuthal angles the corresponding trigonometric terms
are computed afresh. Otherwise, it is expected that the
compute_trigonometric_terms function has been just called.
Args:
thetas ([float]):
polar angles of all neighbors in radians.
phis ([float]):
azimuth angles of all neighbors in radians.
Return:
q2 (float): bond orientational order parameter of weight l=2
corresponding to the input angles thetas and phis.
"""
if thetas is not None and phis is not None:
self.compute_trigonometric_terms(thetas, phis)
nnn = len(self._pow_sin_t[1])
nnn_range = range(nnn)
sqrt_15_2pi = math.sqrt(15.0 / (2.0 * pi))
sqrt_5_pi = math.sqrt(5.0 / pi)
pre_y_2_2 = [0.25 * sqrt_15_2pi * val for val in self._pow_sin_t[2]]
pre_y_2_1 = [0.5 * sqrt_15_2pi * val[0] * val[1]
for val in zip(self._pow_sin_t[1], self._pow_cos_t[1])]
acc = 0.0
# Y_2_-2
real = imag = 0.0
for i in nnn_range:
real += pre_y_2_2[i] * self._cos_n_p[2][i]
imag -= pre_y_2_2[i] * self._sin_n_p[2][i]
acc += (real * real + imag * imag)
# Y_2_-1
real = imag = 0.0
for i in nnn_range:
real += pre_y_2_1[i] * self._cos_n_p[1][i]
imag -= pre_y_2_1[i] * self._sin_n_p[1][i]
acc += (real * real + imag * imag)
# Y_2_0
real = imag = 0.0
for i in nnn_range:
real += 0.25 * sqrt_5_pi * (3.0 * self._pow_cos_t[2][i] - 1.0)
acc += (real * real)
# Y_2_1
real = imag = 0.0
for i in nnn_range:
real -= pre_y_2_1[i] * self._cos_n_p[1][i]
imag -= pre_y_2_1[i] * self._sin_n_p[1][i]
acc += (real * real + imag * imag)
# Y_2_2
real = imag = 0.0
for i in nnn_range:
real += pre_y_2_2[i] * self._cos_n_p[2][i]
imag += pre_y_2_2[i] * self._sin_n_p[2][i]
acc += (real * real + imag * imag)
q2 = math.sqrt(4.0 * pi * acc / (5.0 * float(nnn * nnn)))
return q2
[docs] def get_q4(self, thetas=None, phis=None):
"""
Calculates the value of the bond orientational order parameter of
weight l=4. If the function is called with non-empty lists of
polar and azimuthal angles the corresponding trigonometric terms
are computed afresh. Otherwise, it is expected that the
compute_trigonometric_terms function has been just called.
Args:
thetas ([float]):
polar angles of all neighbors in radians.
phis ([float]):
azimuth angles of all neighbors in radians.
Return:
q4 (float): bond orientational order parameter of weight l=4
corresponding to the input angles thetas and phis.
