"""
Module for storing & accessing symmetry group information, including
- Group class
- Wyckoff_Position class.
- Hall class
"""
# Imports ------------------------------
# Standard Libraries
import numpy as np
from pkg_resources import resource_filename as rf
from copy import deepcopy
import random
import itertools
import re
# External Libraries
from pymatgen.symmetry.analyzer import generate_full_symmops
from pandas import read_csv
from monty.serialization import loadfn
# PyXtal imports
from pyxtal.msg import printx
from pyxtal.operations import (
SymmOp,
apply_ops,
filtered_coords,
filtered_coords_euclidean,
distance,
distance_matrix,
create_matrix,
OperationAnalyzer,
check_images,
)
from pyxtal.constants import letters
# ------------------------------ Constants ---------------------------------------
wyckoff_df = read_csv(rf("pyxtal", "database/wyckoff_list.csv"))
wyckoff_symmetry_df = read_csv(rf("pyxtal", "database/wyckoff_symmetry.csv"))
wyckoff_generators_df = read_csv(rf("pyxtal", "database/wyckoff_generators.csv"))
layer_df = read_csv(rf("pyxtal", "database/layer.csv"))
layer_symmetry_df = read_csv(rf("pyxtal", "database/layer_symmetry.csv"))
layer_generators_df = read_csv(rf("pyxtal", "database/layer_generators.csv"))
rod_df = read_csv(rf("pyxtal", "database/rod.csv"))
rod_symmetry_df = read_csv(rf("pyxtal", "database/rod_symmetry.csv"))
rod_generators_df = read_csv(rf("pyxtal", "database/rod_generators.csv"))
point_df = read_csv(rf("pyxtal", "database/point.csv"))
point_symmetry_df = read_csv(rf("pyxtal", "database/point_symmetry.csv"))
point_generators_df = read_csv(rf("pyxtal", "database/point_generators.csv"))
symbols = loadfn(rf("pyxtal", "database/symbols.json"))
t_subgroup = loadfn(rf("pyxtal",'database/t_subgroup.json'))
k_subgroup = loadfn(rf("pyxtal",'database/k_subgroup.json'))
wyc_sets = loadfn(rf("pyxtal",'database/wyckoff_sets.json'))
hex_cell = np.array([[1, -0.5, 0], [0, np.sqrt(3) / 2, 0], [0, 0, 1]])
hall_table = read_csv(rf("pyxtal", "database/HM_Full.csv"), sep=',')
#The map between spglib default space group and hall numbers
spglib_hall_numbers = [
1, 2, 3, 6, 9, 18, 21, 30, 39, 57, 60, 63, 72, 81, 90,
108, 109, 112, 115, 116, 119, 122, 123, 124, 125, 128, 134, 137, 143, 149,
155, 161, 164, 170, 173, 176, 182, 185, 191, 197, 203, 209, 212, 215, 218,
221, 227, 228, 230, 233, 239, 245, 251, 257, 263, 266, 269, 275, 278, 284,
290, 292, 298, 304, 310, 313, 316, 322, 334, 335, 337, 338, 341, 343, 349,
350, 351, 352, 353, 354, 355, 356, 357, 358, 359, 361, 363, 364, 366, 367,
368, 369, 370, 371, 372, 373, 374, 375, 376, 377, 378, 379, 380, 381, 382,
383, 384, 385, 386, 387, 388, 389, 390, 391, 392, 393, 394, 395, 396, 397,
398, 399, 400, 401, 402, 404, 406, 407, 408, 410, 412, 413, 414, 416, 418,
419, 420, 422, 424, 425, 426, 428, 430, 431, 432, 433, 435, 436, 438, 439,
440, 441, 442, 443, 444, 446, 447, 448, 449, 450, 452, 454, 455, 456, 457,
458, 460, 462, 463, 464, 465, 466, 467, 468, 469, 470, 471, 472, 473, 474,
475, 476, 477, 478, 479, 480, 481, 482, 483, 484, 485, 486, 487, 488, 489,
490, 491, 492, 493, 494, 495, 497, 498, 500, 501, 502, 503, 504, 505, 506,
507, 508, 509, 510, 511, 512, 513, 514, 515, 516, 517, 518, 520, 521, 523,
524, 525, 527, 529, 530]
#The map between standard space group and hall numbers
pyxtal_hall_numbers = [
1, 2, 3, 6, 9, 18, 21, 30, 39, 57, 60, 63, 72, 81, 90,
108, 109, 112, 115, 116, 119, 122, 123, 124, 125, 128, 134, 137, 143, 149,
155, 161, 164, 170, 173, 176, 182, 185, 191, 197, 203, 209, 212, 215, 218,
221, 227, 229, 230, 234, 239, 245, 251, 257, 263, 266, 269, 275, 279, 284,
290, 292, 298, 304, 310, 313, 316, 323, 334, 336, 337, 338, 341, 343, 349,
350, 351, 352, 353, 354, 355, 356, 357, 358, 360, 362, 363, 365, 366, 367,
368, 369, 370, 371, 372, 373, 374, 375, 376, 377, 378, 379, 380, 381, 382,
383, 384, 385, 386, 387, 388, 389, 390, 391, 392, 393, 394, 395, 396, 397,
398, 399, 400, 401, 403, 405, 406, 407, 409, 411, 412, 413, 415, 417, 418,
419, 421, 423, 424, 425, 427, 429, 430, 431, 432, 433, 435, 436, 438, 439,
440, 441, 442, 443, 444, 446, 447, 448, 449, 450, 452, 454, 455, 456, 457,
458, 460, 462, 463, 464, 465, 466, 467, 468, 469, 470, 471, 472, 473, 474,
475, 476, 477, 478, 479, 480, 481, 482, 483, 484, 485, 486, 487, 488, 489,
490, 491, 492, 493, 494, 496, 497, 499, 500, 501, 502, 503, 504, 505, 506,
507, 508, 509, 510, 511, 512, 513, 514, 515, 516, 517, 519, 520, 522, 523,
524, 526, 528, 529, 530,
]
# --------------------------- Hall class -----------------------------
[docs]
class Hall:
"""
Class for conversion between Hall and standard spacegroups
http://cci.lbl.gov/sginfo/itvb_2001_table_a1427_hall_symbols.html
Args:
spg_num: interger number between 1 and 230
style: spglib or pyxtal
permutation: allow permutation or not
"""
def __init__(self, spgnum, style='pyxtal', permutation=False):
self.spg = spgnum
if style == 'pyxtal':
self.hall_default = pyxtal_hall_numbers[spgnum-1]
else:
self.hall_default = spglib_hall_numbers[spgnum-1]
self.hall_numbers = []
self.hall_symbols = []
self.Ps = [] #convertion from standard
self.P1s = [] #inverse convertion to standard
for id in range(len(hall_table['Hall'])):
if hall_table['Spg_num'][id] == spgnum:
if permutation:
include = True
else:
if hall_table['Permutation'][id]==0:
include = True
else:
include = False
if include:
self.hall_numbers.append(hall_table['Hall'][id])
self.hall_symbols.append(hall_table['Symbol'][id])
self.Ps.append(abc2matrix(hall_table['P'][id]))
self.P1s.append(abc2matrix(hall_table['P^-1'][id]))
elif hall_table['Spg_num'][id] > spgnum:
break
if len(self.hall_numbers) == 0:
msg = "hall numbers cannot be found, check input " + spgnum
raise RuntimeError(msg)
# --------------------------- Group class -----------------------------
[docs]
class Group:
"""
Class for storing a set of Wyckoff positions for a symmetry group.
See the documentation for details about settings.
Examples
--------
>>> from pyxtal.symmetry import Group
>>> g = Group(64)
>>> g
-- Spacegroup --# 64 (Cmce)--
16g site symm: 1
8f site symm: m..
8e site symm: .2.
8d site symm: 2..
8c site symm: -1
4b site symm: 2/m..
4a site symm: 2/m..
one can access data such as `symbol`, `number` and `Wyckoff_positions`
>>> g.symbol
'Cmce'
>>> g.number
64
>>> g.Wyckoff_positions[0]
Wyckoff position 16g in space group 64 with site symmetry 1
x, y, z
-x, -y+1/2, z+1/2
-x, y+1/2, -z+1/2
x, -y, -z
-x, -y, -z
x, y+1/2, -z+1/2
x, -y+1/2, z+1/2
-x, y, z
x+1/2, y+1/2, z
-x+1/2, -y+1, z+1/2
-x+1/2, y+1, -z+1/2
x+1/2, -y+1/2, -z
-x+1/2, -y+1/2, -z
x+1/2, y+1, -z+1/2
x+1/2, -y+1, z+1/2
-x+1/2, y+1/2, z
We also provide several utilities functions, e.g.,
one can search the possible wyckoff_combinations by a formula
>>> g.list_wyckoff_combinations([4, 2])
([], [], [])
>>> g.list_wyckoff_combinations([4, 8])
([[['4a'], ['8c']],
[['4a'], ['8d']],
[['4a'], ['8e']],
[['4a'], ['8f']],
[['4b'], ['8c']],
[['4b'], ['8d']],
[['4b'], ['8e']],
[['4b'], ['8f']]],
[False, True, True, True, False, True, True, True],
[[[6], [4]], [[6], [3]], [[6], [2]], [[6], [1]],
[[5], [4]], [[5], [3]], [[5], [2]], [[5], [1]]]
)
or search the subgroup information
>>> g.get_max_t_subgroup()['subgroup']
[12, 14, 15, 20, 36, 39, 41]
or check if a given composition is compatible with Wyckoff positions
>>> g = Group(225)
>>> g.check_compatible([64, 28, 24])
(True, True)
or check the possible transition paths to a given supergroup
>>> g = Group(59)
>>> g.search_supergroup_paths(139, 2)
[[71, 139], [129, 139], [137, 139]]
Args:
group: the group symbol or international number
dim (defult: 3): the periodic dimension of the group
use_hall (default: False): whether or not use the hall number
style (default: `pyxtal`): the choice of hall number ('pyxtal'/'spglib')
quick (defaut: False): whether or not ignore the wyckoff information
"""
def __init__(self, group, dim=3, use_hall=False, style='pyxtal', quick=False):
self.string = None
self.dim = dim
names = ['Point', 'Rod', 'Layer', 'Space']
self.header = "-- " + names[dim] + 'group --'
if not use_hall:
self.symbol, self.number = get_symbol_and_number(group, dim)
else:
self.symbol = hall_table['Symbol'][group-1]
self.number = hall_table['Spg_num'][group-1]
self.PBC, self.lattice_type = get_pbc_and_lattice(self.number, dim)
if dim == 3:
results = get_point_group(self.number)
self.point_group = results[0]
self.pg_number = results[1]
self.polar = results[2]
self.inversion = results[3]
self.chiral = results[4]
if not quick:
if dim == 3:
if not use_hall:
if style == 'pyxtal':
self.hall_number = pyxtal_hall_numbers[self.number-1]
else:
self.hall_number = spglib_hall_numbers[self.number-1]
else:
self.hall_number = group
self.P = abc2matrix(hall_table['P'][self.hall_number-1])
self.P1 = abc2matrix(hall_table['P^-1'][self.hall_number-1])
else:
self.hall_number = None
self.P = None
self.P1 = None
# Wyckoff positions, site_symmetry, generator
self.wyckoffs = get_wyckoffs(self.number, dim=dim)
self.w_symm = get_wyckoff_symmetry(self.number, dim=dim)
wpdicts = [
{
"index": i,
"letter": letter_from_index(i, self.wyckoffs, dim=self.dim),
"ops": self.wyckoffs[i],
"multiplicity": len(self.wyckoffs[i]),
"symmetry": self.w_symm[i],
"PBC": self.PBC,
"dim": self.dim,
"number": self.number,
"symbol": self.symbol,
"P": self.P,
"P1": self.P1,
"hall_number": self.hall_number,
}
for i in range(len(self.wyckoffs))
]
# A list of Wyckoff_positions sorted by descending multiplicity
self.Wyckoff_positions = []
for wpdict in wpdicts:
wp = Wyckoff_position.from_dict(wpdict)
self.Wyckoff_positions.append(wp)
# A 2D list of WP objects, grouped and sorted by multiplicity
self.wyckoffs_organized = organized_wyckoffs(self)
def __str__(self):
if self.string is not None:
return self.string
else:
s = self.header
s += "# " + str(self.number) + " (" + self.symbol + ")--"
for wp in self.Wyckoff_positions:
s += "\n" + wp.get_label()
if not hasattr(wp, "site_symm"): wp.get_site_symmetry()
s += "\tsite symm: " + wp.site_symm
self.string = s
return self.string
def __repr__(self):
return str(self)
def __iter__(self):
yield from self.Wyckoff_positions
def __getitem__(self, index):
return self.get_wyckoff_position(index)
def __len__(self):
return len(self.wyckoffs)
[docs]
def get_ferroelectric_groups(self):
"""
return the list of possible ferroelectric point groups
"""
return para2ferro(self.point_group)
[docs]
def get_site_dof(self, sites):
"""
compute the degree of freedom for each site
Args:
sites: list, e.g. ['4a', '8b'] or ['a', 'b']
Returns:
True or False
"""
dof = np.zeros(len(sites))
for i, site in enumerate(sites):
if len(site) > 1:
site = site[-1]
id = len(self) - letters.index(site) - 1
string = self[id].ops[0].as_xyz_str()
dof[i] = len(set(re.sub('[^a-z]+', '', string)))
return dof
[docs]
def get_lattice_dof(self):
"""
compute the degree of freedom for the lattice
"""
if self.lattice_type in ["triclinic"]:
dof = 6
elif self.lattice_type in ["monoclinic"]:
dof = 4
elif self.lattice_type in ['orthorhombic']:
dof = 3
elif self.lattice_type in ['tetragonal', 'hexagonal', 'trigonal']:
dof = 2
else:
dof = 1
return dof
[docs]
def is_valid_combination(self, sites):
"""
check if the solutions are valid. A special WP with zero freedom (0,0,0)
cannot be occupied twice.
