# coding: utf-8

from __future__ import division, unicode_literals

"""
An interface to the excellent spglib library by Atsushi Togo
(http://spglib.sourceforge.net/) for pymatgen.

v1.0 - Now works with both ordered and disordered structure.
v2.0 - Updated for spglib 1.6.

.. note::
    Not all spglib functions are implemented.
"""

__author__ = "Shyue Ping Ong"
__copyright__ = "Copyright 2012, The Materials Project"
__version__ = "2.0"
__maintainer__ = "Shyue Ping Ong"
__email__ = "shyuep@gmail.com"
__date__ = "Mar 9, 2012"

import re
import itertools
import logging
from collections import defaultdict

import math
from math import cos
from math import sin

import numpy as np

from six.moves import filter, map, zip

from pymatgen.core.structure import Structure
from pymatgen.symmetry.structure import SymmetrizedStructure
from pymatgen.core.lattice import Lattice
from pymatgen.core.structure import PeriodicSite
from pymatgen.core.operations import SymmOp
from pymatgen.util.coord_utils import find_in_coord_list


try:
    import pymatgen._spglib as spg
except ImportError:
    try:
        import pyspglib._spglib as spg
    except ImportError:
        msg = "Spglib required. Please either run python setup.py install" + \
              " for pymatgen, or install pyspglib from spglib."
        raise ImportError(msg)


logger = logging.getLogger(__name__)


class SpacegroupAnalyzer(object):
    """
    Takes a pymatgen.core.structure.Structure object and a symprec.
    Uses pyspglib to perform various symmetry finding operations.

    Args:
        structure (Structure/IStructure): Structure to find symmetry
        symprec (float): Tolerance for symmetry finding. Defaults to 1e-3,
            which is fairly strict and works well for properly refined
            structures with atoms in the proper symmetry coordinates. For
            structures with slight deviations from their proper atomic
            positions (e.g., structures relaxed with electronic structure
            codes), a looser tolerance of 0.1 (the value used in Materials
            Project) is often needed.
        angle_tolerance (float): Angle tolerance for symmetry finding.
    """

    def __init__(self, structure, symprec=1e-3, angle_tolerance=5):
        self._symprec = symprec
        self._angle_tol = angle_tolerance
        self._structure = structure
        #Spglib"s convention for the lattice definition is the transpose of the
        #pymatgen version.
        self._transposed_latt = structure.lattice.matrix.transpose()
        #Spglib requires numpy floats.
        self._transposed_latt = np.array(
            self._transposed_latt, dtype='double', order='C')
        self._positions = np.array(
            [site.frac_coords for site in structure], dtype='double',
            order='C'
        )
        unique_species = []
        zs = []

        for species, g in itertools.groupby(structure,
                                            key=lambda s: s.species_and_occu):
            try:
                ind = unique_species.index(species)
                zs.extend([ind + 1] * len(tuple(g)))
            except ValueError:
                unique_species.append(species)
                zs.extend([len(unique_species)] * len(tuple(g)))
        self._unique_species = unique_species
        self._numbers = np.array(zs, dtype='intc')

        dataset = {}
        keys = ('number',
                'hall_number',
                'international',
                'hall',
                'transformation_matrix',
                'origin_shift',
                'rotations',
                'translations',
                'wyckoffs',
                'equivalent_atoms',
                'brv_lattice',
                'brv_types',
                'brv_positions')
        for key, data in zip(keys, spg.dataset(self._transposed_latt.copy(),
                                               self._positions.copy(),
                                               self._numbers, self._symprec,
                                               self._angle_tol)):

            dataset[key] = data

        dataset['international'] = dataset['international'].strip()
        dataset['hall'] = dataset['hall'].strip()
        dataset['transformation_matrix'] = np.array(
            dataset['transformation_matrix'], dtype='double', order='C')
        dataset['origin_shift'] = np.array(dataset['origin_shift'], dtype='double')
        dataset['rotations'] = np.array(dataset['rotations'],
                                        dtype='intc', order='C')
        dataset['translations'] = np.array(dataset['translations'],
                                           dtype='double', order='C')
        letters = "abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ"
        dataset['wyckoffs'] = [letters[x] for x in dataset['wyckoffs']]
        dataset['equivalent_atoms'] = np.array(dataset['equivalent_atoms'],
                                               dtype='intc')
        dataset['brv_lattice'] = np.array(np.transpose(dataset['brv_lattice']),
                                          dtype='double', order='C')
        dataset['brv_types'] = np.array(dataset['brv_types'], dtype='intc')
        dataset['brv_positions'] = np.array(dataset['brv_positions'],
                                            dtype='double', order='C')
        self._spacegroup_data = dataset

    def get_spacegroup(self):
        """
        Get the Spacegroup for the Structure.

        Returns:
            Spacgroup object.
        """
        # Atomic positions have to be specified by scaled positions for spglib.
        return SpacegroupOperations(self.get_spacegroup_symbol(),
                                    self.get_spacegroup_number(),
                                    self.get_symmetry_operations())

    def get_spacegroup_symbol(self):
        """
        Get the spacegroup symbol (e.g., Pnma) for structure.

        Returns:
            (str): Spacegroup symbol for structure.
        """
        return self._spacegroup_data["international"]

    def get_spacegroup_number(self):
        """
        Get the international spacegroup number (e.g., 62) for structure.

        Returns:
            (int): International spacegroup number for structure.
        """
        return self._spacegroup_data["number"]

    def get_hall(self):
        """
        Returns Hall symbol for structure.

        Returns:
            (str): Hall symbol
        """
        return self._spacegroup_data["hall"]

    def get_point_group(self):
        """
        Get the point group associated with the structure.

        Returns:
            (Pointgroup): Point group for structure.
        """
        return get_point_group(self._spacegroup_data["rotations"])[0].strip()

    def get_crystal_system(self):
        """
        Get the crystal system for the structure, e.g., (triclinic,
        orthorhombic, cubic, etc.).