"""
if thetas is not None and phis is not None:
self.compute_trigonometric_terms(thetas, phis)
nnn = len(self._pow_sin_t[1])
nnn_range = range(nnn)
i16_3 = 3.0 / 16.0
i8_3 = 3.0 / 8.0
sqrt_35_pi = math.sqrt(35.0 / pi)
sqrt_35_2pi = math.sqrt(35.0 / (2.0 * pi))
sqrt_5_pi = math.sqrt(5.0 / pi)
sqrt_5_2pi = math.sqrt(5.0 / (2.0 * pi))
sqrt_1_pi = math.sqrt(1.0 / pi)
pre_y_4_4 = [i16_3 * sqrt_35_2pi * val for val in self._pow_sin_t[4]]
pre_y_4_3 = [i8_3 * sqrt_35_pi * val[0] * val[1] \
for val in zip(self._pow_sin_t[3], self._pow_cos_t[1])]
pre_y_4_2 = [i8_3 * sqrt_5_2pi * val[0] * (7.0 * val[1] - 1.0) \
for val in zip(self._pow_sin_t[2], self._pow_cos_t[2])]
pre_y_4_1 = [i8_3 * sqrt_5_pi * val[0] * (7.0 * val[1] - 3.0 * val[2]) \
for val in zip(self._pow_sin_t[1], self._pow_cos_t[3], \
self._pow_cos_t[1])]
acc = 0.0
# Y_4_-4
real = imag = 0.0
for i in nnn_range:
real += pre_y_4_4[i] * self._cos_n_p[4][i]
imag -= pre_y_4_4[i] * self._sin_n_p[4][i]
acc += (real * real + imag * imag)
# Y_4_-3
real = imag = 0.0
for i in nnn_range:
real += pre_y_4_3[i] * self._cos_n_p[3][i]
imag -= pre_y_4_3[i] * self._sin_n_p[3][i]
acc += (real * real + imag * imag)
# Y_4_-2
real = imag = 0.0
for i in nnn_range:
real += pre_y_4_2[i] * self._cos_n_p[2][i]
imag -= pre_y_4_2[i] * self._sin_n_p[2][i]
acc += (real * real + imag * imag)
# Y_4_-1
real = imag = 0.0
for i in nnn_range:
real += pre_y_4_1[i] * self._cos_n_p[1][i]
imag -= pre_y_4_1[i] * self._sin_n_p[1][i]
acc += (real * real + imag * imag)
# Y_4_0
real = imag = 0.0
for i in nnn_range:
real += i16_3 * sqrt_1_pi * (35.0 * self._pow_cos_t[4][i] - \
30.0 * self._pow_cos_t[2][i] + 3.0)
acc += (real * real)
# Y_4_1
real = imag = 0.0
for i in nnn_range:
real -= pre_y_4_1[i] * self._cos_n_p[1][i]
imag -= pre_y_4_1[i] * self._sin_n_p[1][i]
acc += (real * real + imag * imag)
# Y_4_2
real = imag = 0.0
for i in nnn_range:
real += pre_y_4_2[i] * self._cos_n_p[2][i]
imag += pre_y_4_2[i] * self._sin_n_p[2][i]
acc += (real * real + imag * imag)
# Y_4_3
real = imag = 0.0
for i in nnn_range:
real -= pre_y_4_3[i] * self._cos_n_p[3][i]
imag -= pre_y_4_3[i] * self._sin_n_p[3][i]
acc += (real * real + imag * imag)
# Y_4_4
real = imag = 0.0
for i in nnn_range:
real += pre_y_4_4[i] * self._cos_n_p[4][i]
imag += pre_y_4_4[i] * self._sin_n_p[4][i]
acc += (real * real + imag * imag)
q4 = math.sqrt(4.0 * pi * acc / (9.0 * float(nnn * nnn)))
return q4
[docs] def get_q6(self, thetas=None, phis=None):
"""
Calculates the value of the bond orientational order parameter of
weight l=6. If the function is called with non-empty lists of
polar and azimuthal angles the corresponding trigonometric terms
are computed afresh. Otherwise, it is expected that the
compute_trigonometric_terms function has been just called.
Args:
thetas ([float]):
polar angles of all neighbors in radians.
phis ([float]):
azimuth angles of all neighbors in radians.
Return:
q6 (float): bond orientational order parameter of weight l=6
corresponding to the input angles thetas and phis.