Args:
sites: list, e.g. ['4a', '8b'] or ['a', 'b']
Returns:
True or False
"""
# remove the multiplicity:
for i, site in enumerate(sites):
if len(site) > 1:
sites[i] = site[-1]
for wp in self:
letter = wp.letter
if sites.count(letter)>1:
freedom = np.trace(wp.ops[0].rotation_matrix) > 0
if not freedom:
return False
return True
[docs]
def list_wyckoff_combinations(self, numIons, quick=False, numWp=(None, None), Nmax=10000000):
"""
List all possible wyckoff combinations for the given formula. Note this
is really designed for a light weight calculation. If the solution space
is big, set quick as True.
Args:
numIons (list): [12, 8]
quick ()Boolean): quickly generate some solutions
numWp (tuple): (min_wp, max_wp)
Nmax: maximumly allowed combinations
Returns:
Combinations: list of possible sites
has_freedom: list of boolean numbers
indices: list of wp indices
"""
numIons = np.array(numIons)
(min_wp, max_wp) = numWp
# Must be greater than the number of smallest wp multiplicity
if numIons.min() < self[-1].multiplicity:
return [], [], []
elif max_wp is not None and sum(numIons) > self[0].multiplicity*max_wp:
return [], [], []
basis = [] # [8, 4, 4]
letters = [] # ['c', 'b', 'a']
freedoms = [] # [False, False, False]
ids = [] # [2, 3, 4]
# obtain the basis
for i, wp in enumerate(self):
mul = wp.multiplicity
letter = wp.letter
freedom = np.trace(wp.ops[0].rotation_matrix) > 0
if mul <= max(numIons):
if quick:
if mul in basis and freedom:
pass
#elif mul in basis and basis.count(mul) >= 3:
# pass
else:
basis.append(mul)
letters.append(letter)
freedoms.append(freedom)
else:
basis.append(mul)
letters.append(letter)
freedoms.append(freedom)
ids.append(i)
basis = np.array(basis)
# quickly exit
if np.min(numIons) < np.min(basis):
#print(numIons, basis)
return [], []
# odd and even
elif np.mod(numIons, 2).sum()>0 and np.mod(basis, 2).sum()==0:
#print("odd-even", numIons, basis)
#return None, False
return [], [], []
#print("basis", basis)
#print("numIons", numIons)
# obtain the maximum numbers for each basis
# reset the maximum to 1 if there is no freedom
# find the integer solutions
# reset solutions according to max_wp
max_solutions = np.floor(numIons[:, None]/basis)
for i in range(len(freedoms)):
if not freedoms[i]:
max_solutions[:, i] = 1
if max_wp is not None:
N_max = max_wp - (len(numIons) - 1)
max_solutions[max_solutions > N_max] = N_max
list_solutions = []
for i, numIon in enumerate(numIons):
lists = []
prod = 1
for a in max_solutions[i]:
if prod <= Nmax: #10000000:
d = int(a) + 1
lists.append(list(range(d)))
prod *= d
else:
# If the size is too big, we terminate it asap
lists.append([0])
# Terminate the list
# break
#print(len(lists), prod)
sub_solutions = np.array(list(itertools.product(*lists)))
N = sub_solutions.dot(basis)
sub_solutions = sub_solutions[N == numIon]
list_solutions.append(sub_solutions.tolist())
#print(i)
#print(sub_solutions)#; import sys; sys.exit()
if len(sub_solutions) == 0:
return [], [], []
# Gather all solutions and remove very large number solutions
solutions = np.array(list(itertools.product(*list_solutions)))
dim1 = solutions.shape[0]
dim2 = np.prod(solutions.shape[1:])
solutions = solutions.reshape([dim1, dim2])
if max_wp is not None:
solutions_total = solutions.sum(axis=1)
solutions = solutions[solutions_total <= max_wp]
if min_wp is not None:
solutions_total = solutions.sum(axis=1)
solutions = solutions[solutions_total >= min_wp]
# convert the results to list
combinations = []
has_freedom = []
indices = []
for solution in solutions:
res = solution.reshape([len(numIons), len(basis)])
_com = []
_free = []
_ids = []
# QZ: check what's going on
for i, numIon in enumerate(numIons):
tmp = []
bad_resolution = False
frozen = []
ids_in = []
for j, b in enumerate(basis):
if not freedoms[j] and (res[:, j]).sum() > 1:
bad_resolution = True
break
else:
if res[i, j] > 0:
symbols = [str(b) + letters[j]] * res[i, j]
tmp.extend(symbols)
frozen.extend([freedoms[j]])
ids_in.extend([ids[j]] * res[i, j])
if not bad_resolution:
_com.append(tmp)
_free.extend(frozen)
_ids.append(ids_in)
if len(_com) == len(numIons):
combinations.append(_com)
indices.append(_ids)
if True in _free:
has_freedom.append(True)
else:
has_freedom.append(False)
return combinations, has_freedom, indices
[docs]
def get_wyckoff_position(self, index):
"""
Returns a single Wyckoff_position object.
Args:
index: the index of the Wyckoff position within the group.
Returns: a Wyckoff_position object
"""
if type(index) == str:
# Extract letter from number-letter combinations ("4d"->"d")
for c in index:
if c.isalpha():
letter = c
break
index = index_from_letter(letter, self.wyckoffs, dim=self.dim)
return self.Wyckoff_positions[index]
[docs]
def get_wyckoff_position_from_xyz(self, xyz, decimals=4):
"""
Returns a single Wyckoff_position object.
Args:
xyz: a trial [x, y, z] coordinate
Returns: a Wyckoff_position object
"""
xyz = np.round(np.array(xyz, dtype=float), decimals=decimals)
xyz -= np.floor(xyz)
for wp in self.Wyckoff_positions:
pos = wp.apply_ops(xyz)
pos -= np.floor(pos)
is_present = np.any(np.all(pos == xyz, axis=1))
if is_present:
if len(pos) == len(np.unique(pos, axis=0)):
return wp
print("Cannot find the suitable wp for the given input")
return None
[docs]
def get_alternatives(self):
"""
Get the alternative settings as a dictionary
"""
if self.dim == 3:
return wyc_sets[str(self.number)]
else:
msg = "Only supports the subgroups for space group"
raise NotImplementedError(msg)
[docs]
def get_max_k_subgroup(self):
"""
Returns the maximal k-subgroups as a dictionary
"""
if self.dim == 3:
return k_subgroup[str(self.number)]
else:
msg = "Only supports the subgroups for space group"
raise NotImplementedError(msg)
[docs]
def get_max_t_subgroup(self):
"""
Returns the maximal t-subgroups as a dictionary
"""
if self.dim == 3:
return t_subgroup[str(self.number)]
else:
msg = "Only supports the subgroups for space group"
raise NotImplementedError(msg)
[docs]
def get_max_subgroup(self, H):
"""
Returns the dicts for both t and k subgroup, H is just track the type
QZ: the function name is not instructive, need to revise later
"""
if self.point_group == Group(H, quick=True).point_group:
g_type = 'k'
dicts = self.get_max_k_subgroup()
else:
g_type = 't'
dicts = self.get_max_t_subgroup()
return dicts, g_type
[docs]
def get_wp_list(self, reverse=False):
"""
Get the reversed list of wps
"""
#wp_list = [(str(x.multiplicity)+x.letter) for x in self.Wyckoff_positions]
wp_list = [(x.get_label()) for x in self.Wyckoff_positions]
if reverse: wp_list.reverse()
return wp_list
[docs]
def get_splitters_from_structure(self, struc, group_type='t'):
"""
Get the valid symmetry relations for a structure to its supergroup
e.g.,
Args:
- struc: pyxtal structure
- group_type: `t` or `k`
Returns:
list of valid transitions [(id, (['4a'], ['4b'], [['4a'], ['4c']])]
"""
if group_type=='t':
dicts = self.get_max_t_subgroup()
else:
dicts = self.get_max_k_subgroup()
# search for the compatible solutions
solutions = []
for i, sub in enumerate(dicts['subgroup']):
if sub == struc.group.number:
# extract the relation
relation = dicts['relations'][i]
trans = np.linalg.inv(dicts['transformation'][i][:,:3])
if struc.lattice.check_mismatch(trans, self.lattice_type):
results = self.get_splitters_from_relation(struc, relation)
if results is not None:
sols = list(itertools.product(*results))
trials = self.get_valid_solutions(sols)
solutions.append((i, trials))
return solutions
[docs]
def get_splitters_from_relation(self, struc, relation):
"""
Get the valid symmetry relations for a structure to its supergroup
e.g.,
Args:
- struc: pyxtal structure
- group_type: `t` or `k`
Returns:
list of valid transitions
"""
elements, sites = struc._get_elements_and_sites()
wp_list = self.get_wp_list(reverse=True)
good_splittings_list=[]
# search for all valid compatible relations
# each element is solved one at a time
for site in sites:
# ['4a', '4a', '2b'] -> ['4a', '2b']
_site = np.unique(site)
_site_counts = [site.count(x) for x in _site]
wp_indices = []
# the sum of all positions should be fixed.
total_units = 0
for j, x in enumerate(_site):
total_units += int(x[:-1])*_site_counts[j]
# collect all possible supergroup transitions
for j, split in enumerate(relation):
if np.all([x in site for x in split]):
wp_indices.append(j)
wps = [wp_list[x] for x in wp_indices]
blocks = [np.array([relation[j].count(s) for s in _site]) for j in wp_indices]
block_units = [sum([int(x[:-1])*block[j] for j, x in enumerate(_site)]) for block in blocks]
# below is a brute force search for the valid combinations
combo_storage = [np.zeros(len(block_units))]
good_list = []
while len(combo_storage) > 0:
holder = []
for j, x in enumerate(combo_storage):
for k in range(len(block_units)):
trial = np.array(deepcopy(x)) # trial solution
trial[k] += 1
if trial.tolist() in holder:
continue
sum_units = np.dot(trial, block_units)
if sum_units > total_units:
continue
elif sum_units < total_units:
holder.append(trial.tolist())
else:
tester = np.zeros(len(_site_counts))
for l, z in enumerate(trial):
tester += z*blocks[l]
if np.all(tester == _site_counts):
G_sites = []
for l, number in enumerate(trial):
if number == 0:
continue
elif number == 1:
G_sites.append(wps[l])
else:
for i in range(int(number)):
G_sites.append(wps[l])
if G_sites not in good_list:
good_list.append(G_sites)
combo_storage=holder
if len(good_list) == 0:
return None
else:
good_splittings_list.append(good_list)
return good_splittings_list
[docs]
def get_min_supergroup(self, group_type='t', G=None):
"""
Returns the minimal supergroups as a dictionary
"""
if self.dim == 3:
dicts = {'supergroup': [],
'transformation': [],
'relations': [],
'idx': [],
}
if G is None:
sgs = range(1,231)
else:
sgs = G
for sg in sgs:
subgroups = None
if group_type == 't':
if sg>self.number:
subgroups = Group(sg, quick=True).get_max_t_subgroup()
else:
g1 = Group(sg)
if g1.point_group == self.point_group:
subgroups = Group(sg, quick=True).get_max_k_subgroup()
if subgroups is not None:
for i, sub in enumerate(subgroups['subgroup']):
if sub == self.number:
trans = subgroups['transformation'][i]
relation = subgroups['relations'][i]
dicts['supergroup'].append(sg)
dicts['transformation'].append(trans)
dicts['relations'].append(relation)
dicts['idx'].append(i)
return dicts
else:
msg = "Only supports the supergroups for space group"
raise NotImplementedError(msg)
[docs]
def get_max_subgroup_numbers(self, max_cell=9):
"""
Returns the minimal supergroups as a dictionary
"""
groups = []
if self.dim == 3:
sub_k = k_subgroup[str(self.number)]
sub_t = t_subgroup[str(self.number)]
k = sub_k['subgroup']
t = sub_t['subgroup']
for i, n in enumerate(t):
if np.linalg.det(sub_t['transformation'][i][:3,:3])<=max_cell:
groups.append(n)
for i, n in enumerate(k):
if np.linalg.det(sub_k['transformation'][i][:3,:3])<=max_cell:
groups.append(n)
return groups
else:
msg = "Only supports the subgroups for space group"
raise NotImplementedError(msg)
[docs]
def get_lists(self, numIon, used_indices):
"""
Compute the lists of possible mult/maxn/freedom/i_wp
Args:
numIon: integer number of atoms
used_indices: a list of integer numbers
"""
l_mult0 = []
l_maxn0 = []
l_free0 = []
indices0 = []
for i_wp, wp in enumerate(self):
indices0.append(i_wp)
l_mult0.append(len(wp))
l_maxn0.append(numIon // len(wp))
#check the freedom
if np.allclose(wp[0].rotation_matrix, np.zeros([3, 3])):
l_free0.append(False)
else:
l_free0.append(True)
return self.clean_lists(numIon, l_mult0, l_maxn0, l_free0, indices0, used_indices)
[docs]
def get_lists_mol(self, numIon, used_indices, orientations):
"""
Compute the lists of possible mult/maxn/freedom/i_wp
Args:
numIon: integer number of atoms
used_indices: a list of integer numbers
orientations: list of orientations
"""
l_mult0 = []
l_maxn0 = []
l_free0 = []
indices0 = []
for i_wp, wp in enumerate(self):
# Check that at least one valid orientation exists
j, k = jk_from_i(i_wp, self.wyckoffs_organized)
if len(orientations) > j and len(orientations[j]) > k:
indices0.append(i_wp)
l_mult0.append(len(wp))
l_maxn0.append(numIon // len(wp))
#check the freedom
if np.allclose(wp[0].rotation_matrix, np.zeros([3, 3])):
l_free0.append(False)
else:
l_free0.append(True)
return self.clean_lists(numIon, l_mult0, l_maxn0, l_free0, indices0, used_indices)
[docs]
@staticmethod
def clean_lists(numIon, l_mult0, l_maxn0, l_free0, indices0, used_indices):
# Remove redundant multiplicities:
l_mult = []
l_maxn = []
l_free = []
indices = []
for mult, maxn, free, i_wp in zip(l_mult0, l_maxn0, l_free0, indices0):
if free:
if mult not in l_mult:
l_mult.append(mult)
l_maxn.append(maxn)
l_free.append(True)
indices.append(i_wp)
elif not free and i_wp not in used_indices:
l_mult.append(mult)
indices.append(i_wp)
if mult <= numIon:
l_maxn.append(1)
elif mult > numIon:
l_maxn.append(0)
l_free.append(False)
return l_mult, l_maxn, l_free, indices
[docs]
def check_compatible(self, numIons, valid_orientations=None):
"""
Checks if the number of atoms is compatible with the Wyckoff
positions. Considers the number of degrees of freedom for each Wyckoff
position, and makes sure at least one valid combination of WP's exists.
