Source code for pymatgen.symmetry.groups

# coding: utf-8
# Copyright (c) Pymatgen Development Team.
# Distributed under the terms of the MIT License.

"""
Defines SymmetryGroup parent class and PointGroup and SpaceGroup classes.
Shyue Ping Ong thanks Marc De Graef for his generous sharing of his
SpaceGroup data as published in his textbook "Structure of Materials".
"""

import os
from itertools import product
from fractions import Fraction
from abc import ABCMeta, abstractmethod
from collections.abc import Sequence
import numpy as np
import warnings
import re
from monty.serialization import loadfn

from pymatgen.core.operations import SymmOp
from monty.design_patterns import cached_class


SYMM_DATA = None


def _get_symm_data(name):
    global SYMM_DATA
    if SYMM_DATA is None:
        SYMM_DATA = loadfn(os.path.join(os.path.dirname(__file__),
                                        "symm_data.json"))
    return SYMM_DATA[name]


[docs]class SymmetryGroup(Sequence, metaclass=ABCMeta): """ Abstract class representation a symmetry group. """ @property @abstractmethod def symmetry_ops(self): """ :return: List of symmetry operations """ pass def __contains__(self, item): for i in self.symmetry_ops: if np.allclose(i.affine_matrix, item.affine_matrix): return True return False def __hash__(self): return self.__len__() def __getitem__(self, item): return self.symmetry_ops[item] def __len__(self): return len(self.symmetry_ops)
[docs] def is_subgroup(self, supergroup): """ True if this group is a subgroup of the supplied group. Args: supergroup (SymmetryGroup): Supergroup to test. Returns: True if this group is a subgroup of the supplied group. """ warnings.warn("This is not fully functional. Only trivial subsets are tested right now. ") return set(self.symmetry_ops).issubset(supergroup.symmetry_ops)
[docs] def is_supergroup(self, subgroup): """ True if this group is a supergroup of the supplied group. Args: subgroup (SymmetryGroup): Subgroup to test. Returns: True if this group is a supergroup of the supplied group. """ warnings.warn("This is not fully functional. Only trivial subsets are " "tested right now. ") return set(subgroup.symmetry_ops).issubset(self.symmetry_ops)
[docs]@cached_class class PointGroup(SymmetryGroup): """ Class representing a Point Group, with generators and symmetry operations. .. attribute:: symbol Full International or Hermann-Mauguin Symbol. .. attribute:: generators List of generator matrices. Note that 3x3 matrices are used for Point Groups. .. attribute:: symmetry_ops Full set of symmetry operations as matrices. """ def __init__(self, int_symbol): """ Initializes a Point Group from its international symbol. Args: int_symbol (str): International or Hermann-Mauguin Symbol. """ self.symbol = int_symbol self.generators = [_get_symm_data("generator_matrices")[c] for c in _get_symm_data("point_group_encoding")[int_symbol]] self._symmetry_ops = set([SymmOp.from_rotation_and_translation(m) for m in self._generate_full_symmetry_ops()]) self.order = len(self._symmetry_ops) @property def symmetry_ops(self): """ :return: List of symmetry operations for SpaceGroup """ return self._symmetry_ops def _generate_full_symmetry_ops(self): symm_ops = list(self.generators) new_ops = self.generators while len(new_ops) > 0: gen_ops = [] for g1, g2 in product(new_ops, symm_ops): op = np.dot(g1, g2) if not in_array_list(symm_ops, op): gen_ops.append(op) symm_ops.append(op) new_ops = gen_ops return symm_ops def get_orbit(self, p, tol=1e-5): """ Returns the orbit for a point. Args: p: Point as a 3x1 array. tol: Tolerance for determining if sites are the same. 1e-5 should be sufficient for most purposes. Set to 0 for exact matching (and also needed for symbolic orbits). Returns: ([array]) Orbit for point. """ orbit = [] for o in self.symmetry_ops: pp = o.operate(p) if not in_array_list(orbit, pp, tol=tol): orbit.append(pp) return orbit