"""
if thetas is not None and phis is not None:
self.compute_trigonometric_terms(thetas, phis)
nnn = len(self._pow_sin_t[1])
nnn_range = range(nnn)
i64 = 1.0 / 64.0
i32 = 1.0 / 32.0
i32_3 = 3.0 / 32.0
i16 = 1.0 / 16.0
sqrt_3003_pi = math.sqrt(3003.0 / pi)
sqrt_1001_pi = math.sqrt(1001.0 / pi)
sqrt_91_2pi = math.sqrt(91.0 / (2.0 * pi))
sqrt_1365_pi = math.sqrt(1365.0 / pi)
sqrt_273_2pi = math.sqrt(273.0 / (2.0 * pi))
sqrt_13_pi = math.sqrt(13.0 / pi)
pre_y_6_6 = [i64 * sqrt_3003_pi * val for val in self._pow_sin_t[6]]
pre_y_6_5 = [i32_3 * sqrt_1001_pi * val[0] * val[1] \
for val in zip(self._pow_sin_t[5], self._pow_cos_t[1])]
pre_y_6_4 = [i32_3 * sqrt_91_2pi * val[0] * (11.0 * val[1] - 1.0) \
for val in zip(self._pow_sin_t[4], self._pow_cos_t[2])]
pre_y_6_3 = [
i32 * sqrt_1365_pi * val[0] * (11.0 * val[1] - 3.0 * val[2]) \
for val in zip(self._pow_sin_t[3], self._pow_cos_t[3], \
self._pow_cos_t[1])]
pre_y_6_2 = [i64 * sqrt_1365_pi * val[0] * (33.0 * val[1] - \
18.0 * val[2] + 1.0) for val
in zip(self._pow_sin_t[2], \
self._pow_cos_t[4], self._pow_cos_t[2])]
pre_y_6_1 = [i16 * sqrt_273_2pi * val[0] * (33.0 * val[1] - \
30.0 * val[2] + 5.0 * val[
3]) for val in
zip(self._pow_sin_t[1], \
self._pow_cos_t[5], self._pow_cos_t[3],
self._pow_cos_t[1])]
acc = 0.0
# Y_6_-6
real = imag = 0.0
real = 0.0
imag = 0.0
for i in nnn_range:
real += pre_y_6_6[i] * self._cos_n_p[6][i] # cos(x) = cos(-x)
imag -= pre_y_6_6[i] * self._sin_n_p[6][i] # sin(x) = -sin(-x)
acc += (real * real + imag * imag)
# Y_6_-5
real = imag = 0.0
real = 0.0
imag = 0.0
for i in nnn_range:
real += pre_y_6_5[i] * self._cos_n_p[5][i]
imag -= pre_y_6_5[i] * self._sin_n_p[5][i]
acc += (real * real + imag * imag)
# Y_6_-4
real = imag = 0.0
real = 0.0
imag = 0.0
for i in nnn_range:
real += pre_y_6_4[i] * self._cos_n_p[4][i]
imag -= pre_y_6_4[i] * self._sin_n_p[4][i]
acc += (real * real + imag * imag)
# Y_6_-3
real = imag = 0.0
real = 0.0
imag = 0.0
for i in nnn_range:
real += pre_y_6_3[i] * self._cos_n_p[3][i]
imag -= pre_y_6_3[i] * self._sin_n_p[3][i]
acc += (real * real + imag * imag)
# Y_6_-2
real = imag = 0.0
real = 0.0
imag = 0.0
for i in nnn_range:
real += pre_y_6_2[i] * self._cos_n_p[2][i]
imag -= pre_y_6_2[i] * self._sin_n_p[2][i]
acc += (real * real + imag * imag)
# Y_6_-1
real = imag = 0.0
real = 0.0
imag = 0.0
for i in nnn_range:
real += pre_y_6_1[i] * self._cos_n_p[1][i]
imag -= pre_y_6_1[i] * self._sin_n_p[1][i]
acc += (real * real + imag * imag)
# Y_6_0
real = imag = 0.0
real = 0.0
imag = 0.0
for i in nnn_range:
real += i32 * sqrt_13_pi * (231.0 * self._pow_cos_t[6][i] - \
315.0 * self._pow_cos_t[4][i] + 105.0 *
self._pow_cos_t[2][i] - 5.0)
acc += (real * real)
# Y_6_1
real = imag = 0.0
real = 0.0
imag = 0.0
for i in nnn_range:
real -= pre_y_6_1[i] * self._cos_n_p[1][i]
imag -= pre_y_6_1[i] * self._sin_n_p[1][i]
acc += (real * real + imag * imag)
# Y_6_2
real = imag = 0.0
real = 0.0
imag = 0.0
for i in nnn_range:
real += pre_y_6_2[i] * self._cos_n_p[2][i]
imag += pre_y_6_2[i] * self._sin_n_p[2][i]
acc += (real * real + imag * imag)
# Y_6_3
real = imag = 0.0
real = 0.0
imag = 0.0
for i in nnn_range:
real -= pre_y_6_3[i] * self._cos_n_p[3][i]
imag -= pre_y_6_3[i] * self._sin_n_p[3][i]
acc += (real * real + imag * imag)
# Y_6_4
real = imag = 0.0
real = 0.0
imag = 0.0
for i in nnn_range:
real += pre_y_6_4[i] * self._cos_n_p[4][i]
imag += pre_y_6_4[i] * self._sin_n_p[4][i]