Args:
numIons: list of integers
valid_orientations: list of possible orientations (molecule only)
Returns:
Compatible: True/False
has_freedom: True/False
"""
has_freedom = False #whether or not one degree of freedom exists
used_indices = [] #wp's already used that don't have any freedom
# Loop over species
# Sort the specie from low to high so that the solution can be found ealier
for id in np.argsort(numIons):
# Get lists of multiplicity, maxn and freedom
numIon = numIons[id]
if valid_orientations is None:
l_mult, l_maxn, l_free, indices = self.get_lists(numIon, used_indices)
else:
vo = valid_orientations[id]
l_mult, l_maxn, l_free, indices = self.get_lists_mol(numIon, used_indices, vo)
#print(numIon, l_mult, indices, l_maxn, l_free)
# Loop over possible combinations
p = 0 # Create pointer variable to move through lists
# Store the number of each WP, used across possible WP combinations
n0 = [0] * len(l_mult)
n = deepcopy(n0)
for i, mult in enumerate(l_mult):
if l_maxn[i] != 0:
p = i
n[i] = l_maxn[i]
break
p2 = p
if n == n0:
return False, False
#print(numIon, n, n0, p)
while True:
num = np.dot(n, l_mult)
dobackwards = False
# The combination works: move to next species
if num == numIon:
# Check if at least one degree of freedom exists
for val, free, i_wp in zip(n, l_free, indices):
if val > 0:
if free:
has_freedom = True
else:
used_indices.append(i_wp)
break
# All combinations failed: return False
if n == n0 and p >= len(l_mult) - 1:
#print('All combinations failed', numIon, n, n0)
return False, False
# Too few atoms
if num < numIon:
# Forwards routine
# Move p to the right and max out
if p < len(l_mult) - 1:
p += 1
n[p] = min((numIon - num) // l_mult[p], l_maxn[p])
elif p == len(l_mult) - 1:
# p is already at last position: trigger backwards routine
dobackwards = True
# Too many atoms
if num > numIon or dobackwards:
# Backwards routine
# Set n[p] to 0, move p backwards to non-zero, and decrease by 1
n[p] = 0
while p > 0 and p > p2:
p -= 1
if n[p] != 0:
n[p] -= 1
if n[p] == 0 and p == p2:
p2 = p + 1
break
#print('used_indices', used_indices)
if has_freedom:
# All species passed: return True
return True, True
else:
# All species passed, but no degrees of freedom: return 0
return True, False
[docs]
def search_supergroup_paths(self, H, max_layer=5):
"""
Search paths to transit to super group H. if
- path1 is a>>e
- path2 is a>>b>>c>>e
path 2 will not be counted since path 1 exists
Args:
H: final supergroup number
max_layer: the number of supergroup calculations needed.
Returns:
list of possible paths ordered from G to H
"""
layers = {}
layers[0] = {'groups': [H],
'subgroups': [],
}
final = []
traversed = []
# Searches for every subgroup of the the groups from the previous layer.
# stores the possible groups of each layer and their subgroups
# in a dictinoary to avoid redundant calculations.
for l in range(1, max_layer+1):
previous_layer_groups=layers[l-1]['groups']
groups = []
subgroups = []
for g in previous_layer_groups:
subgroup_numbers=np.unique(Group(g, quick=True).get_max_subgroup_numbers())
# If a subgroup list has been found with H
# trace a path through the dictionary to build the path
if self.number in subgroup_numbers:
paths=[[g]]
for j in reversed(range(l-1)):
holder=[]
for path in paths:
tail_number=path[-1]
indices=[]
for idx, numbers in enumerate(layers[j]['subgroups']):
if tail_number in numbers:
indices.append(idx)
for idx in indices:
holder.append(path+[layers[j]['groups'][idx]])
paths=deepcopy(holder)
final.extend(paths)
subgroups.append([])
# Continue to generate next layer if the path to H has not been found.
else:
subgroups.append(subgroup_numbers)
for x in subgroup_numbers:
if (x not in groups) and (x not in traversed):
groups.append(x)
traversed.extend(groups)
layers[l] = {'groups': deepcopy(groups),
'subgroups':[]}
layers[l-1]['subgroups'] = deepcopy(subgroups)
return final
[docs]
def path_to_subgroup(self, H):
"""
For a given a path, extract the
a list of (g_types, subgroup_id, spg_number, wp_list (optional))
"""
path_list = []
paths = self.search_subgroup_paths(H)
if len(paths) > 0:
path = paths[0]
g0 = Group(path[0], quick=True)
for p in path[1:]:
g1 = Group(p, quick=True)
if g0.point_group == g1.point_group:
g_type = 'k'
spgs = g0.get_max_k_subgroup()['subgroup']
else:
g_type = 't'
spgs = g0.get_max_t_subgroup()['subgroup']
for id, spg in enumerate(spgs):
if spg == p:
break
#print(id, spgs, spgs[id])
path_list.append((g_type, id, p))
g0 = g1
return path_list
[docs]
def search_subgroup_paths(self, G, max_layer=5):
"""
Search paths to transit to subgroup H. if
- path1 is a>>e
- path2 is a>>b>>c>>e
path 2 will not be counted since path 1 exists
Args:
G: final subgroup number
max_layer: the number of supergroup calculations needed.
Returns:
list of possible paths ordered from H to G
"""
tmp = Group(G, quick=True)
paths = tmp.search_supergroup_paths(self.number, max_layer=max_layer)
for p in paths:
p.reverse()
p.append(G)
return paths
[docs]
def add_k_transitions(self, path, n=1):
"""
Adds additional k transitions to a subgroup path. ONLY n = 1 is
supported for now. It will return viable additions in front of each
group in the path.
Args:
path: a single result of search_subgroup_paths function
n: number of extra k transitions to add to the given path
Returns:
a list of maximal subgroup chains with extra k type transitions
"""
if n != 1:
print('only 1 extra k type supported at this time')
return None
solutions=[]
for i in range(len(path[:-1])):
g = path[i]
h = path[i+1]
options = set(k_subgroup[str(g)]['subgroup'] + t_subgroup[str(g)]['subgroup'])
#print(g, h, options)
for _g in options:
ls = k_subgroup[str(_g)]['subgroup'] + t_subgroup[str(_g)]['subgroup']
if h in ls:
sol = deepcopy(path)
sol.insert(i+1, _g)
solutions.append(sol)
#https://stackoverflow.com/questions/2213923/removing-duplicates-from-a-list-of-lists
solutions.sort()
solutions = list(k for k,_ in itertools.groupby(solutions))
return solutions
[docs]
def path_to_general_wp(self, index=1, max_steps=1):
"""
Find the path to transform the special wp into general site
Args:
group: Group object
index: the index of starting wp
max_steps: the number of steps to search
Return:
a list of (g_types, subgroup_id, spg_number, wp_list (optional))
"""
#label = [str(self[index].multiplicity) + self[index].letter]
label = [self[index].get_label()]
potential=[[(None, None, self.number, label)]]
solutions=[]
for step in range(max_steps):
_potential = []
for p in potential:
tail = p[-1]
tdict = t_subgroup[str(tail[2])]; len_t = len(tdict['subgroup'])
kdict = k_subgroup[str(tail[2])]; len_k = len(kdict['subgroup'])
_indexs=[ord(x[-1])-97 for x in tail[3]]
next_steps=[[("t", i, tdict['subgroup'][i], sum([tdict['relations'][i][_index] \
for _index in _indexs],[]))] \
for i in range(len_t)] + \
[[("k", i, kdict['subgroup'][i], sum([kdict['relations'][i][_index] \
for _index in _indexs],[]))] \
for i in range(len_k) if kdict['subgroup'][i]!=tail[2]]
for n in next_steps:
_potential.append(deepcopy(p)+n)
potential=deepcopy(_potential)
for p in deepcopy(potential):
#Check there's only one wp. #Check that the 1 wp is the general position
if (len(set(p[-1][3]))==1) and (p[-1][3][0][-1]==Group(p[-1][2])[0].letter):
solutions.append(deepcopy(p)[1:])
potential.remove(p)
return solutions
[docs]
def short_path_to_general_wp(self, index=1, t_only=False):
"""
Find a short path to turn the spcical wp to general position
Args:
index: index of the wp
t_only: only consider t_spliting
"""
for i in range(1, 5):
paths = self.path_to_general_wp(index, max_steps=i)
if len(paths) > 0:
last_gs = np.array([p[-1][2] for p in paths])
if t_only:
last_gs[last_gs > len(self[0])] = 0
max_id = np.argmax(last_gs)
return paths[max_id]
[docs]
def get_valid_solutions(self, solutions):
"""
check if the solutions are valid
a special WP such as (0,0,0) cannot be occupied twice
Args:
solutions: list of solutions about the distibution of WP sites
Returns:
the filtered solutions that are vaild
"""
valid_solutions = []
for solution in solutions:
sites = []
for s in solution:
sites.extend(s)
if self.is_valid_combination(sites):
valid_solutions.append(solution)
return valid_solutions
[docs]
def cellsize(self):
"""
Returns the number of duplicate atoms in the conventional lattice (in
contrast to the primitive cell). Based on the type of cell centering
(P, A, C, I, R, or F)
"""
if self.dim in [0, 1]:
# Rod and point groups
return 1
elif self.dim == 2:
# Layer groups
if self.number in [10, 13, 18, 22, 26, 35, 36, 47, 48]:
return 2
else:
return 1
else:
# space groups
if self.symbol[0] == 'P':
return 1 # P
elif self.symbol[0] == 'R':
return 3 # R
elif self.symbol[0] == 'F':
return 4 # F
else:
return 2 # A, C, I
[docs]
def get_free_axis(self):
"""
Get the free axis that can perform continus translation
"""
number = self.number
if number == 1:
return [0, 1, 2]
elif number == 2:
return []
elif 3 <= number <= 5:
return [1] # '2'
elif 6 <= number <= 9:
return [0, 2] # 'm'
elif 10 <= number <= 24:
return [] # '2/m', '222'
elif 25 <= number <= 46:
return [2] # 'mm2'
elif 47 <= number <= 74:
return [] # 'mmm'
elif 75 <= number <= 80:
return [2] # '4'
elif 81 <= number <= 98:
return [] # '-4', '4/m', '422'
elif 99 <= number <= 110:
return [2] # '4mm'
elif 111 <= number <= 142:
return [] # '-42m', '4/mmm'
elif 143 <= number <= 146:
return [2] # '3'
elif 147 <= number <= 155:
return [] #'-3', '32'
elif 156 <= number <= 161:
return [2] #'3m'
elif 162 <= number <= 167:
return [] #'-3m'
elif 168 <= number <= 173:
return [2] #'6'
elif 174 <= number <= 182:
return [] # '-6', '6/m', '622'
elif 183 <= number <= 186:
return [2] #'6mm'
elif 187 <= number <= 194:
return [] #'-62m', '6/mmm'
elif 195 <= number <= 230:
return [] #'23', 'm-3', '432', '-43m', 'm-3m',
[docs]
@classmethod
def list_groups(cls, dim=3):
"""
Function for quick print of groups and symbols.
Args:
group: the group symbol or international number
dim: the periodic dimension of the group
"""
import pandas as pd
keys = {
3: "space_group",
2: "layer_group",
1: "rod_group",
0: "point_group",
}
data = symbols[keys[dim]]
index = range(1, len(data) + 1)
df = pd.DataFrame(index=index, data=data, columns=[keys[dim]])
pd.set_option("display.max_rows", len(df))
print(df)
[docs]
def get_index_by_letter(self, letter):
"""
get the wp object by the letter
"""
if len(letter) > 1: letter = letter[-1]
#print(letter); print(letters.index(letter))
return len(self) - letters.index(letter) - 1
[docs]
def get_wp_by_letter(self, letter):
"""
get the wp object by the letter
"""
return self[self.get_index_by_letter(letter)]
#
# ----------------------- Wyckoff Position class ------------------------
[docs]
class Wyckoff_position:
"""
Class for a single Wyckoff position within a symmetry group
Examples
--------
>>> from pyxtal.symmetry import Wyckoff_position as wp
>>> wp.from_group_and_index(19, 0)
Wyckoff position 4a in space group 19 with site symmetry 1
x, y, z
-x+1/2, -y, z+1/2
-x, y+1/2, -z+1/2
x+1/2, -y+1/2, -z
"""
#=============================Initialization===========================
[docs]
def from_dict(dictionary):
"""
Constructs a Wyckoff_position object using a dictionary.