        Returns:
            (str): Crystal system for structure.
        """
        n = self._spacegroup_data["number"]

        f = lambda i, j: i <= n <= j
        cs = {"triclinic": (1, 2), "monoclinic": (3, 15),
              "orthorhombic": (16, 74), "tetragonal": (75, 142),
              "trigonal": (143, 167), "hexagonal": (168, 194),
              "cubic": (195, 230)}

        crystal_sytem = None

        for k, v in cs.items():
            if f(*v):
                crystal_sytem = k
                break
        return crystal_sytem

    def get_lattice_type(self):
        """
        Get the lattice for the structure, e.g., (triclinic,
        orthorhombic, cubic, etc.).This is the same than the
        crystal system with the exception of the hexagonal/rhombohedral
        lattice

        Returns:
            (str): Lattice type for structure.
        """
        n = self._spacegroup_data["number"]
        system = self.get_crystal_system()
        if n in [146, 148, 155, 160, 161, 166, 167]:
            return "rhombohedral"
        elif system == "trigonal":
            return "hexagonal"
        else:
            return system

    def get_symmetry_dataset(self):
        """
        Returns the symmetry dataset as a dict.

        Returns:
            (dict): With the following properties:
            number: International space group number
            international: International symbol
            hall: Hall symbol
            transformation_matrix: Transformation matrix from lattice of
            input cell to Bravais lattice L^bravais = L^original * Tmat
            origin shift: Origin shift in the setting of "Bravais lattice"
            rotations, translations: Rotation matrices and translation
            vectors. Space group operations are obtained by
            [(r,t) for r, t in zip(rotations, translations)]
            wyckoffs: Wyckoff letters
        """
        return self._spacegroup_data

    def _get_symmetry(self):
        """
        Get the symmetry operations associated with the structure.

        Returns:
            Symmetry operations as a tuple of two equal length sequences.
            (rotations, translations). "rotations" is the numpy integer array
            of the rotation matrices for scaled positions
            "translations" gives the numpy float64 array of the translation
            vectors in scaled positions.
        """

        # Get number of symmetry operations and allocate symmetry operations
        # multi = spg.multiplicity(cell, positions, numbers, symprec)
        multi = 48 * self._structure.num_sites
        rotation = np.zeros((multi, 3, 3), dtype='intc')
        translation = np.zeros((multi, 3), dtype='double')

        num_sym = spg.symmetry(rotation, translation,
                               self._transposed_latt.copy(),
                               self._positions, self._numbers, self._symprec,
                               self._angle_tol)
        return rotation[:num_sym], translation[:num_sym]

    def get_symmetry_operations(self, cartesian=False):
        """
        Return symmetry operations as a list of SymmOp objects.
        By default returns fractional coord symmops.
        But cartesian can be returned too.

        Returns:
            ([SymmOp]): List of symmetry operations.
        """
        (rotation, translation) = self._get_symmetry()
        symmops = []
        mat = self._structure.lattice.matrix.T
        invmat = np.linalg.inv(mat)
        for rot, trans in zip(rotation, translation):
            if cartesian:
                rot = np.dot(mat, np.dot(rot, invmat))
                trans = np.dot(trans, self._structure.lattice.matrix)
            op = SymmOp.from_rotation_and_translation(rot, trans)
            symmops.append(op)
        return symmops

    def get_point_group_operations(self, cartesian=False):
        """
        Return symmetry operations as a list of SymmOp objects.
        By default returns fractional coord symmops.
        But cartesian can be returned too.

        Args:
            cartesian (bool): Whether to return SymmOps as cartesian or
                direct coordinate operations.

        Returns:
            ([SymmOp]): List of point group symmetry operations.
        """
        (rotation, translation) = self._get_symmetry()
        symmops = []
        mat = self._structure.lattice.matrix.T
        invmat = np.linalg.inv(mat)
        for rot, trans in zip(rotation, translation):
            if cartesian:
                rot = np.dot(mat, np.dot(rot, invmat))
            op = SymmOp.from_rotation_and_translation(rot, np.array([0, 0, 0]))
            symmops.append(op)
        return symmops

    def get_symmetrized_structure(self):
        """
        Get a symmetrized structure. A symmetrized structure is one where the
        sites have been grouped into symmetrically equivalent groups.

        Returns:
            :class:`pymatgen.symmetry.structure.SymmetrizedStructure` object.
        """
        ds = self.get_symmetry_dataset()
        sg = SpacegroupOperations(self.get_spacegroup_symbol(),
                                  self.get_spacegroup_number(),
                                  self.get_symmetry_operations())
        return SymmetrizedStructure(self._structure, sg,
                                    ds["equivalent_atoms"])

    def get_refined_structure(self):
        """
        Get the refined structure based on detected symmetry. The refined
        structure is a *conventional* cell setting with atoms moved to the
        expected symmetry positions.

        Returns:
            Refined structure.
        """
        # Atomic positions have to be specified by scaled positions for spglib.
        num_atom = self._structure.num_sites
        lattice = self._transposed_latt.copy()
        pos = np.zeros((num_atom * 4, 3), dtype='double')
        pos[:num_atom] = self._positions.copy()

        zs = np.zeros(num_atom * 4, dtype='intc')
        zs[:num_atom] = np.array(self._numbers, dtype='intc')
        num_atom_bravais = spg.refine_cell(
            lattice, pos, zs, num_atom, self._symprec, self._angle_tol)

        zs = zs[:num_atom_bravais]
        species = [self._unique_species[i - 1] for i in zs]
        s = Structure(lattice.T.copy(),
                      species,
                      pos[:num_atom_bravais])
        return s.get_sorted_structure()

    def find_primitive(self):
        """
        Find a primitive version of the unit cell.