acc += (real * real + imag * imag)
# Y_6_5
real = imag = 0.0
real = 0.0
imag = 0.0
for i in nnn_range:
real -= pre_y_6_5[i] * self._cos_n_p[5][i]
imag -= pre_y_6_5[i] * self._sin_n_p[5][i]
acc += (real * real + imag * imag)
# Y_6_6
real = imag = 0.0
real = 0.0
imag = 0.0
for i in nnn_range:
real += pre_y_6_6[i] * self._cos_n_p[6][i]
imag += pre_y_6_6[i] * self._sin_n_p[6][i]
acc += (real * real + imag * imag)
q6 = math.sqrt(4.0 * pi * acc / (13.0 * float(nnn * nnn)))
return q6
[docs] def get_type(self, index):
"""
Return type of order-parameter at the index provided and
represented by a short string.
Args:
index (int):
index of order-parameter for which type is to be returned
"""
if index < 0 or index >= len(self._types):
raise ValueError("Index for getting order-parameter type"
" out-of-bounds!")
return self._types[index]
[docs] def get_parameters(self, index):
"""
Returns list of floats that represents
the parameters associated with calculation of the order
parameter that was defined at the index provided.
Attention: the parameters do not need to equal those originally
inputted because of processing out of efficiency reasons.
Args:
index (int):
index of order-parameter for which associated parameters
are to be returned
"""
if index < 0 or index >= len(self._types):
raise ValueError("Index for getting parameters associated with"
" order-parameter calculation out-of-bounds!")
return self._paras[index]
[docs] def get_order_parameters(self, structure, n, indeces_neighs=None, \
tol=0.0, target_spec=None):
"""
Compute all order parameters of site n.
Args:
structure (Structure):
input structure.
n (int):
index of site in input structure, for which OPs are to be
calculated. Note that we do not use the sites iterator
here, but directly access sites via struct[index].
indeces_neighs ([int]):
list of indeces of those neighbors in Structure object
structure that are to be considered for OP computation.
This optional argument overwrites the way neighbors are
to be determined as defined in the constructor (i.e.,
Voronoi coordination finder via negative cutoff radius
vs constant cutoff radius if cutoff was positive).
We do not use information about the underlying
structure lattice if the neighbor indeces are explicitly
provided. This has two important consequences. First,
the input Structure object can, in fact, be a
simple list of Site objects. Second, no nearest images
of neighbors are determined when providing an index list.
Note furthermore that this neighbor
determination type ignores the optional target_spec
argument.
tol (float):
threshold of weight (= solid angle / maximal solid angle)
to determine if a particular pair is
considered neighbors; this is relevant only in the case
when Voronoi polyhedra are used to determine coordination
target_spec (Specie):
target specie to be considered when calculating the order
parameters of site n; None includes all species of input
structure.
Returns:
list of floats representing order parameters. Should it not be
possible to compute a given OP for a conceptual reason, the
corresponding entry is None instead of a float. For Steinhardt
et al.'s bond orientational OPs and the other geometric OPs
("tet", "oct", "bcc"), this can happen if there is a single
neighbor around site n in the structure because that, obviously,
does not permit calculation of angles between multiple
neighbors.