"""
wp = Wyckoff_position()
for key in dictionary:
setattr(wp, key, dictionary[key])
#wp.get_site_symmetry()
wp.set_euclidean()
# For nonstandard setting only
if wp.P1 is not None and not identity_ops(wp.P1):
wp.set_generators()
wp.set_ops()
return wp
[docs]
def from_group_and_letter(group, letter, dim=3, style='pyxtal', hn=None):
"""
Creates a Wyckoff_position using the space group number and index
Args:
group: the international number of the symmetry group
letter: e.g. 4a
dim: the periodic dimension of the crystal
style: 'pyxtal' or spglib, differing in the choice of origin
hn: hall_number
"""
for c in letter:
if c.isalpha():
letter = c
break
ops_all = get_wyckoffs(group, dim=dim)
index = index_from_letter(letter, ops_all, dim=dim)
if hn is not None:
wp = Wyckoff_position.from_group_and_index(hn, index, dim, use_hall=True, wyckoffs=ops_all)
else:
wp = Wyckoff_position.from_group_and_index(group, index, dim, style=style, wyckoffs=ops_all)
return wp
[docs]
def from_group_and_index(group, index, dim=3, use_hall=False, style='pyxtal', wyckoffs=None):
"""
Creates a Wyckoff_position using the space group number and index
Args:
group: the international number of the symmetry group
index: the index of the Wyckoff position within the group.
dim: the periodic dimension of the crystal
use_hall (default: False): whether or not use the hall number
style (default: `pyxtal`): 'pyxtal' or 'spglib' for hall number
"""
number, hall_number, P, P1 = group, None, None, None
if not use_hall:
symbol, number = get_symbol_and_number(group, dim)
else:
symbol = hall_table['Symbol'][group-1]
number = hall_table['Spg_num'][group-1]
if dim == 3:
PBC = [1, 1, 1]
if not use_hall:
if style == 'pyxtal':
hall_number = pyxtal_hall_numbers[number-1]
else:
hall_number = spglib_hall_numbers[number-1]
P = abc2matrix(hall_table['P'][hall_number-1])
P1 = abc2matrix(hall_table['P^-1'][hall_number-1])
else:
hall_number = group
P = abc2matrix(hall_table['P'][hall_number-1])
P1 = abc2matrix(hall_table['P^-1'][hall_number-1])
elif dim == 2:
PBC = [1, 1, 0]
elif dim == 1:
PBC = [0, 0, 1]
if wyckoffs is None:
wyckoffs = get_wyckoffs(number, dim=dim)
wpdict = {
"index": index,
"letter": letter_from_index(index, wyckoffs, dim=dim),
"ops": wyckoffs[index],
"multiplicity": len(wyckoffs[index]),
"symmetry": get_wyckoff_symmetry(number, dim=dim)[index],
#"generators": get_generators(number, dim=dim)[index],
"PBC": PBC,
"dim": dim,
"number": number,
"P": P,
"P1": P1,
"hall_number": hall_number,
"symbol": symbol,
}
return Wyckoff_position.from_dict(wpdict)
[docs]
def from_symops_wo_group(ops):
"""
search Wyckoff Position by symmetry operations
Now only supports space group symmetry
Assuming general position only
Args:
ops: a list of symmetry operations
Returns:
`Wyckoff_position`
"""
_, spg_num = get_symmetry_from_ops(ops)
wp = Wyckoff_position.from_group_and_index(spg_num, 0)
if isinstance(ops[0], str):
ops = [SymmOp.from_xyz_str(op) for op in ops]
wp.ops = ops
match_spg, match_hm = wp.update()
#print("match_spg", match_spg, "match_hall", match_hm)
return wp
[docs]
def from_symops(ops, G):
"""
search Wyckoff Position by symmetry operations
Args:
ops: a list of symmetry operations
G: the Group object
Returns:
`Wyckoff_position`
"""
if isinstance(ops[0], str):
ops = [SymmOp.from_xyz_str(op) for op in ops]
for wp in G:
if wp.has_equivalent_ops(ops):
return wp
if isinstance(ops[0], str):
print(ops)
else:
for op in ops:
print(op.as_xyz_str())
raise RuntimeError("Cannot find the right wp")
[docs]
def from_index_quick(self, wyckoffs, index, P=None, P1=None):
"""
A short cut to create the WP object from a given index
ignore the site symmetry and generators
Mainly used for the update function
Args:
wyckoffs: wyckoff position
index: index of wp
P: transformation matrix (rot + trans)
"""
if P is None:
P = self.P
P1 = self.P1
wpdict = {
"index": index,
"letter": letter_from_index(index, wyckoffs, dim=self.dim),
"ops": wyckoffs[index],
"multiplicity": len(wyckoffs[index]),
"PBC": self.PBC,
"dim": self.dim,
"number": self.number,
"P": P,
"P1": P1,
"hall_number": self.hall_number,
}
return Wyckoff_position.from_dict(wpdict)
#=============================Fundamentals===========================
def __str__(self, supress=False):
if self.dim not in [0, 1, 2, 3]:
return "invalid crystal dimension. Must be between 0 and 3."
if not hasattr(self, "site_symm"): self.get_site_symmetry()
s = "Wyckoff position " + self.get_label() + " in "
if self.dim == 3:
s += "space "
elif self.dim == 2:
s += "layer "
elif self.dim == 1:
s += "Rod "
elif self.dim == 0:
s += "Point group " + self.symbol
if self.dim != 0:
s += "group " + str(self.number)
s += " with site symmetry " + self.site_symm
if not supress:
for op in self.ops:
s += "\n" + op.as_xyz_str()
self.string = s
return self.string
def __repr__(self):
return str(self)
def __iter__(self):
yield from self.ops
def __getitem__(self, index):
return self.ops[index]
def __len__(self):
return self.multiplicity
[docs]
def copy(self):
"""
Simply copy the structure
"""
return deepcopy(self)
[docs]
def save_dict(self):
dict0 = {
"group": self.number,
"index": self.index,
"dim": self.dim,
#"transformation": self.get_transformation(),
}
return dict0
[docs]
@classmethod
def load_dict(cls, dicts):
g = dicts['group']
index = dicts['index']
dim = dicts['dim']
trans = None #dicts['transformation']
return Wyckoff_position.from_group_and_index(g, index, dim)#, trans=trans)
#=============================Updates===========================
[docs]
def set_ops(self):
self.ops = self.get_ops_from_transformation()
[docs]
def update(self):
"""
update the spacegroup information if needed
"""
match_spg, match_hall = False, False
match_spg = self.update_index()
if not match_spg:
match_hall = self.update_hall()
if not match_spg and not match_hall:
print("match_spg", match_spg, "match_hall", match_hall)
print(self)
print(self.get_hm_symbol())
raise RuntimeError("Cannot find the right hall_number")
return match_spg, match_hall
[docs]
def update_hall(self, hall_numbers=None):
"""
update the Hall number when the symmetry operation changes
Args:
hall_numbers: a list of numbers for consideration
"""
#print("test", self)
if hall_numbers is None:
hall_numbers = Hall(self.number).hall_numbers
candidates = self.process_ops()
success = False
for hall_number in hall_numbers:
P = abc2matrix(hall_table['P'][hall_number-1])
P1 = abc2matrix(hall_table['P^-1'][hall_number-1])
wyckoffs = get_wyckoffs(self.number, dim=self.dim)
# Fist check the original index
wp2 = self.from_index_quick(wyckoffs, self.index, P, P1)
for ops in candidates:
if wp2.has_equivalent_ops(ops):
success = True
#print("same letter") #; import sys; sys.exit()
break
# Check other sites
if not success:
for i in range(len(wyckoffs)):
if i != self.index and len(wyckoffs[i]) == self.multiplicity:
wp2 = self.from_index_quick(wyckoffs, i, P, P1)
for ops in candidates:
if wp2.has_equivalent_ops(ops):
success = True
self.index = i
self.letter = wp2.letter
#print("new letter")
break
if success:
break
if success:
self.hall_number = hall_number
self.P = wp2.P
self.P1 = wp2.P1
self.ops = wp2.ops
return True
return False
[docs]
def update_index(self):
"""
check if needs to update the index due to lattice transformation
"""
wyckoffs = get_wyckoffs(self.number, dim=self.dim)
wp2 = self.from_index_quick(wyckoffs, self.index)
if self.has_equivalent_ops(wp2):
return True
else:
for i in range(len(wyckoffs)):
if i != self.index and len(wyckoffs[i]) == self.multiplicity:
wp2 = self.from_index_quick(wyckoffs, i)
if self.has_equivalent_ops(wp2):
self.index = i
self.letter = wp2.letter
#adjust to normal
self.ops = wp2.ops
return True
return False
[docs]
def process_ops(self):
"""
handle some annoying cases
e.g., in I2, ['1/2, y, 1/2', '0, y+1/2, 0'] can be transfered to
['0, y, 0', '1/2, y+1/2, 1/2']
"""
opss = [self.ops]
if self.number in [5, 12] and self.index > 0:
# replace y with y+1/2
op2 = SymmOp.from_xyz_str('x, y+1/2, z')
ops = [op2*op for op in self.ops]
opss.append(ops)
if self.number in [13] and self.index > 0:
op2 = SymmOp.from_xyz_str('x, -y, z')
ops = [op2*op for op in self.ops]
opss.append(ops)
#for op in ops: print('AAAA', op.as_xyz_str())
return opss
[docs]
def equivalent_set(self, index):
"""
Transform the wp to another equivalent set.
Needs to update both wp and positions
Args:
transformation: index
"""
if self.index > 0:
G = Group(self.number)
if len(G[index]) != len(G[self.index]):
msg = "Spg {:d}, Invalid switch in Wyckoff Pos\n".format(self.number)
msg += str(self)
msg += "\n"+str(G[index])
raise ValueError(msg)
else:
return G[index]
return self
#=============================Get functions===========================
[docs]
def get_site_symm_wo_translation(self):
ops = []
for op in self.symmetry[0]:
op = SymmOp.from_rotation_and_translation(op.rotation_matrix, [0, 0, 0])
ops.append(op)
return ops
[docs]
def get_site_symmetry(self):
if self.euclidean:
ops = self.get_euclidean_symmetries()
else:
ops = self.symmetry[0]
self.site_symm = ss_string_from_ops(ops, self.number, dim=self.dim)
[docs]
def get_hm_number(tol=1e-5):
if self.index == 0:
return get_symmetry_from_ops(self.ops, tol)[0]
else:
print(self)
raise ValueError("input must be general position")
[docs]
def get_hm_symbol(self):
"""
get Hermann-Mauguin symbol
"""
return hall_table['Symbol'][self.hall_number-1]
[docs]
def get_dof(self):
"""
Simply return the degree of freedom
"""
return np.linalg.matrix_rank(self.ops[0].rotation_matrix)
[docs]
def get_label(self):
"""
get the string like 4a
"""
return str(self.multiplicity) + self.letter
[docs]
def get_frozen_axis(self):
if self.index == 0:
return []
elif self.get_dof() == 0:
return [0, 1, 2]
else:
if self.number >=75:
#if self.ops[0].rotation_matrix[2,2] == 1:
# return [0, 1]
#else:
# return [0, 1, 2]
axs = []
for ax in range(3):
if self.ops[0].rotation_matrix[ax, ax] == 0:
axs.append(ax)
return axs
else:
if self.get_dof() == 1:
if self.ops[0].rotation_matrix[2,2] == 1:
return [0, 1]
elif self.ops[0].rotation_matrix[1,1] == 1:
return [0, 2]
elif self.ops[0].rotation_matrix[0,0] == 1:
return [1, 2]
else:
if self.ops[0].rotation_matrix[2,2] != 1:
return [2]
elif self.ops[0].rotation_matrix[1,1] != 1:
return [1]
elif self.ops[0].rotation_matrix[0,0] != 1:
return [0]
[docs]
def get_euclidean_symmetries(self):
"""
return the symmetry operation object at the Euclidean space
Returns:
list of pymatgen SymmOp object
"""
ops = []
for op in self.symmetry[0]:
hat = SymmOp.from_rotation_and_translation(hex_cell, [0, 0, 0])
ops.append(hat * op * hat.inverse)
return ops
[docs]
def get_euclidean_generator(self, cell, idx=0):
"""
return the symmetry operation object at the Euclidean space
Args:
cell: 3*3 cell matrix
idx: the index of wp generator
Returns:
pymatgen SymmOp object
"""
if not hasattr(self, "generators"):
self.set_generators()
op = self.generators[idx]
if self.euclidean:
hat = SymmOp.from_rotation_and_translation(cell.T, [0, 0, 0])
op = hat * op * hat.inverse
return op
[docs]
def get_free_xyzs(self, pos):
"""
return the free xyz paramters from the given xyz position
"""
#print(self.apply_ops(pos)[0])
res = self.apply_ops(pos)[0]
res = np.delete(res, self.get_frozen_axis())
return res
[docs]
def get_position_from_free_xyzs(self, xyz):
"""
generate the full xyz position from the free xyzs
"""
pos = np.zeros(3)
frozen = self.get_frozen_axis()
count = 0
for axis in range(3):
if axis not in frozen:
pos[axis] = xyz[count]
count += 1
pos = self.apply_ops(pos)[0]
pos -= np.floor(pos)
return pos
[docs]
def get_all_positions(self, pos):
"""
return the list of position from any single coordinate from wp.