        Returns:
            A primitive cell in the input cell is searched and returned
            as an Structure object. If no primitive cell is found, None is
            returned.
        """
        # Atomic positions have to be specified by scaled positions for spglib.
        pos = self._positions.copy()
        lattice = self._transposed_latt.copy()
        numbers = self._numbers.copy()
        # lattice is transposed with respect to the definition of Atoms class
        num_atom_prim = spg.primitive(lattice, pos, numbers,
                                      self._symprec, self._angle_tol)
        zs = numbers[:num_atom_prim]
        species = [self._unique_species[i - 1] for i in zs]

        if num_atom_prim > 0:
            return Structure(lattice.T, species, pos[:num_atom_prim],
                             to_unit_cell=True).get_reduced_structure()
        else:
            #Not sure if we should return None or just return the full
            #structure.
            return self._structure.get_reduced_structure()

    def get_ir_reciprocal_mesh(self, mesh=(10, 10, 10), shift=(0, 0, 0),
                               is_time_reversal=True):
        """
        k-point mesh of the Brillouin zone generated taken into account
        symmetry.The method returns the irreducible kpoints of the mesh
        and their weights

        Args:
            mesh (3x1 array): The number of kpoint for the mesh needed in
                each direction
            shift (3x1 array): A shift of the kpoint grid. For instance,
                Monkhorst-Pack is [0.5,0.5,0.5]
            is_time_reversal (bool): Set to True to impose time reversal
                symmetry.

        Returns:
            A list of irreducible kpoints and their weights as a list of
            tuples [(ir_kpoint, weight)], with ir_kpoint given
            in fractional coordinates
        """
        mapping = np.zeros(np.prod(mesh), dtype='intc')
        mesh_points = np.zeros((np.prod(mesh), 3), dtype='intc')
        spg.ir_reciprocal_mesh(
            mesh_points, mapping, np.array(mesh, dtype='intc'),
            np.array(shift, dtype='intc'), is_time_reversal * 1,
            self._transposed_latt, self._positions, self._numbers,
            self._symprec)

        results = []
        tmp_map = list(mapping)
        for i in np.unique(mapping):
            results.append((mesh_points[i] / mesh, tmp_map.count(i)))
        return results

    def get_primitive_standard_structure(self, international_monoclinic=True):
        """
        Gives a structure with a primitive cell according to certain standards
        the standards are defined in Setyawan, W., & Curtarolo, S. (2010).
        High-throughput electronic band structure calculations:
        Challenges and tools. Computational Materials Science,
        49(2), 299-312. doi:10.1016/j.commatsci.2010.05.010

        Returns:
            The structure in a primitive standardized cell
        """
        conv = self.get_conventional_standard_structure(
            international_monoclinic=international_monoclinic)
        lattice = self.get_lattice_type()

        if "P" in self.get_spacegroup_symbol() or lattice == "hexagonal":
            return conv

        if lattice == "rhombohedral":
            conv = self.get_refined_structure()
            lengths, angles = conv.lattice.lengths_and_angles
            a = lengths[0]
            alpha = math.pi * angles[0] / 180
            new_matrix = [
                [a * cos(alpha / 2), -a * sin(alpha / 2), 0],
                [a * cos(alpha / 2), a * sin(alpha / 2), 0],
                [a * cos(alpha) / cos(alpha / 2), 0,
                 a * math.sqrt(1 - (cos(alpha) ** 2 / (cos(alpha / 2) ** 2)))]]
            new_sites = []
            latt = Lattice(new_matrix)
            for s in conv:
                new_s = PeriodicSite(
                    s.specie, s.frac_coords, latt,
                    to_unit_cell=True, properties=s.properties)
                if not any(map(new_s.is_periodic_image, new_sites)):
                    new_sites.append(new_s)
            return Structure.from_sites(new_sites)

        if "I" in self.get_spacegroup_symbol():
            transf = np.array([[-1, 1, 1], [1, -1, 1], [1, 1, -1]],
                              dtype=np.float) / 2
        elif "F" in self.get_spacegroup_symbol():
            transf = np.array([[0, 1, 1], [1, 0, 1], [1, 1, 0]],
                              dtype=np.float) / 2
        elif "C" in self.get_spacegroup_symbol():
            if self.get_crystal_system() == "monoclinic":
                transf = np.array([[1, 1, 0], [-1, 1, 0], [0, 0, 2]],
                                  dtype=np.float) / 2
            else:
                transf = np.array([[1, -1, 0], [1, 1, 0], [0, 0, 2]],
                                  dtype=np.float) / 2
        else:
            transf = np.eye(3)

        new_sites = []
        latt = Lattice(np.dot(transf, conv.lattice.matrix))
        for s in conv:
            new_s = PeriodicSite(
                s.specie, s.coords, latt,
                to_unit_cell=True, coords_are_cartesian=True,
                properties=s.properties)
            if not any(map(new_s.is_periodic_image, new_sites)):
                new_sites.append(new_s)
        return Structure.from_sites(new_sites)

    def get_conventional_standard_structure(
            self, international_monoclinic=True):
        """
        Gives a structure with a conventional cell according to certain
        standards. The standards are defined in Setyawan, W., & Curtarolo,
        S. (2010). High-throughput electronic band structure calculations:
        Challenges and tools. Computational Materials Science,
        49(2), 299-312. doi:10.1016/j.commatsci.2010.05.010
        They basically enforce as much as possible
        norm(a1)<norm(a2)<norm(a3)

        Returns:
            The structure in a conventional standardized cell
        """
        tol = 1e-5
        struct = self.get_refined_structure()
        latt = struct.lattice
        latt_type = self.get_lattice_type()
        sorted_lengths = sorted(latt.abc)
        sorted_dic = sorted([{'vec': latt.matrix[i],
                              'length': latt.abc[i],
                              'orig_index': i} for i in [0, 1, 2]],
                            key=lambda k: k['length'])

        if latt_type in ("orthorhombic", "cubic"):
            #you want to keep the c axis where it is
            #to keep the C- settings
            transf = np.zeros(shape=(3, 3))
            if self.get_spacegroup_symbol().startswith("C"):
                transf[2] = [0, 0, 1]
                a, b = sorted(latt.abc[:2])
                sorted_dic = sorted([{'vec': latt.matrix[i],
                                      'length': latt.abc[i],
                                      'orig_index': i} for i in [0, 1]],
                                    key=lambda k: k['length'])
                for i in range(2):
                    transf[i][sorted_dic[i]['orig_index']] = 1
                c = latt.abc[2]
            else:
                for i in range(len(sorted_dic)):
                    transf[i][sorted_dic[i]['orig_index']] = 1
                a, b, c = sorted_lengths
            latt = Lattice.orthorhombic(a, b, c)

        elif latt_type == "tetragonal":
            #find the "a" vectors
            #it is basically the vector repeated two times
            transf = np.zeros(shape=(3, 3))
            a, b, c = sorted_lengths
            for d in range(len(sorted_dic)):
                transf[d][sorted_dic[d]['orig_index']] = 1

            if abs(b - c) < tol:
                a, c = c, a
                transf = np.dot([[0, 0, 1], [0, 1, 0], [1, 0, 0]], transf)
            latt = Lattice.tetragonal(a, c)
        elif latt_type in ("hexagonal", "rhombohedral"):
            #for the conventional cell representation,
            #we allways show the rhombohedral lattices as hexagonal