"""
# Do error-checking and initialization.
if n < 0:
raise ValueError("Site index smaller zero!")
if n >= len(structure):
raise ValueError("Site index beyond maximum!")
if indeces_neighs is not None:
for index in indeces_neighs:
if index >= len(structure):
raise ValueError("Neighbor site index beyond maximum!")
if tol < 0.0:
raise ValueError("Negative tolerance for weighted solid angle!")
left_of_unity = 1.0 - 1.0e-12
# The following threshold has to be adapted to non-Angstrom units.
very_small = 1.0e-12
# Find central site and its neighbors.
# Note that we adopt the same way of accessing sites here as in
# VoronoiCoordFinder; that is, not via the sites iterator.
centsite = structure[n]
if indeces_neighs is not None:
neighsites = [structure[index] for index in indeces_neighs]
elif self._voroneigh:
vorocf = VoronoiCoordFinder(structure)
neighsites = vorocf.get_coordinated_sites(n, tol, target_spec)
else:
# Structure.get_sites_in_sphere --> also other periodic images
neighsitestmp = [i[0] for i in structure.get_sites_in_sphere(
centsite.coords, self._cutoff)]
neighsites = []
if centsite not in neighsitestmp:
raise ValueError("Could not find center site!")
else:
neighsitestmp.remove(centsite)
if target_spec is None:
neighsites = list(neighsitestmp)
else:
neighsites[:] = [site for site in neighsitestmp \
if site.specie.symbol == target_spec]
nneigh = len(neighsites)
self._last_nneigh = nneigh
# Prepare angle calculations, if applicable.
rij = []
rjk = []
rijnorm = []
rjknorm = []
dist = []
distjk_unique = []
distjk = []
centvec = centsite.coords
if self._computerijs:
for j, neigh in enumerate(neighsites):
rij.append((neigh.coords - centvec))
dist.append(np.linalg.norm(rij[j]))
rijnorm.append((rij[j] / dist[j]))
if self._computerjks:
for j, neigh in enumerate(neighsites):
rjk.append([])
rjknorm.append([])
distjk.append([])
kk = 0
for k in range(len(neighsites)):
if j != k:
rjk[j].append(neighsites[k].coords - neigh.coords)
distjk[j].append(np.linalg.norm(rjk[j][kk]))
if k > j:
distjk_unique.append(distjk[j][kk])
rjknorm[j].append(rjk[j][kk] / distjk[j][kk])
kk = kk + 1
# Initialize OP list and, then, calculate OPs.
ops = [0.0 for t in self._types]
# First, coordination number-based OPs.
for i, t in enumerate(self._types):
if t == "cn":
ops[i] = nneigh / self._paras[i][0]
# Then, bond orientational OPs based on spherical harmonics
# according to Steinhardt et al., Phys. Rev. B, 28, 784-805, 1983.
if self._boops:
thetas = []
phis = []
for j, vec in enumerate(rijnorm):
# z is North pole --> theta between vec and (0, 0, 1)^T.
# Because vec is normalized, dot product is simply vec[2].
thetas.append(math.acos(max(-1.0, min(vec[2], 1.0))))
tmpphi = 0.0
# Compute phi only if it is not (almost) perfectly
# aligned with z-axis.
if vec[2] < left_of_unity and vec[2] > - (left_of_unity):
# x is prime meridian --> phi between projection of vec
# into x-y plane and (1, 0, 0)^T
tmpphi = math.acos(max(
-1.0,
min(vec[0] / (math.sqrt(
vec[0] * vec[0] + vec[1] * vec[1])),
1.0)))
if vec[1] < 0.0:
tmpphi = -tmpphi
phis.append(tmpphi)
# Note that None flags that we have too few neighbors
# for calculating BOOPS.
for i, t in enumerate(self._types):
if t == "q2":
ops[i] = self.get_q2(thetas, phis) if len(
thetas) > 0 else None
elif t == "q4":
ops[i] = self.get_q4(thetas, phis) if len(
thetas) > 0 else None
elif t == "q6":
ops[i] = self.get_q6(thetas, phis) if len(
thetas) > 0 else None
# Then, deal with the Peters-style OPs that are tailor-made
# to recognize common structural motifs
# (Peters, J. Chem. Phys., 131, 244103, 2009;
# Zimmermann et al., J. Am. Chem. Soc., under revision, 2015).
if self._geomops:
gaussthetak = [0.0 for t in self._types] # not used by all OPs
qsptheta = [[] for t in self._types] # not used by all OPs
ipi = 1.0 / pi
piover2 = pi / 2.0
tetangoverpi = math.acos(-1.0 / 3.0) * ipi
itetangminuspihalfoverpi = 1.0 / (tetangoverpi - 0.5)
for j in range(nneigh): # Neighbor j is put to the North pole.