The position does not have to be the 1st number in the wp list
>>> from pyxtal.symmetry import Group
>>> wp2 = Group(62)[-1]
>>> wp2
Wyckoff position 4a in space group 62 with site symmetry -1
0, 0, 0
1/2, 0, 1/2
0, 1/2, 0
1/2, 1/2, 1/2
>>> wp2.get_all_positions([1/2, 1/2, 1/2])
array([[0. , 0. , 0. ],
[0.5, 0. , 0.5],
[0. , 0.5, 0. ],
[0.5, 0.5, 0.5]])
"""
pos0, _, _ = self.merge(pos, np.eye(3), 0.01)
res = self.apply_ops(pos0)
res -= np.floor(res)
return res
#=============================Evaluations===========================
[docs]
def is_standard_setting(self):
"""
Check if the symmetry operation follows the standard setting
"""
G_ops = get_wyckoffs(self.number, dim=self.dim)
for i, ops in enumerate(G_ops):
if self.has_equivalent_ops(ops):
self.ops = ops
self.index = i
self.letter = letter_from_index(i, G_ops, dim=self.dim)
return True
return False
[docs]
def has_equivalent_ops(self, wp2, tol=1e-3):
"""
check if two wps are equivalent
Args:
wp2: wp object or list of operations
"""
if type(wp2) == list:
ops0 = wp2
else:
ops0 = wp2.ops
if len(ops0) == len(self.ops):
count = 0
for i, op0 in enumerate(ops0):
for j, op1 in enumerate(self.ops):
diff0 = op0.translation_vector - op1.translation_vector
diff0 -= np.round(diff0)
diff1 = op0.rotation_matrix - op1.rotation_matrix
if max([np.abs(diff0).sum(), np.abs(diff1).sum()]) < tol:
count += 1
if count == len(ops0):
return True
else:
return False
else:
return False
[docs]
def is_pure_translation(self, id):
"""
Check if the operation is equivalent to pure translation
"""
op = self.generators[id]
diff = op.rotation_matrix - np.eye(3)
if np.sum(diff.flatten()**2) < 1e-4:
return True
else:
ops = self.get_site_symm_wo_translation()
return (op in ops)
[docs]
def swap_axis(self, swap_id):
"""
swap the axis may result in a new wp
"""
if self.index > 0:
perm_id = None
_ops = [self.ops[0]]
trans = [np.zeros(3)]
if self.symbol[0] == "F":
trans.append(np.array([0,0.5,0.5]))
trans.append(np.array([0.5,0,0.5]))
trans.append(np.array([0.5,0.5,0]))
elif self.symbol[0] == "I":
trans.append(np.array([0.5,0.5,0.5]))
elif self.symbol[0] == "A":
trans.append(np.array([0,0.5,0.5]))
elif self.symbol[0] == "B":
trans.append(np.array([0.5,0,0.5]))
elif self.symbol[0] == "C":
trans.append(np.array([0.5,0.5,0]))
op_perm = swap_xyz_ops(_ops, swap_id)[0]
for id, ops in enumerate(Group(self.number)):
if len(ops) == len(self.ops):
for i, tran in enumerate(trans):
if i > 0:
# apply tran
op = op_translation(op_perm, tran)
else:
op = op_perm
#print(id, op.as_xyz_str(),tran)
if are_equivalent_ops(op, ops[0]):
perm_id = id
return Group(self.number)[id], tran
if perm_id is None:
raise ValueError("cannot swap", swap_id, self)
return self, np.zeros(3)
[docs]
def print_ops(self, ops=None):
if ops is None:
ops = self.ops
for op in ops:
print(op.as_xyz_str())
[docs]
def gen_pos(self):
"""
Returns the general Wyckoff position
"""
return self.ops[0]
[docs]
def are_equivalent_pts(self, pt1, pt2, cell=np.eye(3), tol=0.05):
"""
Check if two pts are equivalent
"""
pt1 = self.search_generator(pt1, tol=tol)
pt2 = self.search_generator(pt2, tol=tol)
if pt1 is None or pt2 is None:
return False
else:
pt1 = np.array(pt1); pt1 -= np.floor(pt1)
pt2 = np.array(pt2); pt2 -= np.floor(pt2)
pts = self.apply_ops(pt1); pts -= np.floor(pts)
diffs = pt2 - pts
diffs -= np.round(diffs)
diffs = np.dot(diffs, cell)
dists = np.linalg.norm(diffs, axis=1)
#print(dists)
if len(dists[dists<tol]) > 0:
return True
else:
return False
[docs]
def distance_check(self, pt, lattice, tol):
"""
Given a list of fractional coordinates, merges them within a given
tolerance, and checks if the merged coordinates satisfy a Wyckoff
position.
Args:
pt: the originl point (3-vector)
lattice: a 3x3 matrix representing the unit cell
tol: the cutoff distance for merging coordinates
Returns:
True or False
"""
if len(self.short_distances(pt, lattice, tol)) > 0:
return False
else:
return True
[docs]
def short_distances(self, pt, lattice, tol):
"""
Given a list of fractional coordinates, merges them within a given
tolerance, and checks if the merged coordinates satisfy a Wyckoff
position.
Args:
pt: the originl point (3-vector)
lattice: a 3x3 matrix representing the unit cell
tol: the cutoff distance for merging coordinates
Returns:
a list of short distances
"""
pt = self.project(pt, lattice, self.PBC)
coor = self.apply_ops(pt)
#coor -= np.round(coor)
coor -= np.floor(coor)
dm = distance_matrix([coor[0]], coor, lattice, PBC=self.PBC)[0][1:]
#if len(dm[dm<tol]==0): print('+++++', pt, dm.shape, tol, dm[dm<tol], len(dm[dm<tol]))
return dm[dm<tol]
[docs]
def merge(self, pt, lattice, tol, orientations=None, group=None):
"""
Given a list of fractional coordinates, merges them within a given
tolerance, and checks if the merged coordinates satisfy a Wyckoff
position.
Args:
pt: the originl point (3-vector)
lattice: a 3x3 matrix representing the unit cell
tol: the cutoff distance for merging coordinates
orientations: the valid orientations for a given molecule.
Returns:
pt: 3-vector after merge
wp: a Wyckoff_position object, If no match, returns False.
valid_ori: the valid orientations after merge
"""
wp = self.copy()
PBC = wp.PBC
if group is None: group = Group(wp.number, wp.dim)
pt = self.project(pt, lattice, PBC)
coor = apply_ops(pt, wp)
if orientations is None:
valid_ori = None
else:
j, k = jk_from_i(wp.index, orientations)
valid_ori = orientations[j][k]
# Main loop for merging multiple times
while True:
# Check distances of current WP. If too small, merge
dm = distance_matrix([coor[0]], coor, lattice, PBC=PBC)
passed_distance_check = True
x = np.argwhere(dm < tol)
for y in x:
# Ignore distance from atom to itself
if y[0] == 0 and y[1] == 0:
pass
else:
passed_distance_check = False
break
# for molecular crystal, one more check
if not check_images([coor[0]], [6], lattice, PBC=PBC, tol=tol):
passed_distance_check = False
if not passed_distance_check:
mult1 = wp.multiplicity
# Find possible wp's to merge into
possible = []
for i, wp0 in enumerate(group):
mult2 = wp0.multiplicity
# Check that a valid orientation exists
if orientations is not None:
res = jk_from_i(i, orientations)
if res is None:
continue
else:
j, k = res
if orientations[j][k] == []:
continue
else:
valid_ori = orientations[j][k]
# factor = mult2 / mult1
if (mult2 < mult1) and (mult1 % mult2 == 0):
possible.append(i)
if possible == []:
return None, False, valid_ori
# Calculate minimum separation for each WP
distances = []
pts = []
for i in possible:
#wp = group[i]
p, d = group[i].search_generator_dist(pt.copy(), lattice, group)
distances.append(d)
pts.append(p)
# Choose wp with shortest translation for generating point
tmpindex = np.argmin(distances)
index = possible[tmpindex]
wp = group[index]
pt = pts[tmpindex]
coor = wp.apply_ops(pt)
# Distances were not too small; return True
else:
return pt, wp, valid_ori
[docs]
def set_generators(self):
"""
set up generators, useful for many things
"""
self.generators = get_generators(self.number, dim=self.dim)[self.index]
if self.P is not None and not identity_ops(self.P):
#self.print_ops(self.generators)
ops = transform_ops(self.generators, self.P, self.P1)
self.generators = ops
#self.print_ops(ops)
[docs]
def set_euclidean(self):
"""
For the hexagonal groups, need to consider the euclidean conversion
"""
convert = False
if self.dim == 3:
if 143 <= self.number < 195:
convert = True
elif self.dim == 2:
if self.number >= 65:
convert = True
elif self.dim == 1:
if self.number >= 42:
convert = True
self.euclidean = convert
[docs]
def search_generator_dist(self, pt, lattice=np.eye(3), group=None):
"""
For a given special wp, (e.g., [(x, 0, 1/4), (0, x, 1/4)]),
return the first position and distance
Args:
pt: 1*3 vector
lattice: 3*3 matrix
Returns:
pt: the best matched pt
diff: numerical difference
"""
if self.index == 0: #general sites
return pt, 0
else:
d = []
if self.get_dof == 0: #fixed site like [0, 0, 0]
pts = self.apply_ops(pt)
for p0 in pts:
d.append(distance(p0, lattice, PBC=self.PBC))
else: # sites like (x, 0, 0)
if group is not None:
ops = group[0].ops
else:
ops = get_wyckoffs(self.number, dim=self.dim)[0]
pts = []
for op in ops:
pt0 = op.operate(pt)
pt1 = self.ops[0].operate(pt0)
coord = pt1 - pt0
d.append(distance(coord, lattice, PBC=self.PBC))
pts.append(pt0)
#print(d)
d = np.array(d)
return pts[np.argmin(d)], np.min(d)
[docs]
def search_generator(self, pos, ops=None, tol=1e-2):
"""
search generator for a special Wyckoff position
Args:
pos: initial xyz position
ops: list of symops
tol: tolerance
Return:
pos1: the position that matchs the standard setting
"""
if ops is None:
ops = get_wyckoffs(self.number, dim=self.dim)[0]
match = False
for op in ops:
pos1 = op.operate(pos) #
pos0 = self.ops[0].operate(pos1)
diff = pos1 - pos0
diff -= np.round(diff)
diff = np.abs(diff)
#print(self.letter, "{:24s}".format(op.as_xyz_str()), pos, pos0, pos1, diff)
if diff.sum() < tol:
pos1 -= np.floor(pos1)
match = True
break
if match:
return pos1
else:
#print(pos, wp0, wp)
return None
[docs]
def search_all_generators(self, pos, ops=None, tol=1e-2):
"""
search generator for a special Wyckoff position
Args:
pos: initial xyz position
ops: list of symops
tol: tolerance
Return:
pos1: the position that matchs the standard setting
"""
if ops is None:
ops = get_wyckoffs(self.number, dim=self.dim)[0]
coords = []
for op in ops:
pos1 = op.operate(pos)
pos0 = self.ops[0].operate(pos1)
diff = pos1 - pos0
diff -= np.round(diff)
diff = np.abs(diff)
#print(wp.letter, pos1, pos0, diff)
if diff.sum() < tol:
pos1 -= np.floor(pos1)
coords.append(pos1)
return coords
[docs]
def apply_ops(self, pt):
"""
apply symmetry operation
"""
return apply_ops(pt, self.ops)
[docs]
def project(self, point, cell=np.eye(3), PBC=[1, 1, 1], id=0):
"""
Given a 3-vector and a Wyckoff position operator,
returns the projection onto the axis, plane, or point.
>>> wp.project_point([0,0.3,0.1],
array([0. , 0.3, 0.1])
Args:
point: a 3-vector (numeric list, tuple, or array)
cell: 3x3 matrix describing the unit cell vectors
PBC: A periodic boundary condition list, where 1 means periodic, 0
means not periodic. Ex: [1,1,1] -> full 3d periodicity, [0,0,1]
-> 1d periodicity along the z axis
Returns:
a transformed 3-vector (numpy array)
"""
op = self.ops[id]
rot = op.rotation_matrix
trans = op.translation_vector
point = np.array(point, dtype=float)
def project_single(point, rot, trans):
# move the point in the opposite direction of the translation
point -= trans
new_vector = np.zeros(3)
# Loop over basis vectors of the symmetry element
for basis_vector in rot.T:
# b = np.linalg.norm(basis_vector)
b = np.sqrt(basis_vector.dot(basis_vector)) # a faster version?
#if not np.isclose(b, 0):
if b > 1e-3:
new_vector += basis_vector * (np.dot(point, basis_vector) / (b ** 2))
new_vector += trans
return new_vector
if PBC == [0, 0, 0]:
return project_single(point, rot, trans)
else:
pt = filtered_coords(point)
m = create_matrix(PBC=PBC)
new_vectors = []
distances = []
for v in m:
new_vector = project_single(pt, rot, trans+v)
new_vectors.append(new_vector)
tmp = (new_vector-point).dot(cell)
distances.append(np.linalg.norm(tmp))
i = np.argmin(distances)
return filtered_coords(new_vectors[i], PBC=PBC)
# ----------------- Wyckoff Position selection --------------------------
[docs]
def choose_wyckoff(G, number=None, site=None, dim=3):
"""
Choose a Wyckoff position to fill based on the current number of atoms
needed to be placed within a unit cell
Rules:
0) use the pre-assigned list if this is provided
1) The new position's multiplicity is equal/less than (number).
2) We prefer positions with large multiplicity.
Args:
G: a pyxtal.symmetry.Group object
number: the number of atoms still needed in the unit cell
site: the pre-assigned Wyckoff sites (e.g., 4a)
Returns:
Wyckoff position. If no position is found, returns False
"""
if site is not None:
number = G.number
if G.hall_number is not None:
hn = G.hall_number
else:
hn = None
return Wyckoff_position.from_group_and_letter(number, site, dim, hn=hn)
else:
wyckoffs_organized = G.wyckoffs_organized
if random.uniform(0, 1) > 0.5: # choose from high to low
for wyckoff in wyckoffs_organized:
if len(wyckoff[0]) <= number:
return random.choice(wyckoff)
return False
else:
good_wyckoff = []
for wyckoff in wyckoffs_organized:
if len(wyckoff[0]) <= number:
for w in wyckoff:
good_wyckoff.append(w)
if len(good_wyckoff) > 0:
return random.choice(good_wyckoff)
else:
return False
[docs]
def choose_wyckoff_mol(G, number, site, orientations, gen_site=True, dim=3):
"""
Choose a Wyckoff position to fill based on the current number of molecules
needed to be placed within a unit cell
Rules:
1) The new position's multiplicity is equal/less than (number).