            #check first if we have the refined structure shows a rhombohedral
            #cell
            #if so, make a supercell
            a, b, c = latt.abc
            if np.all(np.abs([a - b, c - b, a - c]) < 0.001):
                struct = Structure.from_sites(struct)
                struct.make_supercell(((1, -1, 0), (0, 1, -1), (1, 1, 1)))
                a, b, c = sorted(struct.lattice.abc)

            if abs(b - c) < 0.001:
                a, c = c, a
            new_matrix = [[a / 2, -a * math.sqrt(3) / 2, 0],
                          [a / 2, a * math.sqrt(3) / 2, 0],
                          [0, 0, c]]
            latt = Lattice(new_matrix)
            transf = np.eye(3, 3)

        elif latt_type == "monoclinic":
            #you want to keep the c axis where it is
            #to keep the C- settings

            if self.get_spacegroup().int_symbol.startswith("C"):
                transf = np.zeros(shape=(3, 3))
                transf[2] = [0, 0, 1]
                sorted_dic = sorted([{'vec': latt.matrix[i],
                                      'length': latt.abc[i],
                                      'orig_index': i} for i in [0, 1]],
                                    key=lambda k: k['length'])
                a = sorted_dic[0]['length']
                b = sorted_dic[1]['length']
                c = latt.abc[2]
                new_matrix = None
                for t in itertools.permutations(list(range(2)), 2):
                    m = latt.matrix
                    landang = Lattice(
                        [m[t[0]], m[t[1]], m[2]]).lengths_and_angles
                    if landang[1][0] > 90:
                        #if the angle is > 90 we invert a and b to get
                        #an angle < 90
                        landang = Lattice(
                            [-m[t[0]], -m[t[1]], m[2]]).lengths_and_angles
                        transf = np.zeros(shape=(3, 3))
                        transf[0][t[0]] = -1
                        transf[1][t[1]] = -1
                        transf[2][2] = 1
                        a, b, c = landang[0]
                        alpha = math.pi * landang[1][0] / 180
                        new_matrix = [[a, 0, 0],
                                      [0, b, 0],
                                      [0, c * cos(alpha), c * sin(alpha)]]
                        continue

                    elif landang[1][0] < 90:
                        transf = np.zeros(shape=(3, 3))
                        transf[0][t[0]] = 1
                        transf[1][t[1]] = 1
                        transf[2][2] = 1
                        a, b, c = landang[0]
                        alpha = math.pi * landang[1][0] / 180
                        new_matrix = [[a, 0, 0],
                                      [0, b, 0],
                                      [0, c * cos(alpha), c * sin(alpha)]]
                if new_matrix is None:
                    #this if is to treat the case
                    #where alpha==90 (but we still have a monoclinic sg
                    new_matrix = [[a, 0, 0],
                                  [0, b, 0],
                                  [0, c * cos(alpha), c * sin(alpha)]]
                if new_matrix is None:
                    #this if is to treat the case
                    #where alpha==90 (but we still have a monoclinic sg
                    new_matrix = [[a, 0, 0],
                                  [0, b, 0],
                                  [0, 0, c]]
                    transf = np.zeros(shape=(3, 3))
                    for c in range(len(sorted_dic)):
                        transf[c][sorted_dic[c]['orig_index']] = 1
            #if not C-setting
            else:
                #try all permutations of the axis
                #keep the ones with the non-90 angle=alpha
                #and b<c
                new_matrix = None
                for t in itertools.permutations(list(range(3)), 3):
                    m = latt.matrix
                    landang = Lattice(
                        [m[t[0]], m[t[1]], m[t[2]]]).lengths_and_angles
                    if landang[1][0] > 90 and landang[0][1] < landang[0][2]:
                        landang = Lattice(
                            [-m[t[0]], -m[t[1]], m[t[2]]]).lengths_and_angles
                        transf = np.zeros(shape=(3, 3))
                        transf[0][t[0]] = -1
                        transf[1][t[1]] = -1
                        transf[2][t[2]] = 1
                        a, b, c = landang[0]
                        alpha = math.pi * landang[1][0] / 180
                        new_matrix = [[a, 0, 0],
                                      [0, b, 0],
                                      [0, c * cos(alpha), c * sin(alpha)]]
                        continue
                    elif landang[1][0] < 90 and landang[0][1] < landang[0][2]:
                        transf = np.zeros(shape=(3, 3))
                        transf[0][t[0]] = 1
                        transf[1][t[1]] = 1
                        transf[2][t[2]] = 1
                        a, b, c = landang[0]
                        alpha = math.pi * landang[1][0] / 180
                        new_matrix = [[a, 0, 0],
                                      [0, b, 0],
                                      [0, c * cos(alpha), c * sin(alpha)]]
                if new_matrix is None:
                    #this if is to treat the case
                    #where alpha==90 (but we still have a monoclinic sg
                    new_matrix = [[sorted_lengths[0], 0, 0],
                                  [0, sorted_lengths[1], 0],
                                  [0, 0, sorted_lengths[2]]]
                    transf = np.zeros(shape=(3, 3))
                    for c in range(len(sorted_dic)):
                        transf[c][sorted_dic[c]['orig_index']] = 1