[docs]@cached_class class SpaceGroup(SymmetryGroup): """ Class representing a SpaceGroup. .. attribute:: symbol Full International or Hermann-Mauguin Symbol. .. attribute:: int_number International number .. attribute:: generators List of generator matrices. Note that 4x4 matrices are used for Space Groups. .. attribute:: order Order of Space Group """ SYMM_OPS = loadfn(os.path.join(os.path.dirname(__file__), "symm_ops.json")) SG_SYMBOLS = set(_get_symm_data("space_group_encoding").keys()) for op in SYMM_OPS: op["hermann_mauguin"] = re.sub(r" ", "", op["hermann_mauguin"]) op["universal_h_m"] = re.sub(r" ", "", op["universal_h_m"]) SG_SYMBOLS.add(op["hermann_mauguin"]) SG_SYMBOLS.add(op["universal_h_m"]) gen_matrices = _get_symm_data("generator_matrices") # POINT_GROUP_ENC = SYMM_DATA["point_group_encoding"] sgencoding = _get_symm_data("space_group_encoding") abbrev_sg_mapping = _get_symm_data("abbreviated_spacegroup_symbols") translations = {k: Fraction(v) for k, v in _get_symm_data( "translations").items()} full_sg_mapping = { v["full_symbol"]: k for k, v in _get_symm_data("space_group_encoding").items()} def __init__(self, int_symbol): """ Initializes a Space Group from its full or abbreviated international symbol. Only standard settings are supported. Args: int_symbol (str): Full International (e.g., "P2/m2/m2/m") or Hermann-Mauguin Symbol ("Pmmm") or abbreviated symbol. The notation is a LaTeX-like string, with screw axes being represented by an underscore. For example, "P6_3/mmc". Alternative settings can be access by adding a ":identifier". For example, the hexagonal setting for rhombohedral cells can be accessed by adding a ":H", e.g., "R-3m:H". To find out all possible settings for a spacegroup, use the get_settings classmethod. Alternative origin choices can be indicated by a translation vector, e.g., 'Fm-3m(a-1/4,b-1/4,c-1/4)'. """ int_symbol = re.sub(r" ", "", int_symbol) if int_symbol in SpaceGroup.abbrev_sg_mapping: int_symbol = SpaceGroup.abbrev_sg_mapping[int_symbol] elif int_symbol in SpaceGroup.full_sg_mapping: int_symbol = SpaceGroup.full_sg_mapping[int_symbol] for spg in SpaceGroup.SYMM_OPS: if int_symbol in [spg["hermann_mauguin"], spg["universal_h_m"]]: ops = [SymmOp.from_xyz_string(s) for s in spg["symops"]] self.symbol = re.sub(r":", "", re.sub(r" ", "", spg["universal_h_m"])) if int_symbol in SpaceGroup.sgencoding: self.full_symbol = SpaceGroup.sgencoding[int_symbol]["full_symbol"] self.point_group = SpaceGroup.sgencoding[int_symbol]["point_group"] else: self.full_symbol = re.sub(r" ", "", spg["universal_h_m"]) self.point_group = spg["schoenflies"] self.int_number = spg["number"] self.order = len(ops) self._symmetry_ops = ops break else: if int_symbol not in SpaceGroup.sgencoding: raise ValueError("Bad international symbol %s" % int_symbol) data = SpaceGroup.sgencoding[int_symbol] self.symbol = int_symbol # TODO: Support different origin choices. enc = list(data["enc"]) inversion = int(enc.pop(0)) ngen = int(enc.pop(0)) symm_ops = [np.eye(4)] if inversion: symm_ops.append(np.array( [[-1, 0, 0, 0], [0, -1, 0, 0], [0, 0, -1, 0], [0, 0, 0, 1]])) for i in range(ngen): m = np.eye(4) m[:3, :3] = SpaceGroup.gen_matrices[enc.pop(0)] m[0, 3] = SpaceGroup.translations[enc.pop(0)] m[1, 3] = SpaceGroup.translations[enc.pop(0)] m[2, 3] = SpaceGroup.translations[enc.pop(0)] symm_ops.append(m) self.generators = symm_ops