zaxis = rijnorm[j]
for i, t in enumerate(self._types):
qsptheta[i].append(0.0)
for k in range(nneigh): # From neighbor k, we construct
if j != k: # the prime meridian.
tmp = max(
-1.0, min(np.inner(zaxis, rijnorm[k]), 1.0))
thetak = math.acos(tmp)
xaxistmp = gramschmidt(rijnorm[k], zaxis)
if np.linalg.norm(xaxistmp) < very_small:
flag_xaxis = True
else:
xaxis = xaxistmp / np.linalg.norm(xaxistmp)
flag_xaxis = False
# Contributions of j-i-k angles, where i represents the central atom
# and j and k two of the neighbors.
for i, t in enumerate(self._types):
if t == "lin":
tmp = self._paras[i][0] * (thetak * ipi - 1.0)
ops[i] += exp(-0.5 * tmp * tmp)
elif t == "bent":
tmp = self._paras[i][1] * (
thetak * ipi - self._paras[i][0])
ops[i] += exp(-0.5 * tmp * tmp)
elif t == "tet" or t == "tet_legacy":
tmp = self._paras[i][0] * (
thetak * ipi - tetangoverpi)
gaussthetak[i] = math.exp(-0.5 * tmp * tmp)
elif t == "oct" or t == "oct_legacy":
if thetak >= self._paras[i][0]:
# k is south pole to j
tmp = self._paras[i][1] * (
thetak * ipi - 1.0)
ops[i] += 3.0 * math.exp(-0.5 * tmp * tmp)
elif t == "bcc" and j < k:
if thetak >= self._paras[i][0]:
# k is south pole to j
tmp = self._paras[i][1] * (
thetak * ipi - 1.0)
ops[i] += 6.0 * math.exp(-0.5 * tmp * tmp)
elif t == "sq_pyr":
tmp = self._paras[i][0] * (thetak * ipi - 0.5)
qsptheta[i][j] = qsptheta[i][j] + exp(-0.5 * tmp * tmp)
elif t == "tri_bipyr":
if thetak >= self._paras[i][0]:
tmp = self._paras[i][1] * (
thetak * ipi - 1.0)
qsptheta[i][j] = 2.0 * math.exp(-0.5 * tmp * tmp)
for m in range(nneigh):
if (m != j) and (m != k) and (not flag_xaxis):
tmp = max(
-1.0, min(np.inner(zaxis, rijnorm[m]), 1.0))
thetam = math.acos(tmp)
xtwoaxistmp = gramschmidt(rijnorm[m], zaxis)
l = np.linalg.norm(xtwoaxistmp)
if l < very_small:
flag_xtwoaxis = True
else:
xtwoaxis = xtwoaxistmp / l
phi = math.acos(max(
-1.0,
min(np.inner(xtwoaxis, xaxis), 1.0)))