            if international_monoclinic:
                # The above code makes alpha the non-right angle.
                # The following will convert to proper international convention
                # that beta is the non-right angle.
                op = [[0, 1, 0], [1, 0, 0], [0, 0, -1]]
                transf = np.dot(op, transf)
                new_matrix = np.dot(op, new_matrix)
                beta = Lattice(new_matrix).beta
                if beta < 90:
                    op = [[-1, 0, 0], [0, -1, 0], [0, 0, 1]]
                    transf = np.dot(op, transf)
                    new_matrix = np.dot(op, new_matrix)

            latt = Lattice(new_matrix)

        elif latt_type == "triclinic":
            #we use a LLL Minkowski-like reduction for the triclinic cells
            struct = struct.get_reduced_structure("LLL")

            a, b, c = latt.lengths_and_angles[0]
            alpha, beta, gamma = [math.pi * i / 180
                                  for i in latt.lengths_and_angles[1]]
            new_matrix = None
            test_matrix = [[a, 0, 0],
                          [b * cos(gamma), b * sin(gamma), 0.0],
                          [c * cos(beta),
                           c * (cos(alpha) - cos(beta) * cos(gamma)) /
                           sin(gamma),
                           c * math.sqrt(sin(gamma) ** 2 - cos(alpha) ** 2
                                         - cos(beta) ** 2
                                         + 2 * cos(alpha) * cos(beta)
                                         * cos(gamma)) / sin(gamma)]]

            def is_all_acute_or_obtuse(m):
                recp_angles = np.array(Lattice(m).reciprocal_lattice.angles)
                return np.all(recp_angles <= 90) or np.all(recp_angles > 90)

            if is_all_acute_or_obtuse(test_matrix):
                transf = np.eye(3)
                new_matrix = test_matrix

            test_matrix = [[-a, 0, 0],
                           [b * cos(gamma), b * sin(gamma), 0.0],
                           [-c * cos(beta),
                            -c * (cos(alpha) - cos(beta) * cos(gamma)) /
                            sin(gamma),
                            -c * math.sqrt(sin(gamma) ** 2 - cos(alpha) ** 2
                                           - cos(beta) ** 2
                                           + 2 * cos(alpha) * cos(beta)
                                           * cos(gamma)) / sin(gamma)]]

            if is_all_acute_or_obtuse(test_matrix):
                transf = [[-1, 0, 0],
                          [0, 1, 0],
                          [0, 0, -1]]
                new_matrix = test_matrix

            test_matrix = [[-a, 0, 0],
                           [-b * cos(gamma), -b * sin(gamma), 0.0],
                           [c * cos(beta),
                            c * (cos(alpha) - cos(beta) * cos(gamma)) /
                            sin(gamma),
                            c * math.sqrt(sin(gamma) ** 2 - cos(alpha) ** 2
                                          - cos(beta) ** 2
                                          + 2 * cos(alpha) * cos(beta)
                                          * cos(gamma)) / sin(gamma)]]

            if is_all_acute_or_obtuse(test_matrix):
                transf = [[-1, 0, 0],
                          [0, -1, 0],
                          [0, 0, 1]]
                new_matrix = test_matrix

            test_matrix = [[a, 0, 0],
                           [-b * cos(gamma), -b * sin(gamma), 0.0],
                           [-c * cos(beta),
                            -c * (cos(alpha) - cos(beta) * cos(gamma)) /
                            sin(gamma),
                            -c * math.sqrt(sin(gamma) ** 2 - cos(alpha) ** 2
                                           - cos(beta) ** 2
                                           + 2 * cos(alpha) * cos(beta)
                                           * cos(gamma)) / sin(gamma)]]
            if is_all_acute_or_obtuse(test_matrix):
                transf = [[1, 0, 0],
                          [0, -1, 0],
                          [0, 0, -1]]
                new_matrix = test_matrix

            latt = Lattice(new_matrix)

        new_coords = np.dot(transf, np.transpose(struct.frac_coords)).T
        new_struct = Structure(latt, struct.species_and_occu, new_coords,
                               site_properties=struct.site_properties,
                               to_unit_cell=True)
        return new_struct.get_sorted_structure()


def get_point_group(rotations):
    """
    Get point group from a set of rotations.

    Returns:
        (pointgroup_symbol, pointgroup_number, transformation_matrix)

    symbol and number are those used in international table.
    The symbols are mapped to the numbers as follows:
    1   "1    "
    2   "-1   "
    3   "2    "
    4   "m    "
    5   "2/m  "
    6   "222  "
    7   "mm2  "
    8   "mmm  "
    9   "4    "
    10  "-4   "
    11  "4/m  "
    12  "422  "
    13  "4mm  "
    14  "-42m "
    15  "4/mmm"
    16  "3    "
    17  "-3   "
    18  "32   "
    19  "3m   "
    20  "-3m  "
    21  "6    "
    22  "-6   "
    23  "6/m  "
    24  "622  "
    25  "6mm  "
    26  "-62m "
    27  "6/mmm"
    28  "23   "
    29  "m-3  "
    30  "432  "
    31  "-43m "
    32  "m-3m "
    """
    return spg.pointgroup(np.array(rotations, dtype='intc', order='C'))


class PointGroupAnalyzer(object):
    """
    A class to analyze the point group of a molecule. The general outline of
    the algorithm is as follows:

    1. Center the molecule around its center of mass.
    2. Compute the inertia tensor and the eigenvalues and eigenvectors.
    3. Handle the symmetry detection based on eigenvalues.

        a. Linear molecules have one zero eigenvalue. Possible symmetry
           operations are C*v or D*v
        b. Asymetric top molecules have all different eigenvalues. The
           maximum rotational symmetry in such molecules is 2
        c. Symmetric top molecules have 1 unique eigenvalue, which gives a
           unique rotation axis.  All axial point groups are possible
           except the cubic groups (T & O) and I.
        d. Spherical top molecules have all three eigenvalues equal. They
           have the rare T, O or I point groups.

    .. attribute:: sch_symbol

        Schoenflies symbol of the detected point group.
    """
    inversion_op = SymmOp.inversion()

    def __init__(self, mol, tolerance=0.3, eigen_tolerance=0.01,
                 matrix_tol=0.1):
        """
        The default settings are usually sufficient.