2) We prefer positions with large multiplicity.
3) The site must admit valid orientations for the desired molecule.
Args:
G: a pyxtal.symmetry.Group object
number: the number of molecules still needed in the unit cell
orientations: the valid orientations for a given molecule.
gen_site: general WP only
Returns:
Wyckoff position. If no position is found, returns False
"""
wyckoffs = G.wyckoffs_organized
if site is not None:
number = G.number
if G.hall_number is not None:
hn = G.hall_number
else:
hn = None
return Wyckoff_position.from_group_and_letter(number, site, dim, hn=hn)
elif gen_site or np.random.random() > 0.5: # choose from high to low
for j, wyckoff in enumerate(wyckoffs):
if len(wyckoff[0]) <= number:
good_wyckoff = []
for k, w in enumerate(wyckoff):
if orientations[j][k] != []:
good_wyckoff.append(w)
if len(good_wyckoff) > 0:
return random.choice(good_wyckoff)
return False
else:
good_wyckoff = []
for j, wyckoff in enumerate(wyckoffs):
if len(wyckoff[0]) <= number:
for k, w in enumerate(wyckoff):
if orientations[j][k] != []:
good_wyckoff.append(w)
if len(good_wyckoff) > 0:
return random.choice(good_wyckoff)
else:
return False
# -------------------- quick utilities for symmetry conversion ----------------
[docs]
def swap_xyz_string(xyzs, permutation):
"""
Permutate the xyz string operation
Args:
xyzs: e.g. ['x', 'y+1/2', '-z']
permuation: list, e.g., [0, 2, 1]
Returns:
the new xyz string after transformation
"""
if permutation == [0,1,2]:
return xyzs
else:
new = []
for xyz in xyzs:
tmp = xyz.replace(" ","").split(',')
tmp = [tmp[it] for it in permutation]
if permutation == [1,0,2]: #a,b
tmp[0] = tmp[0].replace('y','x')
tmp[1] = tmp[1].replace('x','y')
elif permutation == [2,1,0]: #a,c
tmp[0] = tmp[0].replace('z','x')
tmp[2] = tmp[2].replace('x','z')
elif permutation == [0,2,1]: #b,c
tmp[1] = tmp[1].replace('z','y')
tmp[2] = tmp[2].replace('y','z')
elif permutation == [1,2,0]: #b,c
tmp[0] = tmp[0].replace('y','x')
tmp[1] = tmp[1].replace('z','y')
tmp[2] = tmp[2].replace('x','z')
elif permutation == [2,0,1]: #b,c
tmp[0] = tmp[0].replace('z','x')
tmp[1] = tmp[1].replace('x','y')
tmp[2] = tmp[2].replace('y','z')
new.append(tmp[0] + ", " + tmp[1] + ", " + tmp[2])
return new
[docs]
def swap_xyz_ops(ops, permutation):
"""
change the symmetry operation by swaping the axes
Args:
ops: SymmOp object
permutation: list, e.g. [0, 1, 2]
Returns:
the new xyz string after transformation
"""
if permutation == [0,1,2]:
return ops
else:
new = []
for op in ops:
m = op.affine_matrix.copy()
m[:3,:] = m[permutation, :]
for row in range(3):
# [0, y+1/2, 1/2] -> (0, y, 1/2)
if np.abs(m[row,:3]).sum()>0:
m[row, 3] = 0
m[:3,:3] = m[:3, permutation]
new.append(SymmOp(m))
return new
[docs]
def op_translation(op, tran):
m = op.affine_matrix.copy()
m[:3,3] += tran
for row in range(3):
# [0, y+1/2, 1/2] -> (0, y, 1/2)
if np.abs(m[row,:3]).sum()>0:
m[row, 3] = 0
return SymmOp(m)
[docs]
def are_equivalent_ops(op1, op2, tol=1e-2):
"""
check if two ops are equivalent, assuming the same ordering
"""
diff = op1.affine_matrix - op2.affine_matrix
diff[:,3] -= np.round(diff[:,3])
diff = np.abs(diff.flatten())
if np.sum(diff) < tol:
return True
else:
return False
[docs]
def letter_from_index(index, group, dim=3):
"""
Given a Wyckoff position's index within a spacegroup, return its number
and letter e.g. '4a'
Args:
index: WP's index (0 is the general position)
group: an unorganized Wyckoff position array or Group object (preferred)
dim: the periodicity dimension of the symmetry group.
Returns:
the Wyckoff letter corresponding to the Wyckoff position (for example,
for position 4a, the function would return 'a')
"""
letters1 = letters
# See whether the group has an "o" Wyckoff position
checko = False
if type(group) == Group and group.dim == 0:
checko = True
elif dim == 0:
checko = True
if checko is True:
if len(group[-1]) == 1 and group[-1][0] == SymmOp.from_xyz_str("0,0,0"):
# o comes before a
letters1 = "o" + letters
length = len(group)
return letters1[length - 1 - index]
[docs]
def index_from_letter(letter, group, dim=3):
"""
Given the Wyckoff letter, returns the index of a Wyckoff position.
Args:
letter: The wyckoff letter
group: an unorganized Wyckoff position array or Group object (preferred)
dim: the periodicity dimension of the symmetry group.
Returns:
a single index specifying the location of the Wyckoff position.
"""
letters1 = letters
# See whether the group has an "o" Wyckoff position
checko = False
if type(group) == Group and group.dim == 0:
checko = True
elif dim == 0:
checko = True
if checko is True:
if len(group[-1]) == 1 and group[-1][0] == SymmOp.from_xyz_str("0,0,0"):
# o comes before a
letters1 = "o" + letters
length = len(group)
return length - 1 - letters1.index(letter)
[docs]
def jk_from_i(i, olist):
"""
Given an organized list (Wyckoff positions or orientations), determine the
two indices which correspond to a single index for an unorganized list.
Used mainly for organized Wyckoff position lists, but can be used for other
lists organized in a similar way
Args:
i: a single index corresponding to the item's location in the
unorganized list
olist: the organized list
Returns:
[j, k]: two indices corresponding to the item's location in the
organized list
"""
num = -1
found = False
for j, a in enumerate(olist):
for k, b in enumerate(a):
num += 1
if num == i:
return [j, k]
return None
[docs]
def i_from_jk(j, k, olist):
"""
Inverse operation of jk_from_i: gives one list index from 2
Args:
j, k: indices corresponding to the location of an element in the
organized list
olist: the organized list of Wyckoff positions or molecular orientations
Returns:
i: one index corresponding to the item's location in the
unorganized list
"""
num = -1
for x, a in enumerate(olist):
for y, b in enumerate(a):
num += 1
if x == j and y == k:
return num
return None
[docs]
def ss_string_from_ops(ops, number, dim=3, complete=True):
"""
Print the Hermann-Mauguin symbol for a site symmetry group, using a list of
SymmOps as input. Note that the symbol does not necessarily refer to the
x,y,z axes. For information on reading these symbols, see:
http://en.wikipedia.org/wiki/Hermann-Mauguin_notation#Point_groups
Args:
ops: a list of SymmOp objects representing the site symmetry
number: International number of the symmetry group. Used to determine which
axes to show. For example, a 3-fold rotation in a cubic system is
written as ".3.", whereas a 3-fold rotation in a trigonal system is
written as "3.."
dim: the dimension of the crystal. Also used to determine notation type
complete: whether or not all symmetry operations in the group
are present. If False, we generate the rest
Returns:
a string representing the site symmetry (e.g., `2mm`)
"""
# TODO: Automatically detect which symm_type to use based on ops
# Determine which notation to use
symm_type = "high"
if dim == 3:
if number >= 1 and number <= 74:
# Triclinic, monoclinic, orthorhombic
symm_type = "low"
elif number >= 75 and number <= 194:
# Trigonal, Hexagonal, Tetragonal
symm_type = "medium"
elif number >= 195 and number <= 230:
# cubic
symm_type = "high"
if dim == 2:
if number >= 1 and number <= 48:
# Triclinic, monoclinic, orthorhombic
symm_type = "low"
elif number >= 49 and number <= 80:
# Trigonal, Hexagonal, Tetragonal
symm_type = "medium"
if dim == 1:
if number >= 1 and number <= 22:
# Triclinic, monoclinic, orthorhombic
symm_type = "low"
elif number >= 23 and number <= 75:
# Trigonal, Hexagonal, Tetragonal
symm_type = "medium"
# TODO: replace sg with number, add dim variable
# Return the symbol for a single axis
# Will be called later in the function
def get_symbol(opas, order, has_reflection):
# ops: a list of Symmetry operations about the axis
# order: highest order of any symmetry operation about the axis
# has_reflection: whether or not the axis has mirror symmetry
if has_reflection:
# rotations have priority
for opa in opas:
if opa.order == order and opa.type == "rotation":
return str(opa.rotation_order) + "/m"
for opa in opas:
if (
opa.order == order
and opa.type == "rotoinversion"
and opa.order != 2
):
return "-" + str(opa.rotation_order)
return "m"
elif has_reflection is False:
# rotoinversion has priority
for opa in opas:
if opa.order == order and opa.type == "rotoinversion":
return "-" + str(opa.rotation_order)
for opa in opas:
if opa.order == order and opa.type == "rotation":
return str(opa.rotation_order)
return "."
# Given a list of single-axis symbols, return the one with highest symmetry
# Will be called later in the function
def get_highest_symbol(symbols):
symbol_list = [
".",
"2",
"m",
"-2",
"2/m",
"3",
"4",
"-4",
"4/m",
"-3",
"6",
"-6",
"6/m",
]
max_index = 0
use_list = True
for j, symbol in enumerate(symbols):
if symbol in symbol_list:
i = symbol_list.index(symbol)
else:
use_list = False
num_str = "".join(c for c in symbol if c.isdigit())
i1 = int(num_str)
if "m" in symbol or "-" in symbol:
if i1 % 2 == 0:
i = i1
elif i1 % 2 == 1:
i = i1 * 2
else:
i = i1
if i > max_index:
max_j = j
max_index = i
if use_list is True:
return symbol_list[max_index]
else:
return symbols[max_j]
def are_symmetrically_equivalent(index1, index2):
"""
Return whether or not two axes are symmetrically equivalent
It is assumed that both axes possess the same symbol
Will be called within combine_axes
"""
axis1 = axes[index1]
axis2 = axes[index2]
condition1 = False
condition2 = False
# Check for an operation mapping one axis onto the other
for op in ops:
if condition1 is False or condition2 is False:
new1 = op.operate(axis1)
new2 = op.operate(axis2)
if np.isclose(abs(np.dot(new1, axis2)), 1):
condition1 = True
if np.isclose(abs(np.dot(new2, axis1)), 1):
condition2 = True
if condition1 is True and condition2 is True:
return True
else:
return False
def combine_axes(indices):
"""
Given a list of axis indices, return the combined symbol
Axes may or may not be symmetrically equivalent, but must be of the same
type (x/y/z, face-diagonal, body-diagonal)
Will be called for mid- and high-symmetry crystallographic point groups
"""
symbols = {}
for index in deepcopy(indices):
symbol = get_symbol(params[index], orders[index], reflections[index])
if symbol == ".":
indices.remove(index)
else:
symbols[index] = symbol
if len(indices) == 0:
return "."
# Remove redundant axes
for i in deepcopy(indices):
for j in deepcopy(indices):
if j > i:
if symbols[i] == symbols[j]:
if are_symmetrically_equivalent(i, j):
if j in indices:
indices.remove(j)
# Combine symbols for non-equivalent axes
new_symbols = []
for i in indices:
new_symbols.append(symbols[i])
symbol = ""
while new_symbols != []:
highest = get_highest_symbol(new_symbols)
symbol += highest
new_symbols.remove(highest)
if symbol == "":
printx("Error: could not combine site symmetry axes.", priority=1)
return
else:
return symbol
# Generate needed ops
if not complete:
ops = generate_full_symmops(ops, 1e-3)
# Get OperationAnalyzer object for all ops
opas = []
for op in ops:
opas.append(OperationAnalyzer(op))
# Store the symmetry of each axis
params = [[], [], [], [], [], [], [], [], [], [], [], [], []]
has_inversion = False
# Store possible symmetry axes for crystallographic point groups
axes = [
[1, 0, 0],
[0, 1, 0],
[0, 0, 1],
[1, 1, 0],
[0, 1, 1],
[1, 0, 1],
[1, -1, 0],
[0, 1, -1],
[1, 0, -1],
[1, 1, 1],
[-1, 1, 1],
[1, -1, 1],
[1, 1, -1],
]
for i, axis in enumerate(axes):
axes[i] = axis / np.linalg.norm(axis)
for opa in opas:
# Search for the primary rotation axis
if opa.type != "identity" and opa.type != "inversion":
found = False
for i, axis in enumerate(axes):
if np.isclose(abs(np.dot(opa.axis, axis)), 1):
found = True
params[i].append(opa)
# Store uncommon axes for trigonal and hexagonal lattices
if not found: #is False:
axes.append(opa.axis)
# Check that new axis is not symmetrically equivalent to others
unique = True
for i, axis in enumerate(axes):
if i != len(axes) - 1:
if are_symmetrically_equivalent(i, len(axes) - 1):
unique = False
if unique: # is True:
params.append([opa])
else: #if unique is False:
axes.pop()
elif opa.type == "inversion":
has_inversion = True
# Determine how many high-symmetry axes are present
n_axes = 0
# Store the order of each axis
orders = []
# Store whether or not each axis has reflection symmetry
reflections = []
for axis in params:
order = 1
high_symm = False
has_reflection = False
for opa in axis:
if opa.order >= 3:
high_symm = True
if opa.order > order:
order = opa.order
if opa.order == 2 and opa.type == "rotoinversion":
has_reflection = True
orders.append(order)
if high_symm: #== True:
n_axes += 1
reflections.append(has_reflection)
# Triclinic, monoclinic, orthorhombic
# Positions in symbol refer to x,y,z axes respectively
if symm_type == "low":
symbol = (
get_symbol(params[0], orders[0], reflections[0])
+ get_symbol(params[1], orders[1], reflections[1])
+ get_symbol(params[2], orders[2], reflections[2])
)
if symbol != "...":
return symbol
elif symbol == "...":
if has_inversion: #is True:
return "-1"
else:
return "1"
# Trigonal, Hexagonal, Tetragonal
elif symm_type == "medium":
# 1st symbol: z axis
s1 = get_symbol(params[2], orders[2], reflections[2])
# 2nd symbol: x or y axes (whichever have higher symmetry)
s2 = combine_axes([0, 1])
# 3rd symbol: face-diagonal axes (whichever have highest symmetry)
s3 = combine_axes(list(range(3, len(axes))))
symbol = s1 + " " + s2 + " " + s3
if symbol != ". . .":
return symbol
elif symbol == ". . .":
if has_inversion: #is True:
return "-1"
else:
return "1"
# Cubic
elif symm_type == "high":
pass
# 1st symbol: x, y, and/or z axes (whichever have highest symmetry)
s1 = combine_axes([0, 1, 2])
# 2nd symbol: body-diagonal axes (whichever has highest symmetry)
s2 = combine_axes([9, 10, 11, 12])
# 3rd symbol: face-diagonal axes (whichever have highest symmetry)
s3 = combine_axes([3, 4, 5, 6, 7, 8])
symbol = s1 + " " + s2 + " " + s3
if symbol != ". . .":
return symbol
elif symbol == ". . .":
if has_inversion: #is True:
return "-1"
else:
return "1"
else:
printx("Error: invalid spacegroup number", priority=1)
return
[docs]
def organized_wyckoffs(group):
"""
Takes a Group object or unorganized list of Wyckoff positions and returns
a 2D list of Wyckoff positions organized by multiplicity.