self.full_symbol = data["full_symbol"] self.point_group = data["point_group"] self.int_number = data["int_number"] self.order = data["order"] self._symmetry_ops = None def _generate_full_symmetry_ops(self): symm_ops = np.array(self.generators) for op in symm_ops: op[0:3, 3] = np.mod(op[0:3, 3], 1) new_ops = symm_ops while len(new_ops) > 0 and len(symm_ops) < self.order: gen_ops = [] for g in new_ops: temp_ops = np.einsum('ijk,kl', symm_ops, g) for op in temp_ops: op[0:3, 3] = np.mod(op[0:3, 3], 1) ind = np.where(np.abs(1 - op[0:3, 3]) < 1e-5) op[ind, 3] = 0 if not in_array_list(symm_ops, op): gen_ops.append(op) symm_ops = np.append(symm_ops, [op], axis=0) new_ops = gen_ops assert len(symm_ops) == self.order return symm_ops @classmethod def get_settings(cls, int_symbol): """ Returns all the settings for a particular international symbol. Args: int_symbol (str): Full International (e.g., "P2/m2/m2/m") or Hermann-Mauguin Symbol ("Pmmm") or abbreviated symbol. The notation is a LaTeX-like string, with screw axes being represented by an underscore. For example, "P6_3/mmc". """ symbols = [] if int_symbol in SpaceGroup.abbrev_sg_mapping: symbols.append(SpaceGroup.abbrev_sg_mapping[int_symbol]) int_number = SpaceGroup.sgencoding[int_symbol]["int_number"] elif int_symbol in SpaceGroup.full_sg_mapping: symbols.append(SpaceGroup.full_sg_mapping[int_symbol]) int_number = SpaceGroup.sgencoding[int_symbol]["int_number"] else: for spg in SpaceGroup.SYMM_OPS: if int_symbol in [re.split(r"\(|:", spg["hermann_mauguin"])[0], re.split(r"\(|:", spg["universal_h_m"])[0]]: int_number = spg["number"] break for spg in SpaceGroup.SYMM_OPS: if int_number == spg["number"]: symbols.append(spg["hermann_mauguin"]) symbols.append(spg["universal_h_m"]) return set(symbols) @property def symmetry_ops(self): """ Full set of symmetry operations as matrices. Lazily initialized as generation sometimes takes a bit of time. """ if self._symmetry_ops is None: self._symmetry_ops = [ SymmOp(m) for m in self._generate_full_symmetry_ops()] return self._symmetry_ops def get_orbit(self, p, tol=1e-5): """ Returns the orbit for a point. Args: p: Point as a 3x1 array. tol: Tolerance for determining if sites are the same. 1e-5 should be sufficient for most purposes. Set to 0 for exact matching (and also needed for symbolic orbits). Returns: ([array]) Orbit for point. """ orbit = [] for o in self.symmetry_ops: pp = o.operate(p) pp = np.mod(np.round(pp, decimals=10), 1) if not in_array_list(orbit, pp, tol=tol): orbit.append(pp) return orbit def is_compatible(self, lattice, tol=1e-5, angle_tol=5): """ Checks whether a particular lattice is compatible with the *conventional* unit cell. Args: lattice (Lattice): A Lattice. tol (float): The tolerance to check for equality of lengths. angle_tol (float): The tolerance to check for equality of angles in degrees. """ abc = lattice.lengths angles = lattice.angles crys_system = self.crystal_system def check(param, ref, tolerance): return all([abs(i - j) < tolerance for i, j in zip(param, ref) if j is not None]) if crys_system == "cubic": a = abc[0] return check(abc, [a, a, a], tol) and check(angles, [90, 90, 90], angle_tol) elif crys_system == "hexagonal" or ( crys_system == "trigonal" and ( self.symbol.endswith("H") or self.int_number in [143, 144, 145, 147, 149, 150, 151, 152, 153, 154, 156, 157, 158, 159, 162, 163, 164, 165])): a = abc[0] return check(abc, [a, a, None], tol) and check(angles, [90, 90, 120], angle_tol) elif crys_system == "trigonal": a = abc[0] alpha = angles[0] return check(abc, [a, a, a], tol) and check(angles, [alpha, alpha, alpha], angle_tol) elif crys_system == "tetragonal": a = abc[0] return check(abc, [a, a, None], tol) and check(angles, [90, 90, 90], angle_tol) elif crys_system == "orthorhombic": return check(angles, [90, 90, 90], angle_tol) elif crys_system == "monoclinic": return check(angles, [90, None, 90], angle_tol) return True @property def crystal_system(self): """ :return: Crystal system for space group. """ i = self.int_number if i <= 2: return "triclinic" elif i <= 15: return "monoclinic" elif i <= 74: return "orthorhombic" elif i <= 142: return "tetragonal" elif i <= 167: return "trigonal" elif i <= 194: return "hexagonal" else: return "cubic" def is_subgroup(self, supergroup): """ True if this space group is a subgroup of the supplied group. Args: group (Spacegroup): Supergroup to test. Returns: True if this space group is a subgroup of the supplied group. """ if len(supergroup.symmetry_ops) < len(self.symmetry_ops): return False groups = [[supergroup.int_number]] all_groups = [supergroup.int_number] max_subgroups = {int(k): v for k, v in _get_symm_data("maximal_subgroups").items()} while True: new_sub_groups = set() for i in groups[-1]: new_sub_groups.update([j for j in max_subgroups[i] if j not in all_groups]) if self.int_number in new_sub_groups: return True elif len(new_sub_groups) == 0: break else: groups.append(new_sub_groups) all_groups.extend(new_sub_groups) return False def is_supergroup(self, subgroup): """ True if this space group is a supergroup of the supplied group. Args: subgroup (Spacegroup): Subgroup to test. Returns: True if this space group is a supergroup of the supplied group. """ return subgroup.is_subgroup(self) @classmethod def from_int_number(cls, int_number, hexagonal=True): """ Obtains a SpaceGroup from its international number. Args: int_number (int): International number. hexagonal (bool): For rhombohedral groups, whether to return the hexagonal setting (default) or rhombohedral setting. Returns: (SpaceGroup) """ sym = sg_symbol_from_int_number(int_number, hexagonal=hexagonal) if not hexagonal and int_number in [146, 148, 155, 160, 161, 166, 167]: sym += ':R' return SpaceGroup(sym) def __str__(self): return "Spacegroup %s with international number %d and order %d" % ( self.symbol, self.int_number, len(self.symmetry_ops))
[docs]def sg_symbol_from_int_number(int_number, hexagonal=True): """ Obtains a SpaceGroup name from its international number. Args: int_number (int): International number. hexagonal (bool): For rhombohedral groups, whether to return the hexagonal setting (default) or rhombohedral setting. Returns: (str) Spacegroup symbol """ syms = [] for n, v in _get_symm_data("space_group_encoding").items(): if v["int_number"] == int_number: syms.append(n) if len(syms) == 0: raise ValueError("Invalid international number!") if len(syms) == 2: for sym in syms: if "e" in sym: return sym if hexagonal: syms = list(filter(lambda s: s.endswith("H"), syms)) else: syms = list(filter(lambda s: not s.endswith("H"), syms)) return syms.pop()
[docs]def in_array_list(array_list, a, tol=1e-5): """ Extremely efficient nd-array comparison using numpy's broadcasting. This function checks if a particular array a, is present in a list of arrays. It works for arrays of any size, e.g., even matrix searches. Args: array_list ([array]): A list of arrays to compare to. a (array): The test array for comparison. tol (float): The tolerance. Defaults to 1e-5. If 0, an exact match is done. Returns: (bool) """ if len(array_list) == 0: return False axes = tuple(range(1, a.ndim + 1)) if not tol: return np.any(np.all(np.equal(array_list, a[None, :]), axes)) else: return np.any(np.sum(np.abs(array_list - a[None, :]), axes) < tol)