        Args:
            mol (Molecule): Molecule to determine point group for.
            tolerance (float): Distance tolerance to consider sites as
                symmetrically equivalent. Defaults to 0.3 Angstrom.
            eigen_tolerance (float): Tolerance to compare eigen values of
                the inertia tensor. Defaults to 0.01.
            matrix_tol (float): Tolerance used to generate the full set of
                symmetry operations of the point group.
        """
        self.mol = mol
        self.centered_mol = mol.get_centered_molecule()
        self.tol = tolerance
        self.eig_tol = eigen_tolerance
        self.mat_tol = matrix_tol
        self._analyze()

    def _analyze(self):
        if len(self.centered_mol) == 1:
            self.sch_symbol = "Kh"
        else:
            inertia_tensor = np.zeros((3, 3))
            total_inertia = 0
            for site in self.mol:
                c = site.coords
                wt = site.species_and_occu.weight
                for i in range(3):
                    inertia_tensor[i, i] += wt * (c[(i + 1) % 3] ** 2
                                                  + c[(i + 2) % 3] ** 2)
                for i, j in itertools.combinations(list(range(3)), 2):
                    inertia_tensor[i, j] += -wt * c[i] * c[j]
                    inertia_tensor[j, i] += -wt * c[j] * c[i]
                total_inertia += wt * np.dot(c, c)

            # Normalize the inertia tensor so that it does not scale with size
            # of the system.  This mitigates the problem of choosing a proper
            # comparison tolerance for the eigenvalues.
            inertia_tensor /= total_inertia
            eigvals, eigvecs = np.linalg.eig(inertia_tensor)
            self.principal_axes = eigvecs.T
            self.eigvals = eigvals
            v1, v2, v3 = eigvals
            eig_zero = abs(v1 * v2 * v3) < self.eig_tol ** 3
            eig_all_same = abs(v1 - v2) < self.eig_tol and abs(
                v1 - v3) < self.eig_tol
            eig_all_diff = abs(v1 - v2) > self.eig_tol and abs(
                v1 - v2) > self.eig_tol and abs(v2 - v3) > self.eig_tol

            self.rot_sym = []
            self.symmops = [SymmOp(np.eye(4))]

            if eig_zero:
                logger.debug("Linear molecule detected")
                self._proc_linear()
            elif eig_all_same:
                logger.debug("Spherical top molecule detected")
                self._proc_sph_top()
            elif eig_all_diff:
                logger.debug("Asymmetric top molecule detected")
                self._proc_asym_top()
            else:
                logger.debug("Symmetric top molecule detected")
                self._proc_sym_top()

    def _proc_linear(self):
        if self.is_valid_op(PointGroupAnalyzer.inversion_op):
            self.sch_symbol = "D*h"
            self.symmops.append(PointGroupAnalyzer.inversion_op)
        else:
            self.sch_symbol = "C*v"

    def _proc_asym_top(self):
        """
        Handles assymetric top molecules, which cannot contain rotational
        symmetry larger than 2.
        """
        self._check_R2_axes_asym()
        if len(self.rot_sym) == 0:
            logger.debug("No rotation symmetries detected.")
            self._proc_no_rot_sym()
        elif len(self.rot_sym) == 3:
            logger.debug("Dihedral group detected.")
            self._proc_dihedral()
        else:
            logger.debug("Cyclic group detected.")
            self._proc_cyclic()

    def _proc_sym_top(self):
        """
        Handles symetric top molecules which has one unique eigenvalue whose
        corresponding principal axis is a unique rotational axis.  More complex
        handling required to look for R2 axes perpendicular to this unique
        axis.
        """
        if abs(self.eigvals[0] - self.eigvals[1]) < self.eig_tol:
            ind = 2
        elif abs(self.eigvals[1] - self.eigvals[2]) < self.eig_tol:
            ind = 0
        else:
            ind = 1

        unique_axis = self.principal_axes[ind]
        self._check_rot_sym(unique_axis)
        if len(self.rot_sym) > 0:
            self._check_perpendicular_r2_axis(unique_axis)

        if len(self.rot_sym) >= 2:
            self._proc_dihedral()
        elif len(self.rot_sym) == 1:
            self._proc_cyclic()
        else:
            self._proc_no_rot_sym()

    def _proc_no_rot_sym(self):
        """
        Handles molecules with no rotational symmetry. Only possible point
        groups are C1, Cs and Ci.
        """
        self.sch_symbol = "C1"
        if self.is_valid_op(PointGroupAnalyzer.inversion_op):
            self.sch_symbol = "Ci"
            self.symmops.append(PointGroupAnalyzer.inversion_op)
        else:
            for v in self.principal_axes:
                mirror_type = self._find_mirror(v)
                if not mirror_type == "":
                    self.sch_symbol = "Cs"
                    break

    def _proc_cyclic(self):
        """
        Handles cyclic group molecules.
        """
        main_axis, rot = max(self.rot_sym, key=lambda v: v[1])
        self.sch_symbol = "C{}".format(rot)
        mirror_type = self._find_mirror(main_axis)
        if mirror_type == "h":
            self.sch_symbol += "h"
        elif mirror_type == "v":
            self.sch_symbol += "v"
        elif mirror_type == "":
            if self.is_valid_op(SymmOp.rotoreflection(main_axis,
                                                      angle=180 / rot)):
                self.sch_symbol = "S{}".format(2 * rot)

    def _proc_dihedral(self):
        """
        Handles dihedral group molecules, i.e those with intersecting R2 axes
        and a main axis.
        """
        main_axis, rot = max(self.rot_sym, key=lambda v: v[1])
        self.sch_symbol = "D{}".format(rot)
        mirror_type = self._find_mirror(main_axis)
        if mirror_type == "h":
            self.sch_symbol += "h"
        elif not mirror_type == "":
            self.sch_symbol += "d"

    def _check_R2_axes_asym(self):
        """
        Test for 2-fold rotation along the principal axes. Used to handle
        asymetric top molecules.
        """
        for v in self.principal_axes:
            op = SymmOp.from_axis_angle_and_translation(v, 180)
            if self.is_valid_op(op):
                self.symmops.append(op)
                self.rot_sym.append((v, 2))

    def _find_mirror(self, axis):
        """
        Looks for mirror symmetry of specified type about axis.  Possible
        types are "h" or "vd".  Horizontal (h) mirrors are perpendicular to
        the axis while vertical (v) or diagonal (d) mirrors are parallel.  v
        mirrors has atoms lying on the mirror plane while d mirrors do
        not.
        """
        mirror_type = ""