Args:
group: a pyxtal.symmetry.Group object
Returns:
a 2D list of Wyckoff_position objects if group is a Group object.
a 3D list of SymmOp objects if group is a 2D list of SymmOps
"""
if type(group) == Group:
wyckoffs = group.Wyckoff_positions
else:
wyckoffs = group
wyckoffs_organized = [[]] # 2D Array of WP's organized by multiplicity
old = len(wyckoffs[0])
for wp in wyckoffs:
mult = len(wp)
if mult != old:
wyckoffs_organized.append([])
old = mult
wyckoffs_organized[-1].append(wp)
return wyckoffs_organized
[docs]
def symmetry_element_from_axis(axis):
"""
Given an axis, returns a SymmOp representing a symmetry element on the axis.
For example, the symmetry element for the vector (0,0,2) would be (0,0,z).
Args:
axis: a 3-vector representing the symmetry element
Returns:
a SymmOp object of form (ax, bx, cx), (ay, by, cy), or (az, bz, cz)
"""
if len(axis) != 3:
return
# Vector must be non-zero
if axis.dot(axis) < 1e-6:
return
v = np.array(axis) / np.linalg.norm(axis)
# Find largest component (x, y, or z)
abs_vals = [abs(a) for a in v]
f1 = max(abs_vals)
index1 = list(abs_vals).index(f1)
# Initialize an affine matrix
m = np.eye(4)
m[:3] = [0.0, 0.0, 0.0, 0.0]
# Set values for affine matrix
m[:3, index1] = v
return SymmOp(m)
[docs]
def get_wyckoffs(num, organized=False, dim=3):
"""
Returns a list of Wyckoff positions for a given group. Has option to
organize the list based on multiplicity (this is used for
random_crystal.wyckoffs)
For an unorganized list:
- 1st index: index of WP in sg (0 is the WP with largest multiplicity)
- 2nd index: a SymmOp object in the WP
For an organized list:
- 1st index: specifies multiplicity (0 is the largest multiplicity)
- 2nd index: a WP within the group of equal multiplicity.
- 3nd index: a SymmOp object within the Wyckoff position
You may switch between organized and unorganized lists using the methods
i_from_jk and jk_from_i. For example, if a Wyckoff position is the [i]
entry in an unorganized list, it will be the [j][k] entry in an organized
list.
Args:
num: the international group number
dim: dimension [0, 1, 2, 3]
organized: whether or not to organize the list based on multiplicity
Returns:
a list of Wyckoff positions, each of which is a list of SymmOp's
"""
if dim == 3:
wyckoff_strings = eval(wyckoff_df["0"][num])
elif dim == 2:
wyckoff_strings = eval(layer_df["0"][num])
elif dim == 1:
wyckoff_strings = eval(rod_df["0"][num])
elif dim == 0:
wyckoff_strings = eval(point_df["0"][num])
wyckoffs = []
for x in wyckoff_strings:
wyckoffs.append([])
for y in x:
if dim == 0:
wyckoffs[-1].append(SymmOp(y))
else:
wyckoffs[-1].append(SymmOp.from_xyz_str(y))
if organized:
wyckoffs_organized = [[]] # 2D Array of WP's organized by multiplicity
old = len(wyckoffs[0])
for wp in wyckoffs:
mult = len(wp)
if mult != old:
wyckoffs_organized.append([])
old = mult
wyckoffs_organized[-1].append(wp)
return wyckoffs_organized
else:
return wyckoffs
[docs]
def get_wyckoff_symmetry(num, dim=3):
"""
Returns a list of site symmetry for a given group.
1st index: index of WP in sg (0 is the WP with largest multiplicity)
2nd index: a point within the WP
3rd index: a site symmetry SymmOp of the point
Args:
sg: the international spacegroup number
dim: 0, 1, 2, 3
Returns:
a 3d list of SymmOp objects representing the site symmetry of each
point in each Wyckoff position
"""
if dim == 3:
symmetry_strings = eval(wyckoff_symmetry_df["0"][num])
elif dim == 2:
symmetry_strings = eval(layer_symmetry_df["0"][num])
elif dim == 1:
symmetry_strings = eval(rod_symmetry_df["0"][num])
elif dim == 0:
symmetry_strings = eval(point_symmetry_df["0"][num])
symmetry = []
# Loop over Wyckoff positions
for x in symmetry_strings:
symmetry.append([])
# Loop over points in WP
for y in x:
symmetry[-1].append([])
# Loop over ops
for z in y:
if dim == 0:
op = SymmOp(z)
else:
op = SymmOp.from_xyz_str(z)
symmetry[-1][-1].append(op)
return symmetry
[docs]
def get_generators(num, dim=3):
"""
Returns a list of Wyckoff generators for a given group.
1st index: index of WP in sg (0 is the WP with largest multiplicity)
2nd index: a generator for the WP
This function is useful for rotating molecules based on Wyckoff position,
since special Wyckoff positions only encode positional information, but not
information about the orientation. The generators for each Wyckoff position
form a subset of the spacegroup's general Wyckoff position.
Args:
num: the international spacegroup number
dim: dimension
Returns:
a 2d list of symmop objects [[wp0], [wp1], ... ]
"""
generators = []
if dim == 3:
generator_strings = eval(wyckoff_generators_df["0"][num])
elif dim == 2:
generator_strings = eval(layer_generators_df["0"][num])
elif dim == 1:
generator_strings = eval(rod_generators_df["0"][num])
elif dim == 0:
generator_strings = eval(point_generators_df["0"][num])
# Loop over Wyckoff positions
for x in generator_strings:
generators.append([])
# Loop over ops
for y in x:
if dim > 0:
op = SymmOp.from_xyz_str(y)
else:
op = SymmOp(y)
generators[-1].append(op)
return generators
[docs]
def site_symm(point, gen_pos, tol=1e-3, lattice=np.eye(3), PBC=None):
"""
Given a point and a general Wyckoff position, return the list of symmetry
operations leaving the point (coordinate or SymmOp) invariant. The returned
SymmOps are a subset of the general position. The site symmetry can be used
for determining the Wyckoff position for a set of points, or for
determining the valid orientations of a molecule within a given Wyckoff
position.
Args:
point: a 1x3 coordinate or SymmOp object to find the symmetry of. If a
SymmOp is given, the returned symmetries must also preserve the
point's orientaion
gen_pos: the general position of the spacegroup. Can be a Wyckoff_position
object or list of SymmOp objects.
tol:
the numberical tolerance for determining equivalent positions and
orientations.
lattice:
a 3x3 matrix representing the lattice vectors of the unit cell
PBC: A periodic boundary condition list, 1 means periodic, 0 means not periodic.
Ex: [1,1,1] -> full 3d periodicity, [0,0,1] -> periodicity along the z axis.
Need not be defined here if gen_pos is a Wyckoff_position object.
Returns:
a list of SymmOp objects which leave the given point invariant
"""
if PBC is None:
if type(gen_pos) == Wyckoff_position:
PBC = gen_pos.PBC
else:
PBC = [1, 1, 1]
# Convert point into a SymmOp
if type(point) != SymmOp:
point = SymmOp.from_rotation_and_translation(
[[0, 0, 0], [0, 0, 0], [0, 0, 0]], np.array(point)
)
symmetry = []
for op in gen_pos:
is_symmetry = True
# Calculate the effect of applying op to point
difference = SymmOp((op * point).affine_matrix - point.affine_matrix)
# Check that the rotation matrix is unaltered by op
if not np.allclose(
difference.rotation_matrix, np.zeros((3, 3)), rtol=1e-3, atol=1e-3
):
is_symmetry = False
# Check that the displacement is less than tol
displacement = difference.translation_vector
if distance(displacement, lattice, PBC=PBC) > tol:
is_symmetry = False
if is_symmetry:
"""
The actual site symmetry's translation vector may vary from op by
a factor of +1 or -1 (especially when op contains +-1/2).
We record this to distinguish between special Wyckoff positions.
As an example, consider the point (-x+1/2,-x,x+1/2) in position 16c
of space group Ia-3(206). The site symmetry includes the operations
(-z+1,x-1/2,-y+1/2) and (y+1/2,-z+1/2,-x+1). These operations are
not listed in the general position, but correspond to the operations
(-z,x+1/2,-y+1/2) and (y+1/2,-z+1/2,-x), respectively, just shifted
by (+1,-1,0) and (0,0,+1), respectively.
"""
el = SymmOp.from_rotation_and_translation(
op.rotation_matrix, op.translation_vector - np.round(displacement)
)
symmetry.append(el)
return symmetry
[docs]
def check_wyckoff_position(points, group, tol=1e-3):
"""
Given a list of points, returns a single index of a matching Wyckoff
position in the space group. Checks the site symmetry of each supplied
point against the site symmetry for each point in the Wyckoff position.
Also returns a point which can be used to generate the rest using the
Wyckoff position operators.
Args:
points: a list of 3d coordinates or SymmOps to check
group: a Group object
tol: the max distance between equivalent points
Returns:
index, p: index is a single index for the Wyckoff position within
the sg. If no matching WP is found, returns False. point is a
coordinate taken from the list points. When plugged into the Wyckoff
position, it will generate all the other points.
"""
points = np.array(points)
wyckoffs = group.wyckoffs
w_symm_all = group.w_symm
PBC = group.PBC
# new method
# Store the squared distance tolerance
t = tol ** 2
# Loop over Wyckoff positions
for i, wp in enumerate(wyckoffs):
# Check that length of points and wp are equal
if len(wp) != len(points):
continue
failed = False
# Search for a generating point
for p in points:
failed = False
# Check that point works as x,y,z value for wp
xyz = filtered_coords_euclidean(wp[0].operate(p) - p, PBC=PBC)
if xyz.dot(xyz) > t:
continue
# Calculate distances between original and generated points
pw = np.array([op.operate(p) for op in wp])
dw = distance_matrix(points, pw, None, PBC=PBC, metric="sqeuclidean")
# Check each row for a zero
for row in dw:
num = (row < t).sum()
if num < 1:
failed = True
break
if failed:
continue
# Check each column for a zero
for column in dw.T:
num = (column < t).sum()
if num < 1:
failed = True
break
# Calculate distance between original and generated points
ps = np.array([op.operate(p) for op in w_symm_all[i][0]])
ds = distance_matrix([p], ps, None, PBC=PBC, metric="sqeuclidean")
# Check whether any generated points are too far away
num = (ds > t).sum()
if num > 0:
failed = True
if failed:
continue
return i, p
return False, None
[docs]
def get_symbol_and_number(input_group, dim=3):
"""
Function for quick conversion between symbols and numbers.