flag_xtwoaxis = False
# Contributions of j-i-m angle and
# angles between plane j-i-k and i-m vector.
if not flag_xaxis and not flag_xtwoaxis:
for i, t in enumerate(self._types):
if t == "tet_legacy":
tmp = self._paras[i][0] * (
thetam * ipi - tetangoverpi)
ops[i] += gaussthetak[i] * math.exp(
-0.5 * tmp * tmp) * math.cos(
3.0 * phi)
elif t == "tet":
tmp = self._paras[i][0] * (
thetam * ipi - tetangoverpi)
tmp2 = math.cos(1.5 * phi)
ops[i] += gaussthetak[i] * math.exp(
-0.5 * tmp * tmp) * tmp2 * tmp2
elif t == "oct_legacy":
if thetak < self._paras[i][0] and \
thetam < \
self._paras[i][0]:
tmp = math.cos(2.0 * phi)
tmp2 = self._paras[i][2] * (
thetam * ipi - 0.5)
ops[i] += tmp * tmp * \
self._paras[i][4] * (
math.exp(
-0.5 * tmp2 * tmp2) - \
self._paras[i][3])
elif t == "oct":
if thetak < self._paras[i][0] and \
thetam < self._paras[i][0]:
tmp = math.cos(2.0 * phi)
tmp2 = self._paras[i][2] * (
thetam * ipi - 0.5)
ops[i] += tmp * tmp * \
math.exp(
-0.5 * tmp2 * tmp2)
elif t == "bcc" and j < k:
if thetak < self._paras[i][0]:
if thetak > piover2:
fac = 1.0
else:
fac = -1.0
tmp = (thetam - piover2) / (
19.47 * pi / 180.0)
ops[i] += fac * math.cos(
3.0 * phi) * \
1.6 * tmp * \
math.exp(
-0.5 * tmp * tmp)
elif t == "tri_bipyr":
if thetak < self._paras[i][0] and \
thetam < self._paras[i][0]:
tmp = math.cos(1.5 * phi)
tmp2 = self._paras[i][2] * (
thetam * ipi - 0.5)
qsptheta[i][j] += \
tmp * tmp * math.exp( \
-0.5 * tmp2 * tmp2)
# Normalize Peters-style OPs.
for i, t in enumerate(self._types):
if t == "lin":
ops[i] = ops[i] / float(nneigh * (
nneigh - 1)) if nneigh > 1 else None
elif t == "bent":
ops[i] = ops[i] / float(nneigh * (
nneigh - 1)) if nneigh > 1 else None
elif t == "tet" or t == "tet_legacy":
ops[i] = ops[i] / float(nneigh * (nneigh - 1) * (
nneigh - 2)) if nneigh > 2 else None
elif t == "oct" or t == "oct_legacy":
ops[i] = ops[i] / float(nneigh * (3 + (nneigh - 2) * (
nneigh - 3))) if nneigh > 3 else None
elif t == "bcc":
ops[i] = ops[i] / float(0.5 * float(
nneigh * (6 + (nneigh - 2) * (nneigh - 3)))) \
if nneigh > 3 else None
elif t == "sq_pyr":
if nneigh > 1:
dmean = np.mean(dist)
acc = 0.0
for d in dist:
tmp = self._paras[i][1] * (d - dmean)
acc = acc + exp(-0.5 * tmp * tmp)
ops[i] = acc * max(qsptheta[i]) / float(
nneigh * (nneigh - 1))
else:
ops[i] = None
elif t == "tri_bipyr":
ops[i] = max(qsptheta[i]) / float(
2 + (nneigh - 2) * (nneigh - 3)) if nneigh > 3 \
else None
# Then, deal with the new-style OPs that require vectors between
# neighbors.
if self._geomops2:
# Compute all (unique) angles and sort the resulting list.
aij = []
for ir, r in enumerate(rijnorm):
for j in range(ir+1, len(rijnorm)):
aij.append(math.acos(max(-1.0, min(np.inner(
r, rijnorm[j]), 1.0))))
aijs = sorted(aij)
# Compute height, side and diagonal length estimates.
neighscent = np.array([0.0, 0.0, 0.0])
for j, neigh in enumerate(neighsites):
neighscent = neighscent + neigh.coords
if nneigh > 0:
neighscent = (neighscent / float(nneigh))
h = np.linalg.norm(neighscent - centvec)
b = min(distjk_unique) if len(distjk_unique) > 0 else 0
dhalf = max(distjk_unique) / 2.0 if len(distjk_unique) > 0 else 0
for i, t in enumerate(self._types):
if t == "reg_tri" or t == "sq":
if nneigh < 3:
ops[i] = None
else:
ops[i] = 1.0
if t == "reg_tri":
a = 2.0 * asin(b / (2.0 * sqrt(h*h + (b / (
2.0 * cos(3.0 * pi / 18.0)))**2.0)))
nmax = 3
else:
a = 2.0 * asin(b / (2.0 * sqrt(h*h + dhalf*dhalf)))
nmax = 4
for j in range(min([nneigh,nmax])):
ops[i] = ops[i] * exp(-0.5 * ((
aijs[j] - a) * self._paras[i][0])**2)
return ops