        # First test whether the axis itself is the normal to a mirror plane.
        if self.is_valid_op(SymmOp.reflection(axis)):
            self.symmops.append(SymmOp.reflection(axis))
            mirror_type = "h"
        else:
            # Iterate through all pairs of atoms to find mirror
            for s1, s2 in itertools.combinations(self.centered_mol, 2):
                if s1.species_and_occu == s2.species_and_occu:
                    normal = s1.coords - s2.coords
                    if np.dot(normal, axis) < self.tol:
                        op = SymmOp.reflection(normal)
                        if self.is_valid_op(op):
                            self.symmops.append(op)
                            if len(self.rot_sym) > 1:
                                mirror_type = "d"
                                for v, r in self.rot_sym:
                                    if not np.linalg.norm(v - axis) < self.tol:
                                        if np.dot(v, normal) < self.tol:
                                            mirror_type = "v"
                                            break
                            else:
                                mirror_type = "v"
                            break

        return mirror_type

    def _get_smallest_set_not_on_axis(self, axis):
        """
        Returns the smallest list of atoms with the same species and
        distance from origin AND does not lie on the specified axis.  This
        maximal set limits the possible rotational symmetry operations,
        since atoms lying on a test axis is irrelevant in testing rotational
        symmetryOperations.
        """

        def not_on_axis(site):
            v = np.cross(site.coords, axis)
            return np.linalg.norm(v) > self.tol

        valid_sets = []
        origin_site, dist_el_sites = cluster_sites(self.centered_mol, self.tol)
        for test_set in dist_el_sites.values():
            valid_set = list(filter(not_on_axis, test_set))
            if len(valid_set) > 0:
                valid_sets.append(valid_set)

        return min(valid_sets, key=lambda s: len(s))

    def _check_rot_sym(self, axis):
        """
        Determines the rotational symmetry about supplied axis.  Used only for
        symmetric top molecules which has possible rotational symmetry
        operations > 2.
        """
        min_set = self._get_smallest_set_not_on_axis(axis)
        max_sym = len(min_set)
        for i in range(max_sym, 0, -1):
            if max_sym % i != 0:
                continue
            op = SymmOp.from_axis_angle_and_translation(axis, 360 / i)
            rotvalid = self.is_valid_op(op)
            if rotvalid:
                self.symmops.append(op)
                self.rot_sym.append((axis, i))
                return i
        return 1

    def _check_perpendicular_r2_axis(self, axis):
        """
        Checks for R2 axes perpendicular to unique axis.  For handling
        symmetric top molecules.
        """
        min_set = self._get_smallest_set_not_on_axis(axis)
        for s1, s2 in itertools.combinations(min_set, 2):
            test_axis = np.cross(s1.coords - s2.coords, axis)
            if np.linalg.norm(test_axis) > self.tol:
                op = SymmOp.from_axis_angle_and_translation(test_axis, 180)
                r2present = self.is_valid_op(op)
                if r2present:
                    self.symmops.append(op)
                    self.rot_sym.append((test_axis, 2))
                    return True

    def _proc_sph_top(self):
        """
        Handles Sperhical Top Molecules, which belongs to the T, O or I point
        groups.
        """
        self._find_spherical_axes()
        main_axis, rot = max(self.rot_sym, key=lambda v: v[1])
        if len(self.rot_sym) == 0 or rot < 3:
            logger.debug("Accidental speherical top!")
            self._proc_sym_top()
        elif rot == 3:
            mirror_type = self._find_mirror(main_axis)
            if mirror_type != "":
                if self.is_valid_op(PointGroupAnalyzer.inversion_op):
                    self.symmops.append(PointGroupAnalyzer.inversion_op)
                    self.sch_symbol = "Th"
                else:
                    self.sch_symbol = "Td"
            else:
                self.sch_symbol = "T"
        elif rot == 4:
            if self.is_valid_op(PointGroupAnalyzer.inversion_op):
                self.symmops.append(PointGroupAnalyzer.inversion_op)
                self.sch_symbol = "Oh"
            else:
                self.sch_symbol = "O"
        elif rot == 5:
            if self.is_valid_op(PointGroupAnalyzer.inversion_op):
                self.symmops.append(PointGroupAnalyzer.inversion_op)
                self.sch_symbol = "Ih"
            else:
                self.sch_symbol = "I"

    def _find_spherical_axes(self):
        """
        Looks for R5, R4, R3 and R2 axes in speherical top molecules.  Point
        group T molecules have only one unique 3-fold and one unique 2-fold
        axis. O molecules have one unique 4, 3 and 2-fold axes. I molecules
        have a unique 5-fold axis.
        """
        rot_present = defaultdict(bool)
        origin_site, dist_el_sites = cluster_sites(self.centered_mol, self.tol)
        test_set = min(dist_el_sites.values(), key=lambda s: len(s))
        coords = [s.coords for s in test_set]
        for c1, c2, c3 in itertools.combinations(coords, 3):
            for cc1, cc2 in itertools.combinations([c1, c2, c3], 2):
                if not rot_present[2]:
                    test_axis = cc1 + cc2
                    if np.linalg.norm(test_axis) > self.tol:
                        op = SymmOp.from_axis_angle_and_translation(test_axis,
                                                                    180)
                        rot_present[2] = self.is_valid_op(op)
                        if rot_present[2]:
                            self.symmops.append(op)
                            self.rot_sym.append((test_axis, 2))

            test_axis = np.cross(c2 - c1, c3 - c1)
            if np.linalg.norm(test_axis) > self.tol:
                for r in (3, 4, 5):
                    if not rot_present[r]:
                        op = SymmOp.from_axis_angle_and_translation(
                            test_axis, 360 / r)
                        rot_present[r] = self.is_valid_op(op)
                        if rot_present[r]:
                            self.symmops.append(op)
                            self.rot_sym.append((test_axis, r))
                            break
            if rot_present[2] and rot_present[3] and (
                        rot_present[4] or rot_present[5]):
                break

    def get_pointgroup(self):
        """
        Returns a PointGroup object for the molecule.
        """
        return PointGroupOperations(self.sch_symbol, self.symmops, self.mat_tol)

    def is_valid_op(self, symmop):
        """
        Check if a particular symmetry operation is a valid symmetry operation
        for a molecule, i.e., the operation maps all atoms to another
        equivalent atom.