Args:
input_group: the group symbol or international number
dim: the periodic dimension of the group
"""
keys = {
3: "space_group",
2: "layer_group",
1: "rod_group",
0: "point_group",
}
found = False
lists = symbols[keys[dim]]
number = None
symbol = None
if dim not in [0, 1, 2, 3]:
raise ValueError("Dimension ({:d}) should in [0, 1, 2, 3] ".format(dim))
if type(input_group) == str:
for i, _symbol in enumerate(lists):
if _symbol == input_group:
number = i + 1
symbol = input_group
return symbol, number
msg = "({:s}) not found in {:s} ".format(input_group, keys[dim])
raise ValueError(msg)
else:
valid, msg = check_symmetry_and_dim(input_group, dim)
if not valid:
raise ValueError(msg)
else:
number = input_group
symbol = lists[number - 1]
return symbol, number
[docs]
def check_symmetry_and_dim(number, dim=3):
"""
check if it is a valid number for the given symmetry
Args:
number: int
dim: 0, 1, 2, 3
"""
valid = True
msg = 'This is a valid group number'
numbers = [56, 75, 80, 230]
if dim not in [0, 1, 2, 3]:
msg = "invalid dimension {:d}".format()
valid = False
else:
max_num = numbers[dim]
if number not in range(1, max_num+1):
valid = False
msg = "invalid symmetry group {:d}".format(number)
msg += " in dimension {:d}".format(dim)
return valid, msg
[docs]
def get_pbc_and_lattice(number, dim):
if dim == 3:
PBC = [1, 1, 1]
if number <= 2:
lattice_type = "triclinic"
elif number <= 15:
lattice_type = "monoclinic"
elif number <= 74:
lattice_type = "orthorhombic"
elif number <= 142:
lattice_type = "tetragonal"
elif number <= 194:
lattice_type = "hexagonal"
elif number <= 230:
lattice_type = "cubic"
elif dim == 2:
PBC = [1, 1, 0]
if number <= 2:
lattice_type = "triclinic"
elif number <= 18:
lattice_type = "monoclinic"
elif number <= 48:
lattice_type = "orthorhombic"
elif number <= 64:
lattice_type = "tetragonal"
elif number <= 80:
lattice_type = "hexagonal"
elif dim == 1:
PBC = [0, 0, 1]
if number <= 2:
lattice_type = "triclinic"
elif number <= 12:
lattice_type = "monoclinic"
elif number <= 22:
lattice_type = "orthorhombic"
elif number <= 41:
lattice_type = "tetragonal"
elif number <= 75:
lattice_type = "hexagonal"
elif dim == 0:
PBC = [0, 0, 0]
# "C1", "Ci", "D2", "D2h", "T", "Th",
# "O", "Td", "Oh", "I", "Ih",
if number in [1, 2, 6, 8, 28, 29, 30, 31, 32, 55, 56]:
lattice_type = "spherical"
else:
lattice_type = "ellipsoidal"
return PBC, lattice_type
[docs]
def search_cloest_wp(G, wp, op, pos):
"""
For a given position, search for the cloest wp which satisfies the desired
symmetry relation, e.g., for pos (0.1, 0.12, 0.2) and op (x, x, z) the
closest match is (0.11, 0.11, 0.2)
Args:
G: space group number
wp: Wyckoff object
op: symmetry operation belonging to wp
pos: initial xyz position
Return:
pos1: the position that matchs symmetry operation
"""
#G = Group(wp.number)
if np.linalg.matrix_rank(op.rotation_matrix) == 0:
# fixed point (e.g, 1/2, 1/2, 1/2)
return op.translation_vector
elif np.linalg.matrix_rank(op.rotation_matrix) == 3:
# fully independent, e.g., (x,y,z), (-x,y,z)
return pos
else:
# check if this is already matched
wp0 = G[0]
coords = wp.search_all_generators(pos, wp0)
if len(coords)>0:
diffs = []
for coord in coords:
tmp = op.operate(coord)
diff1 = tmp - pos
diff1 -= np.round(diff1)
dist = np.linalg.norm(diff1)
if dist < 1e-3:
return tmp
else:
diffs.append(dist)
minID = np.argmin(diffs)
return op.operate(coords[minID])
# if not match, search for the closest solution
else:
# extract all possible xyzs
all_xyz = apply_ops(pos, wp0)[1:]
dists = all_xyz - pos
dists -= np.round(dists)
ds = np.linalg.norm(dists, axis=1)
ids = np.argsort(ds)
for id in ids:
d = all_xyz[id] - pos
d -= np.round(d)
res = pos + d/2
if wp.search_generator(res, wp0) is not None:
#print(ds[id], pos, res)
return res
return op.operate(pos)
[docs]
def get_point_group(number):
"""
Parse the point group symmetry info from space group. According to
http://img.chem.ucl.ac.uk/sgp/misc/pointgrp.htm, among 32(230) point(space)
groups, there are
- 10(68) polar groups,
- 11(92) centrosymmetric groups,
- 11(65) enantiomorphic groups
Args:
number: space group number
Return:
point group symbol
polar: 1, 2, m, mm2, 3, 3m, 4, 4mm, 6, 6mm
centrosymmetry: -1, 2/m, mmm, 4/m, 4/mmm, -3, -3m, 6/m, 6/mmm, m-3, m-3m
enantiomorphic: 1, 2, 222, 4, 422, 3, 32, 6, 622, 23, 432
"""
if number == 1:
return '1', 1, True, False, True
elif number == 2:
return '-1', 2, False, True, False
elif 3 <= number <= 5:
return '2', 3, True, False, True
elif 6 <= number <= 9:
return 'm', 4, True, False, False
elif 10 <= number <= 15:
return '2/m', 5, False, True, False
elif 16 <= number <= 24:
return '222', 6, False, False, True
elif 25 <= number <= 46:
return 'mm2', 7, True, False, False
elif 47 <= number <= 74:
return 'mmm', 8, False, True, False
elif 75 <= number <= 80:
return '4', 9, True, False, True
elif 81 <= number <= 82:
return '-4', 10, False, False, False
elif 83 <= number <= 88:
return '4/m', 11, False, True, False
elif 89 <= number <= 98:
return '422', 12, False, False, True
elif 99 <= number <= 110:
return '4mm', 13, True, False, False
elif 111 <= number <= 122:
return '-42m', 14, False, False, False
elif 123 <= number <= 142:
return '4/mmm', 15, False, True, False
elif 143 <= number <= 146:
return '3', 16, True, False, True
elif 147 <= number <= 148:
return '-3', 17, False, True, False
elif 149 <= number <= 155:
return '32', 18, False, False, True
elif 156 <= number <= 161:
return '3m', 19, True, False, False
elif 162 <= number <= 167:
return '-3m', 20, False, True, False
elif 168 <= number <= 173:
return '6', 21, True, False, True
elif number == 174:
return '-6', 22, False, False, False
elif 175 <= number <= 176:
return '6/m', 23, False, True, False
elif 177 <= number <= 182:
return '622', 24, False, False, True
elif 183 <= number <= 186:
return '6mm', 25, True, False, False
elif 187 <= number <= 190:
return '-62m', 26, False, False, False
elif 191 <= number <= 194:
return '6/mmm', 27, False, True, False
elif 195 <= number <= 199:
return '23', 28, False, False, True
elif 200 <= number <= 206:
return 'm-3', 29, False, True, False
elif 207 <= number <= 214:
return '432', 30, False, False, True
elif 215 <= number <= 220:
return '-43m', 31, False, False, False
elif 221 <= number <= 230:
return 'm-3m', 32, False, True, False
[docs]
def get_close_packed_groups(pg):
"""
List the close packed groups based on the molcular symmetry
Compiled from AIK Book, Table 2 P34
Args:
pg: point group symbol
Return:
list of space group numbers
"""
if pg == '1':
return [1, 2, 4, 14, 19, 29, 33, 51, 54, 61, 62]
elif pg == '2':
return [1, 15, 18, 60]
elif pg == 'm':
return [1, 26, 36, 63, 64]
elif pg == 'I':
return [1, 2, 14, 15, 61]
elif pg == 'mm':
return [42, 51, 59]
elif pg == '2/m':
return [12, 54, 64]
elif pg == '222':
return [21, 22, 23, 68]
elif pg == 'mmm':
return [65, 69, 71]
[docs]
def para2ferro(pg):
"""
88 potential paraelectric-to-ferroelectric phase transitions
https://journals.aps.org/prb/abstract/10.1103/PhysRevB.2.754
https://pubs.rsc.org/en/content/articlelanding/2016/cs/c5cs00308c
Args:
paraelectric point group
Returns:
list of ferroelectric point groups
"""
#Triclinic: 1
if pg == '-1': #2
return ['1']
#Monoclinic: 5
elif pg in ['2', 'm']: #2
return '1'
elif pg == '2/m': #3
return ['1', 'm', '2']
#Orthorhombic: #7
elif pg == '222': #2
return ['1', '2']
elif pg == 'mm2': #2
return ['1', 'm']
elif pg == 'mmm': #3
return ['1', 'm', 'mm2']
#Tetragonal: 20
elif pg == '4': #1
return ['1']
elif pg == '-4': #2
return ['1', '2']
elif pg == '4/m': #3
return ['1', '2', '4']
elif pg == '422': #3
#return ['1', '2(s)', '4']
return ['1', '2', '4']
elif pg == '4mm': #2
return ['1', 'm']
elif pg == '-42m': #4
#return ['1', '2(s)', 'm', 'mm2']
return ['1', '2', 'm', 'mm2']
elif pg == '4/mmm': #5
#return ['1', 'm(s)', 'm(p)', 'mm2(s)', '4mm']
return ['1', 'm', 'mm2', '4mm']
#Trigonal: 12
elif pg == '3': #1
return ['1']
elif pg == '-3': #2
return ['1', '3']
elif pg == '32': #3
return ['1', '2', '3']
elif pg == '3m': #2
return ['1', 'm']
elif pg == '-3m': #4
return ['1', '2', 'm', '3m']
#Hexagonal: 22
elif pg == '6': #1
return ['1']
elif pg == '-6': #3
return ['1', 'm', '3']
elif pg == '6/m': #3
return ['1', 'm', '6']
elif pg == '622': #3
#return ['1', '2(s)', '6']
return ['1', '2', '6']
elif pg == '6mm': #2
return ['1', '2']
elif pg in ['-62m', '-6m2']: #5
#return ['1', 'm(s)', 'm(p)', 'mm2', '3m']
return ['1', 'm', 'mm2', '3m']
elif pg == '6/mmm': #5
#return ['1', 'm(s)', 'm(p)', 'mm2(s)', '6mm']
return ['1', 'm', 'mm2', '6mm']
#Cubic: 21
elif pg == '23': #3
return ['1', '2', '3']
elif pg == 'm-3': #4
return ['1', 'm', 'mm2', '3']
elif pg == '432': #4
#return ['1', '2(s)', '4', '3']
return ['1', '2', '4', '3']
elif pg == '-43m': #4
return ['1', 'm', 'mm2', '3m']
elif pg == 'm-3m': #6
#return ['1', 'm(s)', 'm(p)', 'mm2', '4mm', '3m']
return ['1', 'm', 'mm2', '4mm', '3m']
[docs]
def get_all_polar_space_groups():
ps, nps = [], []
for i in range(1,231):
g = Group(i, quick=True)
if g.polar:
ps.append(i)
else:
nps.append(i)
return ps, nps
[docs]
def abc2matrix(abc):
"""
convert the abc string representation to matrix
Args:
abc: string like 'a, b, c' or 'a+c, b, c' or 'a+1/4, b+1/4, c'
Returns:
4*4 affine matrix
"""
rot_matrix = np.zeros((3, 3))
trans = np.zeros(3)
toks = abc.strip().replace(" ", "").lower().split(",")
re_rot = re.compile(r"([+-]?)([\d\.]*)/?([\d\.]*)([a-c])")
re_trans = re.compile(r"([+-]?)([\d\.]+)/?([\d\.]*)(?![a-c])")
for i, tok in enumerate(toks):
# build the rotation matrix
for m in re_rot.finditer(tok):
factor = -1.0 if m.group(1) == "-" else 1.0
if m.group(2) != "":
if m.group(3) != "":
factor *= float(m.group(2)) / float(m.group(3))
else:
factor *= float(m.group(2))
j = ord(m.group(4)) - 97
try:
rot_matrix[i, j] = factor
except:
print(abc); import sys; sys.exit()
# build the translation vector
for m in re_trans.finditer(tok):
factor = -1 if m.group(1) == "-" else 1
if m.group(3) != "":
num = float(m.group(2)) / float(m.group(3))
else:
num = float(m.group(2))
trans[i] = num * factor
return (rot_matrix, trans)
[docs]
def get_symmetry_from_ops(ops, tol=1e-5):
"""
get the hall number from symmetry operations.
Args:
ops: tuple of (rotation, translation) or list of strings
tol: tolerance
"""
from spglib import get_hall_number_from_symmetry
if isinstance(ops[0], str):
ops = [SymmOp.from_xyz_str(op) for op in ops]
rot = [op.rotation_matrix for op in ops]
tran = [op.translation_vector for op in ops]
hall_number = get_hall_number_from_symmetry(rot, tran, tol)
spg_number = hall_table['Spg_num'][hall_number-1]
return hall_number, spg_number
[docs]
def identity_ops(op):
"""
check if the operation is the identity.
"""
(rot, trans) = op
if np.allclose(rot, np.eye(3)) and np.sum(np.abs(trans))<1e-3:
return True
else:
return False
[docs]
def trim_ops(ops):
"""
Convert the operation to the simplest form. e.g.,
- 'x+1/8, y+1/8, z+1/8' -> 'x, y, z'
- '1/8 y+1/8 -y+1/8' -> '1/8, y, -y+1/4'
Args:
ops: a list of symmetry opertions.
"""
def in_base(op, base):
for b in base:
if abs(op-b[:3]).sum()<1e-2 or abs(op+b[:3]).sum()<1e-2:
return b
return None
ops1 = []
for i, op in enumerate(ops):
rot = op.rotation_matrix
tran = op.translation_vector
if i == 0:
base = [] # e.g., [1, 0, 0, 1/2] means (x+1/2)
for i in range(3):
tmp = rot[i, :]
if np.linalg.norm(tmp) > 1e-3:
b = in_base(tmp, base)
if b is None:
_base = np.zeros(4)
_base[:3] = tmp
_base[3] = tran[i]
base.append(_base)
tran[i] = 0
else:
for j in range(3):
if abs(b[j]) > 0:
coef = tmp[j]/b[j]
break
tran[i] -= coef*b[3]
else:
for i in range(3):
tmp = rot[i, :]
if np.linalg.norm(tmp) > 1e-3:
b = in_base(tmp, base)
for j in range(3):
if abs(b[j]) > 0:
coef = tmp[j]/b[j]
break
tran[i] -= coef*b[3]
ops1.append(SymmOp.from_rotation_and_translation(rot, tran))
return ops
#op = SymmOp.from_xyz_str('y+1/8, -y+1/8, 0')
#op = SymmOp.from_xyz_str('1/8, y+1/8, -y+1/8')
#op = SymmOp.from_xyz_str(['x+1/8,x+1/8,z+1/8', '')
#print(trim_op(op).as_xyz_str())
#from_symops(ops, group=None, permutation=True)