        Args:
            symmop (SymmOp): Symmetry operation to test.

        Returns:
            (bool): Whether SymmOp is valid for Molecule.
        """
        coords = self.centered_mol.cart_coords
        for site in self.centered_mol:
            coord = symmop.operate(site.coords)
            ind = find_in_coord_list(coords, coord, self.tol)
            if not (len(ind) == 1 and
                            self.centered_mol[ind[0]].species_and_occu
                            == site.species_and_occu):
                return False
        return True


def cluster_sites(mol, tol):
    """
    Cluster sites based on distance and species type.

    Args:
        mol (Molecule): Molecule **with origin at center of mass**.
        tol (float): Tolerance to use.

    Returns:
        (origin_site, clustered_sites): origin_site is a site at the center
        of mass (None if there are no origin atoms). clustered_sites is a
        dict of {(avg_dist, species_and_occu): [list of sites]}
    """
    # Cluster works for dim > 2 data. We just add a dummy 0 for second
    # coordinate.
    dists = [[np.linalg.norm(site.coords), 0] for site in mol]
    import scipy.cluster as spcluster
    f = spcluster.hierarchy.fclusterdata(dists, tol, criterion='distance')
    clustered_dists = defaultdict(list)
    for i, site in enumerate(mol):
        clustered_dists[f[i]].append(dists[i])
    avg_dist = {label: np.mean(val) for label, val in clustered_dists.items()}
    clustered_sites = defaultdict(list)
    origin_site = None
    for i, site in enumerate(mol):
        if avg_dist[f[i]] < tol:
            origin_site = site
        else:
            clustered_sites[(avg_dist[f[i]],
                             site.species_and_occu)].append(site)
    return origin_site, clustered_sites


def generate_full_symmops(symmops, tol):
    """
    Recursive algorithm to permute through all possible combinations of the
    initially supplied symmetry operations to arrive at a complete set of
    operations mapping a single atom to all other equivalent atoms in the
    point group.  This assumes that the initial number already uniquely
    identifies all operations.

    Args:
        symmops ([SymmOp]): Initial set of symmetry operations.

    Returns:
        Full set of symmetry operations.
    """

    a = [o.affine_matrix for o in symmops]

    if len(symmops) > 300:
        logger.debug("Generation of symmetry operations in infinite loop.  " +
                     "Possible error in initial operations or tolerance too "
                     "low.")
    else:
        for op1, op2 in itertools.product(symmops, symmops):
            m = np.dot(op1.affine_matrix, op2.affine_matrix)
            d = np.abs(a - m) < tol
            if not np.any(np.all(np.all(d, axis=2), axis=1)):
                return generate_full_symmops(symmops + [SymmOp(m)], tol)

    return symmops


class SpacegroupOperations(list):
    """
    Represents a space group, which is a collection of symmetry operations.

    Args:
        int_symbol (str): International symbol of the spacegroup.
        int_number (int): International number of the spacegroup.
        symmops ([SymmOp]): Symmetry operations associated with the
            spacegroup.
    """

    def __init__(self, int_symbol, int_number, symmops):
        self.int_symbol = int_symbol
        self.int_number = int_number
        super(SpacegroupOperations, self).__init__(symmops)

    def are_symmetrically_equivalent(self, sites1, sites2, symm_prec=1e-3):
        """
        Given two sets of PeriodicSites, test if they are actually
        symmetrically equivalent under this space group.  Useful, for example,
        if you want to test if selecting atoms 1 and 2 out of a set of 4 atoms
        are symmetrically the same as selecting atoms 3 and 4, etc.

        One use is in PartialRemoveSpecie transformation to return only
        symmetrically distinct arrangements of atoms.

        Args:
            sites1 ([Site]): 1st set of sites
            sites2 ([Site]): 2nd set of sites
            symm_prec (float): Tolerance in atomic distance to test if atoms
                are symmetrically similar.

        Returns:
            (bool): Whether the two sets of sites are symmetrically
            equivalent.
        """
        def in_sites(site):
            for test_site in sites1:
                if test_site.is_periodic_image(site, symm_prec, False):
                    return True
            return False
        for op in self:
            newsites2 = [PeriodicSite(site.species_and_occu,
                                      op.operate(site.frac_coords),
                                      site.lattice) for site in sites2]
            ismapping = True
            for site in newsites2:
                if not in_sites(site):
                    ismapping = False
                    break
            if ismapping:
                return True
        return False

    def __str__(self):
        return "{} ({}) spacegroup".format(self.int_symbol, self.int_number)


class PointGroupOperations(list):
    """
    Defines a point group, which is essentially a sequence of symmetry
    operations.

    Args:
        sch_symbol (str): Schoenflies symbol of the point group.
        operations ([SymmOp]): Initial set of symmetry operations. It is
            sufficient to provide only just enough operations to generate
            the full set of symmetries.
        tol (float): Tolerance to generate the full set of symmetry
            operations.

    .. attribute:: sch_symbol

        Schoenflies symbol of the point group.
    """
    def __init__(self, sch_symbol, operations, tol=0.1):
        self.sch_symbol = sch_symbol
        super(PointGroupOperations, self).__init__(
            generate_full_symmops(operations, tol))

    def __str__(self):
        return self.sch_symbol

    def __repr__(self):
        return self.__str__()